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dual configuration. The torus partition function can be recast into the one without vacuum charges$$\tilde{Z}(\alpha,\beta)=\int_{\Delta_0}^{\infty}\int_{\xi_0}^{\infty}e^{-\alpha\Delta-\beta\xi}\rho(\Delta,\xi)$$ where the $\tilde{Z}$ is the partition function defined by $$\tilde{Z}(\alpha,\beta)=e^{\alpha M_v+\beta H_v}Z(\alpha,\beta).$$ Here $M_v$ and $H_v$ are the translation charges of the vacuum. They can be non-vanishing in various theories. The spectrum density could be read via inverse Laplace transformation $$\rho(\Delta,\xi)=-\frac{1}{(2\pi)^2}\int d\alpha d\beta e^{\alpha \Delta+\beta\xi} \tilde{Z}(\alpha,\beta).$$ Just like in 2D CFT case[@Cardy:1986ie], the key point is to use the modular invariance to reexpress the equation above, and do the saddle point approximation at large $\Delta,\xi$. The modular invariance suggests $$\tilde{Z}(\alpha,\beta)=e^{(\alpha-\alpha')M_v+(\beta-\beta')H_v}\tilde{Z}(\alpha',\beta').$$ For non-vanishing $H_v$ and $M_v$, the saddle point is at $$\alpha=\frac{2\pi}{(1+\frac{\xi}{H_v})^{\frac{1}{d+1}}},$$ and the microcanonical entropy is $$S(\Delta,\xi)=\log \rho(\Delta,\xi)=\frac{2\pi(\Delta+M_v)}{(\frac{\xi}{H_v}-1)^{\frac{1}{d+1}}}+2\pi(\frac{\xi}{H_v}-1)^{\frac{1}{d+1}}M_v.$$ For $$\xi>>H_v,\ \ \ \ \Delta>>M_v, \nn$$ the entropy is $$S(\Delta,\xi)=2\pi\Delta(\frac{H_v}{\xi})^{\frac{1}{d+1}}+2\pi M_v(\frac{\xi}{H_v})^{\frac{1}{d+1}}. \label{entropy}$$ When $d=1,\ M_v=0$, it matches with the result in [@Bagchi:2012xr] in which the entropy reproduces the Bekenstein-Hawking entropy of flat cosmological horizon. For general $d=1$ case, our result match with the one in [@Bagchi:2013qva].
Several remarks are in order:\
1. We assume that there is no $M$ extension such that the theory is anomaly-free and modular invariant.\
2.The saddle point approximation is valid. The $\tilde{Z}(\alpha,\beta)$ does vary slowly around the putative saddle point. Near the saddle point, $\alpha$ is small. The dominant part of the partition function is the contribution from vacuum module. We
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In our case, we will work in conventions in which our Fock vacuum $| 0 \rangle$ has the properties $$\lambda_{-,0}^{a} | 0 \rangle \: = \: 0 \: = \:
\psi_{+,0}^{\overline{\imath}} | 0 \rangle .$$
As before, reflecting the fact that the $\lambda$’s and $\psi$’s couple to nontrivial bundles, this Fock vacuum is itself a section of a line bundle. From those periodic fermions, the Fock vacuum behaves as a section of a square root of the determinant of the periodic modes, specifically, $$\label{eq:squareroot}
\sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 },$$ (square root chosen with periodic boundary conditions), where $$K_{\alpha} \: = \: \det (T^{\alpha}_0)^*$$ [*i.e.*]{} the canonical bundle of the $\alpha$ component of $I_{\mathfrak{X}}$. (Note that in an (NS,R) sector, the ‘invariant’ subbundle ${\cal E}_0$ is defined to be invariant under the combination of spacetime group action and spin state boundary condition, and hence will be different from the ${\cal E}_0$ in an (R,R) sector.) If the square root above does not exist, then the orbifold is not well-defined, which we shall come back to after we derive the expression above.
We can derive the result above for periodic fermions as follows. Different choices of Fock vacua act as sections of different line bundles, related by fermions acting as raising and lowering operators. Just as in fractional charges, the square root and bundles above are constrained by the fact that the set of Fock vacua must be consistent with those raising and lowering operations. For example, the ‘opposite’ Fock vacuum $| 0 \rangle^{\rm op}$ is defined by applying raising operators maximally: $$| 0 \rangle^{\rm op} \: = \: \lambda_{-,0}^{\overline{a}_1} \cdots
\lambda_{-,0}^{\overline{a}_r}
\psi_{+,0}^{i_1} \cdots
\psi_{+,0}^{i_d} | 0 \rangle ,$$ (where $r$ is the rank of ${\cal E}_0^{\alpha}$ and $d$ the rank of $T_0^{\alpha}$), so if our Fock vacuum $| 0 \rangle$ couples to a line bundle ${\cal L}$, then the opposite or dual Fock vacuum above must couple to $$\lef
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covering of $A_4$ [@Liu:2019khw], in which masses, mixing, and CP phases for quark and lepton are predicted [^1]. A possible correction from Kähler potential is also discussed in Ref. [@Chen:2019ewa]. Furthermore, a systematic approach to understand the origin of CP transformations is recently discussed in ref. [@Baur:2019iai], and CP violation in models with modular symmetry is discussed in Ref. [@Kobayashi:2019uyt]. Another big mystery in the SM is the lack of a dark matter (DM) candidate. Even though many experiments from different aspects are going on to search for DM signatures, we have not obtained any decisive proofs yet. However, there are a lot of nice scenarios of DM that are connected to other observables. One of interesting models is known as the radiative seesaw model [@Ma:2006km]. This scenario not only explains the neutrino sector and DM at the same time but also provides a lot of new phenomena at a low energy scale such as lepton flavor violations, muon anomalous magnetic moment, collider signatures, etc. Since such a model connects the DM sector and the neutrino sector, the understanding of the neutrino nature leads to the understanding of the DM nature, and vise versa. In this paper, we work on a radiative seesaw scenario with a Dirac DM candidate based on our previous work [@Okada:2020oxh], applying a modular $A_4$ flavor symmetry. Then, we try to find several predictions in the lepton sector.
The manuscript is organized as follows. [In Sec. \[sec:realization\], we give our model setup under the $A_4$ modular symmetry, in which we review the modular $A_4$ symmetry and define relevant interactions needed to formulate the neutrino mass matrix and lepton flavor violations (LFVs). Then, we execute a numerical analysis and give several predictions in the lepton sector in Sec. III. Finally, we give our conclusion and discussion in Sec. \[sec:conclusion\].]{}
Model {#sec:realization}
=====
In this section, we introduce our model, which is based on a modular $A_4$ symmetry. The leptonic
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ght parameters can be extracted from eight observables that can be used to completely determine them. Additional observables can then be used in order to overconstrain the system. We divide the eight observables that we use to determine the system into four categories:
$(i)$ Branching ratio measurements (3 observables) [@Tanabashi:2018oca]. They are used to calculate the squared matrix elements. We neglect the tiny effects of order $|\lambda_b/\Sigma|$ and we get $$\begin{aligned}
\vert A_{\Sigma}(KK)\vert^2 &=
\frac{\mathcal{B}( \overline{D}^0\rightarrow K^+K^-)) }{ |\Sigma|^2 \mathcal{P}(D^0,K^+,K^-) }\,, \\
\vert A_{\Sigma}( \pi\pi)\vert^2 &=
\frac{\mathcal{B}( \overline{D}^0\rightarrow \pi^+\pi^-) }{ |\Sigma|^2 \mathcal{P}(D^0,\pi^+,\pi^-) }\,, \\
\vert A( K\pi)\vert^2 &=
\frac{\mathcal{B}( \overline{D}^0\rightarrow K^+\pi^-) }{ |V_{cs} V_{ud}^*|^2 \mathcal{P}(D^0, K^+, \pi^-) }\,, \\
\vert A(\pi K)\vert^2 &=
\frac{\mathcal{B}( \overline{D}^0\rightarrow K^-\pi^+) }{ |V_{cd} V_{us}^*|^2 \mathcal{P}(D^0,K^-,\pi^+ ) }\,.\end{aligned}$$ We consider three ratios of combinations of the four branching ratios, which are $$\begin{aligned}
R_{K\pi} &\equiv \frac{
\vert A(K\pi)\vert^2 -
\vert A(\pi K)\vert^2
}{
\vert A(K\pi) \vert^2 +
\vert A(\pi K) \vert^2
}\,, \label{eq:br-ratio-1}\\
R_{KK,\pi\pi} &\equiv \frac{
\vert A(KK)\vert^2 -
\vert A(\pi\pi)\vert^2
}{
\vert A(KK)\vert^2 +
\vert A(\pi \pi)\vert^2
}\,, \label{eq:br-ratio-2} \\
R_{KK,\pi\pi,K\pi} &\equiv \frac{
\vert A( KK )\vert^2
+ \vert A( \pi\pi) \vert^2
- \vert A( K\pi) \vert^2
- \vert A( \pi K) \vert^2
}{
\vert A( KK) \vert^2
+ \vert A( \pi \pi) \vert^2
+ \vert A( K \pi) \vert^2
+ \vert A( \pi K) \vert^2
}\,. \label{eq:br-ratio-3}\end{aligned}$$
$(ii)$ Strong phase which does not require CP violation (1 observable). The relative strong phase between CF and DCS decay modes $$\begin{aligned}
\delta_{K\pi} &\equiv \mathrm{arg}\left( \frac{\mathcal{A}(\overline{D}^0\rightarrow K^-\pi^+)}{\mathcal{A}(D^0\rightarrow K^-\pi^+)}\right)
= \m
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4 &&&&&\\
& & & & & \\
\hline
& & & & & \\
P\Omega_{2n}^{+}(q) & q_{2(n-1)} & q_{n-1} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} &\frac{(q^{n-1}-1)(q-1)}{(4, q^n-1)} & n \mbox{ even } \\
& & &&&(n,q)\neq (4,2)\\
n \geq 4 & q_{2(n-1)} & q_{n} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} & \frac{q^n-1}{(4, q^n-1)} & n \mbox{ odd}\\
& & & & & \\
\hline
\end{array}$$
$$\begin{array}{|c|c|} \hline
\, N \, & \, s \, \\
\hline
\, L_3(4) \, & \quad 7 \quad \\
\hline
\, L_6(2) \, & \quad 31 \quad \\
\hline
\, L_7(2) \, & \quad 127 \quad \\
\hline
\, U_6(2) \, & \quad 11 \quad \\
\hline
\, U_7(2) \, & \quad 43 \quad \\
\hline
\, PSp_{4}(4) \, & \quad 17 \quad \\
\hline
\, PSp_{6}(2) \, & \quad 7 \quad \\
\hline
\, PSp_{8}(2) \, & \quad 17 \quad \\
\hline
\, P\Omega_{8}^{-}(2) \, & \quad 17 \quad \\
\hline
\, P\Omega_{8}^{+}(2) \, & \quad 7 \quad \\
\hline
\end{array}$$
We recall that a torus is an abelian $t'$-group. The existence of the subgroups $T_1$ and $T_2$ appearing in Table 1 can be derived from the known facts about the maximal tori in these groups (see, [@Car Propositions 7-10] or [@VV Lemma 1.2]). The fact that the corresponding orders of the tori are coprime in each case can be deduced easily from Lemma \[cuentas\], while the assertion regarding the primitive prime divisors is deduced from Lemma \[Zsi\].
The information in Table 2 can be found either in [@Atl], or from the orders of maximal tori for the corresponding groups.
Note that the case $PSp_{4}(2)\cong \Sigma_6$ has already been considered in Lemma \[a6\].
The assertion on $L_2(q)$ is well known (see for instance [@Car Proposition 7]).
The existence of tori of the corresponding orders in $L_3(q)$ and $U_3(q)$ can be found in [@VV Lemma 1.2], and the claim on the prime divisors is easily deduced applying Lemma \[cuentas\].
\[excep\] For $N=G(q)$ an exceptional simple group of Lie type of characteristic $t$ and $q=t^e$, there exist primes $r, \, s \in \pi(N) \setminus (\pi({{\operatorname}{\textup{Out}}({N})}) \cup \
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]). It is important to remark that Eq. (\[hs\]) indeed describes quite well the system when $\omega_0 - \omega_L=\pm m\nu$ and $\Omega$ is moderately weak, which are conditions easily implemented in the laboratories [@meekhof1996; @roos]. Before the interaction with the laser, the trapped ion is found to be in thermal equilibrium with the environment (at inverse temperature $\beta$). This is described by the Gibbs state associated with Hamiltonian Eq. (\[h0\]), i.e., $$\label{initstate}
\hat{\rho}_i = \frac{\text{e}^{-\beta\hbar\nu\hat{n}}}{(\bar n + 1)} \otimes
\frac{\text{e}^{-\frac{\beta\hbar\omega_{0}}{2}\hat{\sigma}_{z}}}
{2\cosh \frac{\beta\hbar\omega_{0}}{2}},$$ where $\hat n = \hat a^\dag \hat a$ is the number operator and $$\label{mocn}
\bar n = {\rm Tr}(\hat n \hat \rho_i) = ({\rm e}^{\beta\hbar\nu} - 1)^{-1}$$ is the thermal occupation number of the CM motion. In spite of the difficulties found in dealing with the full Hamiltonian Eq. (\[hf\]), we were able to find the first moments of the work distribution. This is already valuable information because to obtain the full distribution we would need the whole set of eigenvalues and eigenvectors of Eq. (\[hf\]) which are not possible to be obtained, except numerically and to a restricted precision giving the complexity of the Hamiltonian. We then use Eq. (\[mom\]), appropriate to a sudden change, to calculate a few first moments of the work distribution and get some insight of it.
The first moment, $n=1$, using Eq. (\[hf\]) and Eq. (\[initstate\]), turns out to be $$\label{1mom}
\langle W \rangle =
{\rm Tr}\left[ \hat{\mathcal{H}}_{\rm I} \, \hat \rho_i \right]
\propto {\rm Tr}[\hat \sigma_{\pm} \, \text{e}^{-\frac{\beta\hbar\omega_{0}}{2}
\hat{\sigma}_{z}}] = 0.$$ As for the second, we now find $$\left\langle W^{2}\right\rangle = \hbar^2\Omega^2/4,$$ which, interesting enough, depends only on the magnitude of the work parameter $\Omega$ (controlled by laser power) and it is completely independent of the temperature.
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|
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|
tarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,
x)$}}},\end{aligned}$$ to which we can apply the bound discussed between [(\[eq:Theta’-2ndindbd2\])]{} and [(\[eq:Theta’-2ndindbd8\])]{}.
**(d-2)** If $v\notin{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for any $i=1,\dots,j$, then there exists a $v'\in{{\cal D}}_{{{\bf n}};l}$ for some $l\in\{0,\dots,T\}$ such that $v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v$ and ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\cap{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)={\varnothing}$ for any $i$. In addition, since all connections from $y$ to $x$ on the graph $\tilde{{\cal D}}\cup{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-2pt} {$\scriptstyle
j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)$ have to go through ${{\cal A}}$, there is an $h\in\{1,\dots,j\}$ such that $z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h$. Therefore, the contribution from this case to [(\[eq:Theta”-2ndindrewr\])]{} is bounded by $$\begin{gathered}
\sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\!\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\sum_{j=1}^T\sum_{\{s_it_i\}_{i
=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\sum_{\substack{v',z_1,\dots,z_j\\ z'_1,\dots,
z'_j}}\!\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};s_i},\;z'_i\in{{\cal D}}_{{{\bf n}};
t_i}\}$}}}\bigg)\sum_{l=0}^T{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'\in{{\cal D}}_{{{\bf n}};l}\}$}}}{\nonumber}\\
\times\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf k}}={\varnothing}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}
\bigg(\sum_{h=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\under
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f the dissipative GPE coupled to the impurity, and the final section summarizes our conclusions.
Modeling approach {#sec:model}
=================
We model the interaction between the impurity and a two-dimensional BEC through a Gaussian repulsive potential which can be reduced to a delta-function limit similar to previous studies [@astrakharchik2004motion; @pinsker2017gaussian]. The BEC itself, which is at a finite temperature, is described by a macroscopic wavefunction $\psi({\boldsymbol{r}},t)$ that evolves according to the damped Gross Pitaevskii equation (dGPE) [@Bradley_2012; @Reeves_2013; @skaugen2016vortex]: $$\begin{aligned}
\label{eq:GPe}
&& i\hbar\partial_t\psi = \nonumber \\
&& (1-i\gamma)\left(-\frac{\hbar^2}{2m}\nabla^2+g|\psi|^2-\mu+ V_{ext}+g_p{\cal U}_p\right)\psi,\end{aligned}$$ where $g$ is an effective scattering parameter between condensate atoms. $V_{ext}$ is any external potential used to confine the condensate atoms or to stir them. The damping coefficient $\gamma>0$ is related to finite-temperature effects due to thermal drag between the condensate and the stationary thermal reservoir of excited atoms at fixed chemical potential $\mu$. This damping $\gamma$ is very small at low temperatures and can be expressed as function of temperature $T$, chemical potential $\mu$ and the energy of the thermal cloud [@bradley2008bose]. The dGPE can be derived from the stochastic projected Gross-Pitaevskii equation in the low-temperature regime [@blakie2008dynamics] and has been used to study different quantum turbulence regimes and vortex dynamics [@Reeves_2014; @billam2015spectral; @Reeves_2013; @Bradley_2012] that are also observed in recent experiments [@neely2013characteristics]. An hydrodynamic description in terms of density and velocity of the BEC can be developed using the Madelung transformation of the wavefunction: $\psi=|\psi|e^{i\phi}$. The macroscopic number density is $\rho({\boldsymbol{r}},t)=|\psi({\boldsymbol{r}},t)|^2$ and the condensate velocity is ${\boldsymbol{v}}({\boldsymb
| 3,308
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Lambda )$, since ${\Omega }(u)v \in I^\chi (\Lambda )\subset
\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$. Thus by Eq. , $\Lambda {\mathrm{Sh}}$ induces a symmetric bilinear form on $L^\chi (\Lambda )$, also denoted by $\Lambda {\mathrm{Sh}}$. The radical of this form is a ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $L^\chi (\Lambda )$, but does not contain $1{\otimes }1_\Lambda $, and hence it is zero. Thus $\Lambda {\mathrm{Sh}}$ is a nondegenerate symmetric bilinear form on $L^\chi (\Lambda )$.
[*For the rest of this section let $\chi \in {\mathcal{X}}_4$, $n=|R^\chi _+|$, and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$.*]{} For all $\nu \in \{1,2,\dots ,n\}$ let $$\beta _\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }),\qquad
\chi _\nu =r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi ).$$ For all $\nu \in \{1,2,\dots ,n\}$, ${\alpha }\in {\mathbb{N}}_0^I$, and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$ let $$\begin{aligned}
{P}^\chi ({\alpha },\beta _\nu ;t)=\Big|\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
\sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,m_\nu \ge t,\quad &\\
m_\mu <{b^{\chi}} (\beta _\mu )\quad \text{for all $\mu \in \{1,2,\dots ,n\}$}
\Big\}\Big|.&
\label{eq:PF}
\end{aligned}$$
We will use two important facts on the function ${P}^\chi $.
\[le:P1\] For all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, $$\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}=
\frac {e^{-t\beta _\nu}-e^{-{b^{\chi}} (\beta _\nu )\beta _\nu }}
{1-e^{-\beta _\nu }}
\prod _{\mu \in \{1,\dots ,n\},\, \mu \not=\nu }
\frac {1-e^{-{b^{\chi}} (\beta _\mu )\beta _\mu }}
{1-e^{-\beta _\mu }}.$$
By Thm. \[th:PBWtau\], the two sides of the equation are two different expressions for the formal character of the subspace of $U^-(\chi ){\otimes }{\mathbb{K}}$ spanned by the elements $$\prod _{ {m_1,\dots ,m_n \atop
| 3,309
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when we apply the results after sample splitting (as is our goal), we need to define $u$ as $u = \min_{|S|\leq k} \lambda_{\rm min}(\Sigma_S)$. As $d$ increases, $u$ can get smaller and smaller even with fixed $k$. Hence, the usual fixed $k$ asymptotics may be misleading.
[**Remark:**]{} We only ever report inferences for the selected parameters. The bootstrap provides uniform coverage over all parameters in $S$. There is no need for a Bonferroni correction. This is because the bootstrap is applied to $||\hat\beta_{{\widehat{S}}}^* - \hat\beta_{{\widehat{S}}}||_\infty$. However, we also show that univariate Normal approximations together with Bonferroni adjustments leads valid hyper-rectangular regions; see Theorem \[thm::bonf\].
LOCO Parameters {#sec:loco.parameters}
---------------
Now we turn to the LOCO parameter $\gamma_{{\widehat{S}}} \in \mathbb{R}^{{\widehat{S}}}$, where ${\widehat{S}}$ is the model selected on the first half of the data. Recall that $j^{\mathrm{th}}$ coordinate of this parameter is $$\gamma_{{\widehat{S}}}(j) = \mathbb{E}_{X,Y}\Biggl[|Y-\hat\beta_{{\widehat{S}}(j)}^\top X_{{\widehat{S}}(j)}|-
|Y-\hat\beta_{{\widehat{S}}}^\top X_{{\widehat{S}}}| \Big| \mathcal{D}_{1,n} \Biggr],$$ where $\hat\beta_{{\widehat{S}}} \in \mathbb{R}^{{\widehat{S}}}$ is any estimator of $\beta_{{\widehat{S}}}$, and $\hat\beta_{{\widehat{S}}(j)}$ is obtained by re-computing the same estimator on the set of covariates ${\widehat{S}}(j)$ resulting from re-running the same model selection procedure after removing covariate $X_j$. The model selections ${\widehat{S}}$ and ${\widehat{S}}(j)$ and the estimators $\hat{\beta}_{{\widehat{S}}}$ and $\hat{\beta}_{{\widehat{S}}}(j)$ are all computed using half of the sample, $\mathcal{D}_{1,n}$.
In order to derive confidence sets for $\gamma_{{\widehat{S}}}$ we will assume that the data generating distribution belongs to the class ${\cal P}_n'$ of all distributions on $\mathbb{R}^{d+1}$ supported on $[-A,A]^{d+1}$, for some fixed constant $A>0$. Clearly the class $\mathca
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ice product is a relation between $\prod\Theta \times \prod\Phi$ and $\prod\Theta\Phi$. For this relation to be a mapping, it must yet be established that any member of the domain is related to exactly one member of the co-domain.
Again with $(\theta, \phi) \in \prod\Theta \times \prod\Phi$, suppose $\alpha \in \prod\Theta\Phi$ and $\beta \in \prod\Theta\Phi$ are $\Theta\Phi$-choices such that $(\theta, \phi) \mapsto \alpha$ and $(\theta, \phi) \mapsto \beta$. The hypothesis $\alpha \not= \beta$ now leads to the contradiction $\theta \cup \phi \not= \theta \cup \phi$, so the hypothesis is false and $\alpha = \beta$. Therefore the dyadic choice product is a mapping.
To be bijective, this mapping must be injective and surjective.
To assess injection, let $\theta$, $\theta' \in \Theta$ and $\phi$, $\phi' \in \prod\Phi$, and hypothesize $\theta\phi = \theta'\phi'$. With $\theta\phi = \theta'\phi'$, then the restrictions $\theta\phi\mid{{\operatorname{dom}{\Theta}}} = \theta'\phi'\mid{{\operatorname{dom}{\Theta}}}$. By \[D:DYADIC\_CHOICE\_PROD\], $\theta\phi\mid{{\operatorname{dom}{\Theta}}} = \theta$ and $\theta'\phi'\mid{{\operatorname{dom}{\Theta}}} = \theta'$. This establishes $\theta = \theta'$ in $\prod\Theta$. A similar approach, using restriction by ${{\operatorname{dom}{\Phi}}}$, establishes $\phi = \phi'$ in $\prod\Phi$. Since the assumption $\theta\phi = \theta'\phi'$ implies $\theta = \theta'$ and $\phi = \phi'$, then the dyadic choice product is injective.
To assess surjection, consider a general $\gamma \in \prod\Theta\Phi$. Define $\theta = \gamma \mid {{\operatorname{dom}{\Theta}}}$. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], the non-empty sets $\Theta(i) = \Theta\Phi(i)$ for $i \in {{\operatorname{dom}{\Theta}}}$. Since $\theta(i) \in \Theta\Phi(i)$ by construction, and $\Theta(i) = \Theta\Phi(i)$, then $\theta(i) \in \Theta(i)$ for $i \in {{\operatorname{dom}{\Theta}}}$ – that is, $\theta$ is a choice in $\prod\Theta$.
A similar tactic shows the existence of $\phi \in \prod\Phi$. Construc
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eta_k\|}_{p+1}^{p+1} + C(p,k){\|\theta_k\|}_p^p + C(k,p,M). \end{aligned}$$ Using the Gagliardo-Nirenberg-Sobolev inequality (Lemma \[lem:GNS\]) implies, $$\frac{d}{d\tau}{\|\theta_k\|}_p^p \leq -\frac{C(p)}{{\|\theta_k\|}^{\alpha_2}_1}{\|\theta_k\|}_{p+1}^{p+1} + C(p){\|\theta_k\|}_{p+1}^{p+1} + C(p,k){\|\theta_k\|}_p^p + C(k,p,M),$$ where $\alpha_2 = 1 - \alpha_1(p+m-1)/2 > 0$ and $$\alpha_1 = \frac{2d(1 - 1/(p+1))}{2 - d + dp + d(m-1)}.$$ Note ${\|\theta_k\|}_p \leq M^{1/p^2}{\|\theta_k\|}_{p+1}^{(p^2 -1)/p^2}$, which by weighted Young’s inequality implies, $$\frac{d}{d\tau}{\|\theta_k\|}_p^p \leq -\frac{C(p)}{{\|\theta_k\|}^{\alpha_2}_1}{\|\theta_k\|}_{p+1}^{p+1} + C(p){\|\theta_k\|}_{p+1}^{p+1} + C(k,p,M).$$ Using we may make the leading order terms as negative as we want and interpolating $L^p$ against $L^1$ and $L^{p+1}$ again implies there is a $\delta > 0$ such that if $k$ is sufficiently large we have, $$\frac{d}{d\tau}{\|\theta_k\|}_p^p \leq -\delta{\|\theta_k\|}_p^p + C(k,p,M).$$ By and conservation of mass, this concludes the proof of Lemma \[lem:finite\_p\_bounded\_unifint\].
Finite Length-Scales
====================
We begin by proving Theorem \[thm:Decay\] for the case ${\nabla}{\mathcal{K}}\in L^1$. Alikakos iteration [@Alikakos] is a standard method for using a result such as Lemma \[lem:finite\_p\_bounded\] to imply a result of the following form.
\[lem:rescaled\_inftybdd\] Let ${\nabla}{\mathcal{K}}\in L^1$. Then there exists $C_{\overline{q}} = C_{\overline{q}}(M)$ and $C_M = C_M({\|\theta_0\|}_{\overline{q}})$ such that if ${\|\theta_0\|}_{\overline{q}} < C_{\overline{q}}$ and $M < C_M$, then ${\|\theta(\tau)\|}_{\infty} \in L_\tau^\infty({\mathbb R}^+)$.
Standard iteration implies ${\|\theta(\tau)\|}_\infty \in L_\tau^\infty({\mathbb R}^+)$, provided $$\vec{v} := e^{(1 - \alpha - \beta)\beta^{-1}\tau}{e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot) \ast \theta} \in L^\infty_{\tau,\eta}({\mathbb R}^+\times {\mathbb R}^d).$$ See [@JagerLuckhaus92; @CalvezCarrillo06; @BRB10; @Kowalczyk05;
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thermore, randomly and uniformly select 2000 sets of data with irradiance between *G* = 150 W/m^2^--1000 W/m² and temperature between 15--40 °C as the test set.
Two layers of MLP structure are used and the activation function is set as LeakyRelu. In the MLP model, the learning rate is set as 1.75 × 10^-3^, the learning decay is 0.9, the training generation is 30, the small batch size is 500, the first layer of neurons is set as 150 and the second layer of neurons is set as 100.
Two layers of CNN structure are used, and the activation function is also set as LeakyRelu. In the CNN model, the learning rate is set as 3.5 × 10^-4^, the learning decay is 0.9, the training generation is 30, the small batch size is 25, the convolution kernel dimension is 50, the convolution kernel size is 3 and the hidden layer dimension is 100.
MLP and CNN models are trained according to the program flow as shown in [Figure 3](#sensors-20-02119-f003){ref-type="fig"}. The training process is illustrated in [Figure 4](#sensors-20-02119-f004){ref-type="fig"}. [Figure 4](#sensors-20-02119-f004){ref-type="fig"}a,b indicate the changes of loss function and accuracy in MLP training while [Figure 4](#sensors-20-02119-f004){ref-type="fig"}c,d indicate the changes of loss function and accuracy in CNN training. In [Figure 4](#sensors-20-02119-f004){ref-type="fig"}, *Acc* is defined as $$Acc = 1 - \frac{\left| {y_{i}^{\prime} - y_{i}} \right|}{y}$$
Curves 1 and 2 in [Figure 4](#sensors-20-02119-f004){ref-type="fig"}b,d show the accuracy in training set and test set respectively. It can be seen from [Figure 4](#sensors-20-02119-f004){ref-type="fig"} that the CNN model has a fast convergence rate and a stable process while the MLP model has a relatively slow convergence rate and has a series of oscillating links. However, the final convergence values of the two models are similar.
4. Results Analysis {#sec4-sensors-20-02119}
===================
4.1. The Accuracy Analysis {#sec4dot1-sensors-20-02119}
--------------------------
MLP and CNN a
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e $SU(N)$ gauge symmetry.) The gauge algebra is non-Abelian even for the Abelian group $U(1)$.
To describe the noncommutative $U(N)$ gauge algebra properly, it is a necessity to use the generators (\[factor\]) including functional dependence on the base space. Nevertheless, the noncommutative $U(N)$ gauge symmetry still assumes that the generators can be factorized (\[factor\]), and that $e^{ip\cdot x}$ always commutes with $T_a$. These are unnecessary assumptions for most algebraic calculations in the gauge theory. After all, in field theories, only the coefficients $\tilde{A}^a_{\mu}(p)$ are operators (observables), while the space-time and Lie algebra dependence are to be integrated out (summed over) in the action.
It is thus natural to slightly extend the formulation of gauge symmetry (and YM theory) to allow the Lie algebra to be directly defined in terms of the generators $T_a(p)$, without even assuming its factorization into a Lie algebra factor $T_a$ and a function $e^{p\cdot x}$. The integration over space-time and trace of the internal space in the action can be replaced by the Killing form of the Lie algebra of $T_a(p)$. The distinction between internal space and external space is reduced in this description.
In short, we propose to study gauge symmetries without assuming its factorization into two associative algebras (an algebra for the functions on the base space and a finite dimensional Lie algebra). Even when it is possible to factorize the generators formally as (\[factor\]), we will not assume that the space-time functions to commute with the algebraic elements $T_a$. (In general, we do not have to use the Fourier basis, and the argument $p$ of $T_a(p)$ can represent labels of any complete basis of functions on the base space.) One of the goals of this paper is to show that this generalization is beneficial, for bringing in new insights into gravity theories.
Algebraically, this generalization is very natural. A corresponding geometric notion is however absent at this moment. (It is not clea
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d $E_{\mathrm{th}}=4.44$ and 4.51 MeV. []{data-label="28Ne_strength"}](fig8-2.eps "fig:")
The central panel in Fig. \[response\] shows the response function in $^{28}$Ne. In the low-energy region, we can see a two-bump structure at around 7 and 8 MeV. Because the deformation is small as in $^{26}$Ne, we cannot see a splitting of the giant resonance. In Fig. \[28Ne\_strength\], we show the low-energy part of the strength functions. In the $K^{\pi}=0^{-}$ states, there is a prominent peak at 8.1 MeV with a strength of 0.098 $e^{2}$fm$^{2}$. The strength distribution is fragmented for the $K^{\pi}=1^{-}$ mode, but correspondingly, we can see an eigenmode at 8.9 MeV with the largest transition strength of 0.058 $e^{2}$fm$^{2}$.
We show in Table \[28Ne\_0-\] the QRPA amplitude for the $K^{\pi}=0^{-}$ state at 8.14 MeV in $^{28}$Ne. The main component is the neutron two-quasiparticle excitation of $\nu([310]1/2 \otimes [211]1/2)$ corresponding to $\nu(2s^{-1}_{1/2}2p_{3/2})$. Two quasiparticle excitations of (b) and (c) in Table \[28Ne\_0-\] correspond to $\nu(1d^{-1}_{5/2}1f_{7/2})$, and (d): $\nu(2s^{-1}_{1/2}1f_{7/2})$, (e) and (f): $\nu(1d^{-1}_{3/2}2p_{3/2})$, (g): $\nu(1d^{-1}_{3/2}1f_{7/2})$, (h) and (i): $\nu(1d^{-1}_{5/2}2p_{3/2})$, and (j): $\nu(1d^{-1}_{5/2}2p_{1/2})$ excitations, respectively. The proton excitation of $\pi(1p_{1/2}\otimes 1d_{5/2})$ has an appreciable contribution as in $^{26}$Ne.
----- --------------- --------------- ------------------------ -------------------------------------------- ----------------------
$E_{\alpha}+E_{\beta}$ $Q_{10,\alpha\beta}$
$\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm)
(a) $\nu[310]1/2$ $\nu[211]1/2$ 8.27 0.569 $-0.303$
(b) $\nu[330]1/2$ $\nu[220]1/2$ 11.2 0.055
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the usual open-closed intervals in $\mathbb{R}$[^13]. The subbases $_{\textrm{T}}\mathcal{S}_{1}=\{(a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{2}=\{[a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{3}=\{(a,\infty),(-\infty,b]\}$ and $_{\textrm{T}}\mathcal{S}_{4}=\{[a,\infty),(-\infty,b]\}$ give the respective bases $_{\textrm{T}}\mathcal{B}_{1}=\{(a,b)\}$, $_{\textrm{T}}\mathcal{B}_{2}=\{[a,b)\}$, $_{\textrm{T}}\mathcal{B}_{3}=\{(a,b]\}$ and $_{\textrm{T}}\mathcal{B}_{4}=\{[a,b]\}$, $a\leq b\in\mathbb{R}$, leading to the *standard (usual*)*, lower limit (Sorgenfrey*)*, upper limit,* and *discrete* (take $a=b$) topologies on $\mathbb{R}$. Bases of the type $(a,\infty)$ and $(-\infty,b)$ provide the *right* and *left ray* topologies on $\mathbb{R}$.
*This feasibility of generating different topologies on a set can be of great practical significance because open sets determine convergence characteristics of nets and continuity characteristics of functions, thereby making it possible for nature to play around with the structure of its working space in its kitchen to its best possible advantage.*[^14] **
Here are a few essential concepts and terminology for topological spaces.
**Definition 2.2.** ***Boundary, Closure, Interior*.** *The* *boundary of $A$ in $X$* *is the set of points $x\in X$ such that every neighbourhood $N$ of $x$ intersects both $A$ and $X-A$:* $${\textstyle \textrm{Bdy}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})((N\bigcap A\neq\emptyset)\wedge(N\bigcap(X-A)\neq\emptyset))\}}\label{Eqn: Def: Boundary}$$ *where $\mathcal{N}_{x}$ is the neighbourhood system of Eq. (\[Eqn: Def: nbd system\]) at $x$.*
*The* *closure of $A$* *is the set of all points $x\in X$ such that each neighbourhood of $x$ contains at least one point of $A$* ***that may be $\boldmath{x}$ itself****. Thus the set* $${\textstyle \textrm{Cl}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap A\neq\emptyset)\}}\label{Eqn: Def: Closure}$$ *of all points in $X$
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following we derive the desired properties for applying Lemma \[thelemma\]. First we write also the potential term in in a quadratic way.
\[magneticL\] The operator matrix $$\begin{aligned}
{\mathbf{L}}=\mathbf{P}_{[0,t)}\left(\begin{matrix} 0 & ik\left(A-A^*\right)\\ik\left(A^* - A\right) & 0\end{matrix}\right)\mathbf{P}_{[0,t)},\label{magnetiLformula}\end{aligned}$$ fulfills $$\begin{aligned}
\frac12\langle\mathbf{f},{\mathbf{L}}\mathbf{f}\rangle &=-ik\int\limits_{0}^t \left(\int_0^{\tau} f_1(s) \, ds f_2(\tau)-f_1(\tau) \int_0^{\tau}f_2(s) \, ds \,\right) d\tau, \quad 0\leq t<\infty,\end{aligned}$$ where $\mathbf{f} = (f_1,f_2) \in L_2^2({\mathbb{R}})$ and operator $A$ is defined by $$A f(\tau) = {\mathbf{1}}_{[0,t)}(\tau) \int_{[0,\tau)} f(s) \, ds,\quad f \in L^2({\mathbb{R}}), \tau\in {\mathbb{R}}.$$ $A^*$ denotes its adjoint w.r.t. the bilinear dual pairing $ \langle \cdot,\cdot \rangle$. Moreover ${\mathbf{L}}$ is symmetric w.r.t. $\langle \cdot , \cdot \rangle$.
[**Proof:**]{} With ${\mathbf{L}}$ as above we have by the symmetry of the dual pairing $$\begin{aligned}
\langle\mathbf{f},{\mathbf{L}}\mathbf{f}\rangle&=\left\langle\left(\begin{matrix}f_1\\ f_2\end{matrix}\right),
\left(\begin{matrix} 0 & ikP_{[0,t)}\left(A-A^*\right)\\ikP_{[0,t)}\left(A^*-A\right) & 0\end{matrix}
\right)\left(\begin{matrix}f_1\\\ f_2\end{matrix}\right)
\right\rangle\\
&=\left\langle f_1,ikP_{[0,t)}Af_2\right\rangle-\left\langle f_1 , ikP_{[0,t)}A^*f_2\right\rangle\\
&\hspace{35mm}+\left\langle f_2,ikP_{[0,t)}A^*f_1\right\rangle-\left\langle f_2, ikP_{[0,t)}A f_1\right\rangle\\
&=2\left\langle f_1,ikP_{[0,t)} A f_2\right\rangle-2\left\langle f_2,ikP_{[0,t)}A f_1\right\rangle\\
&=2ik\int\limits_{0}^t \left(\int_0^{\tau} f_1(s) \, ds f_2(\tau)-f_1(\tau) \int_0^{\tau}f_2(s) \, ds\right),\end{aligned}$$ since $P_{[0,t)}$ and $A$ commute. $\blacksquare$
If we extend $\langle \cdot, {\mathbf{L}}\cdot \rangle$ informally to an element $\boldsymbol{\omega}\in S'_2({\mathbb{R}})$ we have $$\label{80}
\frac{1}{2} \langle \boldsy
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with the sole assumption (besides $g_p$ sufficiently small) of smallness of the unsteady and/or inhomogeneous part $\delta\psi_0$ of the wavefunction, which allows linearization. Eq. (\[eq:Fi\_th\]) assumes in addition weak inhomogeneities below scales $a$ and $\xi$, and finally Eqs. (\[eq:InertialSimpleDimless\]) and (\[eq:InertialSimpleDim\]) (equivalent under the previous linearization approximation) completely neglects such inhomogeneities (or equivalently, they correspond to $a,\xi\rightarrow 0$). We show as black lines in Fig. (\[fig:inertial\_force\]) the prediction of this last approximation, similar to the most standard classical expressions. Since we have computed the wavefunction $\psi=1+\delta\psi_0$ in the comoving frame from Eq. (\[eq:ComovingdGPE\]), we actually use expression (\[eq:Fi\_th\]) without the Faxén Laplacian terms, with $\delta{\boldsymbol{\omega}}^{(0)}=\nabla(\delta \psi_0-\delta
\psi_0^*)/(2i) - {\boldsymbol{V}}_p$, and ${\boldsymbol{\dot V_p}}=0$. Fig. (\[eq:ComovingdGPE\]) shows that the full force computed from Eq. (\[eq:fp2\]) is well-captured by the approximate expression of the inertial force for small test-particle size $a'$. Accuracy progressively deteriorates for increasing $a'$, and also for increasing $V_p$, but this classical expression remains a reasonable approximation until $a'\approx 1$. The accuracy can be improved by considering higher-order Faxén corrections, Eq. (\[eq:Fi\_th\]), or even better, by considering the integral form in Eq. (\[eq:F0full\]). We have explicitly checked that keeping the full Gaussian integration in Eq. (\[eq:F0full\]) but approximating the integrand in the Bessel integral by its value at the particle position gives a very good approximation to the exact force even for $a'=1$.
Numerical evaluation of the drag force {#sec:numerical-drag}
--------------------------------------
![Plot of the drag force in the steady-state regime as function of the constant speed $V_p$. Dashed lines are the analytical predictions based on Eq. (\[eq:forcegamm
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relative to the KL loss (\[eqn:loss\]) and has a constant risk.
Recently, Matsuda and Komaki (2015) constructed an improved Bayesian predictive density on $\ph_U(Y\mid X)$ by using a prior density of the form $$\label{eqn:pr_em}
\pi_{EM}(\Th)=|\Th\Th^\top|^{-\al^{EM}/2},\quad \al^{EM}=q-r-1.$$ The prior (\[eqn:pr\_em\]) is interpreted as an extension of Stein’s (1973, 1981) harmonic prior $$\label{eqn:pr_js}
\pi_{JS}(\Th)=\Vert\Th\Vert^{-\be^{JS}}=\{\tr(\Th\Th^\top)\}^{-\be^{JS}/2},\quad \be^{JS}=qr-2.$$ In the context of Bayesian estimation for mean matrix, (\[eqn:pr\_em\]) yields a matricial shrinkage estimator, while (\[eqn:pr\_js\]) does a scalar shrinkage one. Note that, when $X\sim\Nc_{r\times q}(\Th, v_x I_r\otimes I_q)$, typical examples of the matricial and the scalar shrinkage estimators for $\Th$ are, respectively, the Efron-Morris (1972) estimator $$\label{eqn:EM}
\Thh_{EM}=\{I_r-\al^{EM}v_x(XX^\top)^{-1}\}X
\quad \textup{for $\al^{EM}\geq 1$ (i.e., $q\geq r+2$)}$$ and the James-Stein (1961) like estimator $$\label{eqn:JS}
\Thh_{JS}=\Big\{1-\frac{\be^{JS}v_x}{\tr(XX^\top)}\Big\}X
\quad \textup{for $\be^{JS}\geq 1$ (i.e., $qr\geq 3$)}.$$ The two estimators $\Thh_{EM}$ and $\Thh_{JS}$ are minimax relative to a quadratic loss. Also, $\Thh_{EM}$ and $\Thh_{JS}$ are characterized as empirical Bayes estimators, but they are not generalized Bayes estimators which minimize the posterior expected quadratic loss.
The purposes of this paper are to construct some Bayesian predictive densities with different priors from (\[eqn:pr\_em\]) and (\[eqn:pr\_js\]) and to discuss their decision-theoretic properties such as admissibility and minimaxity. Section \[sec:preliminaries\] first lists some results on the Kullback-Leibler risk and the differentiation operators. Section \[sec:properminimax\] applies an extended Faith’s (1978) prior to our predictive density estimation problem and provides sufficient conditions for minimaxity of the resulting Bayesian predictive densities. Also, an admissible and minimax pred
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$p_0$ and $q_0$ and, by Proposition \[1stpart3rdrequire\], there is an open $O$ in $S$ containing $\Lambda_0$ such that, for all $\Lambda=(p, k^\mu)$ in $O$, there is a conjugate point $q$. By the proof of Proposition \[1stpart3rdrequire\] and the discussion above, one may find a value $\lambda_i$ such that for all the geodesics defined by every $\Lambda$ in $O$, the inequality (\[inicial\]) holds. Note that the value $\lambda_i$ is strictly larger than zero if the origin is defined such that $\gamma_\Lambda(0)=p$.
The length parameter $\lambda_e$ that defines the conjugate point $q_0$ in the geodesic $\gamma_0$ is given by (\[explota\]) which, adapted to the present situation, is given by =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_i<\lambda_1<\lambda_e$ is an initial parameter, and p\_0()=\[R\_k\^k\^+\_\^\](),R\_0(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_0()+c)\^2 d. The length parameter defining the conjugate point $q$ for another geodesic $\gamma_\Lambda(\lambda)$ is $\lambda_e+\Delta\lambda_e$, and is given by =\_[\_1]{}\^[\_e+\_e]{}{1+}\^2e\^[c]{}d, with p\_()=\[R\_k\^k\^+\_\^\]\_(),R\_(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_()+c)\^2 d. Note that $\Delta \lambda_e$ depends on the choice of the geodesic, that is, $\Delta \lambda_e=f(\Lambda)$. This dependence would be implicitly understood in the following reasoning. The task is to show that $|\Delta \lambda_e|=|f(\Lambda))|<\epsilon$ when $\Lambda$ is in an open $O$ of $S$ small enough, containing $\Lambda_0$.
A point that might cause confusion is that, in principle, $\lambda_e+\Delta\lambda_e$ may be such that $\lambda_e+\Delta \lambda_e<\lambda_1$, as $\Delta\lambda_e$ may be negative. But the formula (\[mat2\]) is true for the opposite case. Thus, $\lambda_1$ should not be chosen so arbitrary. However, it has been mentioned that, from the continuity of $\theta_\Lambda(\lambda)$ in $S$ outside a conjugate point, one has that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small enough. In fact $|\theta_\Lambda(\lambda)-\theta_0(\lambda_i
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A \to A$ a map for all $i\in I$, and $C^a=(c^a_{jk})_{j,k \in I}$ a generalized Cartan matrix in ${\mathbb{Z}}^{I \times I}$ for all $a\in A$. The quadruple $${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$$ is called a *Cartan scheme* if
1. ${r}_i^2 = {\operatorname{id}}$ for all $i \in I$,
2. $c^a_{ij} = c^{{r}_i(a)}_{ij}$ for all $a\in A$ and $i,j\in I$.
Let $A=\{a\}$ be a set with a single element, and let $C$ be a generalized Cartan matrix. Then $r_i={\operatorname{id}}$ for all $i\in I$, and ${\mathcal{C}}$ becomes a Cartan scheme.
One says that a Cartan scheme ${\mathcal{C}}$ is *connected*, if the group $\langle {r}_i\,|\,i\in I\rangle \subset {\mathrm{Aut}}(A)$ acts transitively on $A$, that is, if for all $a,b\in A$ with $a\not=b$ there exist $n\in {\mathbb{N}}$ and $i_1,i_2,\ldots ,i_n\in I$ such that $b=r_{i_n}\cdots r_{i_2}
r_{i_1}(a)$. Two Cartan schemes ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ and ${\mathcal{C}}'={\mathcal{C}}'(I',A',$ $({r}'_i)_{i\in I'},({C'}^a)_{a\in A'})$ are called *equivalent*, if there are bijections $\varphi _0:I\to I'$ and $\varphi _1:A\to A'$ such that $$\begin{aligned}
\label{eq:equivCS}
\varphi _1({r}_i(a))={r}'_{\varphi _0(i)}(\varphi _1(a)),
\qquad
c^{\varphi _1(a)}_{\varphi _0(i) \varphi _0(j)}=c^a_{i j}
\end{aligned}$$ for all $i,j\in I$ and $a\in A$.
Let ${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$ be a Cartan scheme. For all $i \in I$ and $a \in A$ define ${\sigma }_i^a \in
\Aut({\mathbb{Z}}^I)$ by $$\begin{aligned}
{\sigma }_i^a ({\alpha }_j) = {\alpha }_j - c_{ij}^a{\alpha }_i \qquad
\text{for all $j \in I$.}
\label{eq:sia}
\end{aligned}$$ This map is a reflection. The *Weyl groupoid of* ${\mathcal{C}}$ is the category ${\mathcal{W}}({\mathcal{C}})$ such that ${\mathrm{Ob}}({\mathcal{W}}({\mathcal{C}}))=A$ and the morphisms are generated by the maps ${\sigma }_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$ with $i\in I$, $a\in A$. Formally, for $a,b\in A$ the se
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in $A$ (similarly in $B$), its degree is at most $\frac n 2$, and at least $\frac n 2 - O(\sqrt{n})$. It has $\frac n 2 - O(\sqrt{n})$ neighbors in $B$, so the number of its neighbors in $A$, and the number of its non-neighbors in $B$ is $O(\sqrt{n})$. By deleting and adding $O(\sqrt{n})$ edges to each vertex, we get a complete bipartite graph.
Alternatively, we could have defined $\G$ as a family of $\frac n 2$-regular graphs with $\lambda_2$ bounded, and $\lambda_n(G) = -\frac n 2 + O(1)$. It’s interesting to note that in this case it follows that $\lambda_{n-1}$ is bounded. For $G \in G$, if $G$ is bipartite, then it is complete bipartite, and $\lambda_{n-1}(G) = 0$. Otherwise, $\chi(G) > 2$, and by a theorem of Hoffman ([@Hoff]) $\lambda_n(G) + \lambda_{n-1}(G) + \lambda_1(G) \geq 0$. By our assumption, $\lambda_n(G) + \lambda_1(G) = O(1)$, and since $\lambda_{n-1}(G) < 0$ (otherwise the eigenvalues won’t sum up to 0), it follows that $\lambda_{n-1}(G) = -O(1)$.
Conclusion and Open Problems
============================
The only explicit examples known so far for graphs that have linear sphericity are $K_{n,n}$ and small modifications of it. We conjecture that more complicated graphs, such as the Paley graph, also have linear sphericity. Note that the lower bound presented here only shows a bound of $\Omega(\sqrt{n})$. It is also interesting to know if the bound can be improved, either as a pure spectral bound, or with some further assumptions on the structure of the graph.\
What is the largest sphericity, $d=d(n)$, of an $n$-vertex graph? We know that $\frac n 2 \leq d \leq n-1$. Can this gap be closed? For a seemingly related question, the smallest dimension required to realize a sign matrix (see [@AFR85]) the answer is known to be $\frac n 2 \pm o(n)$. We have also seen a similar gap for $d(n,l_2)$ and $d(n,l_\infty)$. Can this be closed? Can some kind of interpolation arguments generalize the bounds we know for these two numbers to bounds on $d(n,l_p)$ for $p>2$?\
Our interest in sphericity arose fr
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(\mathcal{L(H)},\subset )$ and/or the lattice $(\mathcal{L(S)},\subset )$ can be identified with Birkhoff and von Neumann’s lattice of *experimental propositions*, which was introduced in the 1936 paper that started the research on QL.$^{(31)}$ This identification is impossible, however, if $\mathcal{H}$ is not finite-dimensional, since Birkhoff and von Neumann’s lattice is modular, not simply weakly modular. The requirement of modularity has deep roots in the von Neumann concept of probability in QM according to some authors.$^{(2)}$
[^7]: One can provide an intuitive support to this definition by noticing that the result obtained in a test of $E$ on a physical object $x$ in the state $S$ can be attributed to $x$ only whenever $S$ is not modified by the test. Moreover, only in this case the test is *repeatable*, i.e., it can be performed again obtaining the same result.
It is well known that classical physics assumes that tests which do not modify the state $S$ are always possible, at least as ideal limits of concrete procedures, while this assumption does not hold in QM.
[^8]: Assumption A$_{4}$ can be stated unchanged whenever the standard interpretation of QM is adopted instead of the SR interpretation. In this case, however, for every $\xi $, $\sigma (\xi )$ is defined only on a subset of rfs, not on the whole $\psi _{R}^{Q}$ (which requires a weakening of the assumptions on $\sigma $ if one wants to recover this case within the general perspective in Sec. 3.1). Furthermore, $\Sigma _{S}$ reduces to a singleton. Indeed, for every interpretation $\xi $, a state $S=S(\xi )$ exists such that $\xi (x)\in S$. Then, $\sigma (\xi )$ is defined on a rf $E(x)$ iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (Sec. 2.4), and does not change if $\xi $ is substituted by an interpretation $\xi ^{\prime
} $ such that $\xi ^{\prime }(x)\in S$.
[^9]: Assumption A$_{5}$ in Sec. 3.2 can be stated unchanged if the standard interpretation of QM is adopted instead of the SR interpretation. In this case, how
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ictive density is obtained by considering a proper hierarchical prior. In Section \[sec:superharmonic\], we utilize Stein’s (1973, 1981) ideas for deriving some minimax predictive densities with superharmonic priors. Section \[sec:MCstudies\] investigates numerical performance in risk of some Bayesian minimax predictive densities.
Preliminaries {#sec:preliminaries}
=============
The Kullback-Leibler risk
-------------------------
First, we state some useful lemmas in terms of the Kullback-Leibler (KL) risk. The lemmas are based on Stein (1973, 1981), George et al. (2006) and Brown et al. (2008) and play important roles in studying decision-theoretic properties of a Bayesian predictive density.
From George et al. (2006, Lemma 3), we observe that $m_\pi(W;v_w)<\infi$ for all $W\in\Re^{r\times q}$ if $m_\pi(X;v_x)<\infi$ for all $X\in\Re^{r\times q}$. Note also that $\int_{\Re^{r\times q}}\ph_\pi(Y\mid X)\dd Y=1$ and $$\int_{\Re^{r\times q}}Y\ph_\pi(Y\mid X)\dd Y
=\frac{\int_{\Re^{r\times q}} \Th p(X\mid \Th)\pi(\Th)\dd\Th}{\int_{\Re^{r\times q}} p(X\mid \Th)\pi(\Th)\dd\Th},$$ namely, the mean of a predictive distribution for $Y$ is the same as the posterior mean of $\Th$ given $X$ or, equivalently, the generalized Bayes estimator relative to a quadratic loss for a mean of $X$.
Hereafter denote by $p(W|\Th)$ a density of $W|\Th\sim\Nc_{r\times q}(\Th, vI_r\otimes I_q)$ with a positive value $v$. In order to prove minimaxity of a Bayesian predictive density, we require the following lemma, which implies that our Bayesian prediction problem can be reduced to the Bayesian estimation problem for the normal mean matrix relative to a quadratic loss.
\[lem:identity\] The KL risk difference between $\ph_U(Y\mid X)$ and $\ph_\pi(Y\mid X)$ can be written as $$R_{KL}(\Th,\ph_U) - R_{KL}(\Th,\ph_\pi)
=\frac{1}{2}\int_{v_w}^{v_x}\frac{1}{v^2}\{\Er^{W|\Th}[\Vert W-\Th\Vert^2]-\Er^{W|\Th}[\Vert\Thh_\pi-\Th\Vert^2]\}\dd v,$$ where $\Er^{W|\Th}$ stands for expectation with respect to $W$ and $$\Thh_\pi=\Thh_\pi(W)=\frac{\in
| 3,324
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stant $C_1>0$ such that *a priori* estimate \[diss-co-es-ad\] [\^\*]{}\_[L\^2(GSI)\^3]{} C\_1(\_[L\^2(GSI)\^3]{}+ \_[T\^2(\_+)H\^1(I,T\^2(\_+’))\^2]{}), holds.
The existence result analogous to Theorem \[coupthev\] (based on the theory of evolution operators) holds also for the adjoint problem, and it guarantees that $\psi^*
\in\tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$. In this case, one assumes that $K$ takes the form , and that consequently its adjoint version $K^*=(K_1^*, K_2^*, K_3^*)$ is, $$\begin{aligned}
\label{ec1moda:ad}
(K_j^*\psi)(x,\omega,E)=\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega,\omega',E)\psi_k(x,\omega',E)d\omega',\quad j=1,2,3.\end{aligned}$$
The adjoint version of Theorem \[coupthev\] can be formulated as follows.
\[coupthevad\] Suppose that the adjoint collision operator is of the form , and that the assumptions (\[ass1-aa\])-(\[ec9-aa\]) of Theorem \[coupthev\] are valid for $\Sigma_j,\ \sigma_{jk}$ and $S_j$. Furthermore, suppose that $f^*\in C^1(I,L^2(G\times S)^3)$ and $g^*\in C^2(I,T^2(\Gamma_+')^3)$ which satisfies the compatibility condition $$g_j^*(0)=0,\quad j=2,3.$$ Then the problem - has a unique solution $\psi^*\in \tilde W^2(G\times S\times I)\times \big( C(I,\tilde W^2(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)\big)$. In particular, $\psi^*\in\tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap W_1^2(G\times S\times I)\big)^2$.
If, in addition, the conditions , are valid, then the solution $\psi^*$ satisfies the estimate (\[adjoint11b\]).
It is clear that a Sobolev space version of the above theorem analogous to Corollary \[coupcoev\], holds for the adjoint problem as well.
\[exgreenf1\]
In radiation therapy the absorbed *dose* from the particle field $\psi=(\psi_1,\psi_2,\psi_3)$ is defined by the functional D(x)=(D)(x):=\_[j=1]{}\^3\_[SI]{}\_j(x,E)\_j(x,,E) ddE, where $\psi$ is the solution of (\[desol10\])-(\[desol12\]). We see that (D)(x)=\_[j=1]{}\^3\_j(x,),\_j(x,,)\_[L\^2(SI)]{} =
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e globally for all $i=1,2,\cdots,I$ is assured leading to the conclusion that $f_{\alpha}(x_{i})\in V_{i}$ eventually for every $i=1,2,\cdots,I$. Hence $f_{\alpha}\in B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ eventually; this completes the demonstration that $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$, and thus of the proof.$\qquad\blacksquare$
***End Tutorial6***
***3.2. The Extension*** **Multi$_{\mid}$(*X,Y*)** ***of*** **Map(*X,Y*)**
In this Section we show how the topological treatment of pointwise convergence of functions to functions given in Example A1.1 of Appendix 1 can be generalized to generate the boundary $\textrm{Multi}_{\mid}(X,Y)$ between $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$; here $X$ and $Y$ are Hausdorff spaces and $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$ are respectively the sets of all functional and non-functional relations between $X$ and $Y$. The generalization we seek defines neighbourhoods of $f\in\textrm{Map}(X,Y)$ to consist of those functional relations in $\textrm{Multi}(X,Y)$ whose images at any point $x\in X$ lies not only arbitrarily close to $f(x)$ (this generates the usual topology of pointwise convergence $\mathcal{T}_{Y}$ of Example A1.1) but whose inverse images at $y=f(x)\in Y$ contain points arbitrarily close to $x$. Thus the graph of $f$ must not only lie close enough to $f(x)$ at $x$ in $V$, but must additionally be such that $f^{-}(y)$ has at least branch in $U$ about $x$; thus $f$ is constrained to cling to $f$ as the number of points on the graph of $f$ increases with convergence and, unlike in the situation of simple pointwise convergence, no gaps in the graph of the limit object is permitted not only, as in Example A1.1 on the domain of $f$, but simultaneously on it range too. We call the resulting generated topology the *topology of pointwise biconvergence on* $\textrm{Map}(X,Y)$, to be denoted by $\mathcal{T}$. Thus for any given integer $I\geq1$, the generalization of Eq. (\[Eqn: point\]) gives for $i=1,2,\cdots,I$, the open sets
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sheff and Scott Wilson for useful comments and remarks regarding this topic. The second author was partially supported by the Max-Planck Institute in Bonn.
$\widehat{\mathcal Comm}_\infty$ structure for Poincaré duality spaces {#Comm-section}
======================================================================
Before going into the details of the construction of homotopy inner products over a general cyclic quadratic operad $\mathcal O$, we give an application for the case of the commutative operad $\mathcal Comm$. More precisely, we show how a homotopy $\mathcal Comm$-inner product arises on the chain level of a Poincaré duality space $X$. In fact, the construction for the homotopy $\mathcal Comm$-algebra is taken from R. Lawrence and D. Sullivan’s paper [@S] on the construction of local infinity structures. In [@TZ], M. Zeinalian and the second author construct homotopy $\mathcal Assoc$-inner products on a Poincaré duality space $X$. The same reasoning may in fact be used to construct homotopy $\mathcal Comm$-inner products on $X$. The proof of the next proposition will be a sketch using these arguments.
\[prop:comm-pd\] Let $X$ be a closed, finitely triangulated Poincaré duality space, such that the closure of every simplex is contractible. Denote by $C=C_\ast(X)$ the simplicial chains on $X$. Then its dual space $A:=C^*=Hom(C_*(X),k)$ has the structure of a $\widehat{\mathcal Comm}_\infty$ algebra, such that the lowest multiplication is the symmetrized Alexander-Whitney multiplication and the lowest inner product is given by capping with the fundamental cycle $\mu\in C$.
Let $\mathcal Lie$ denote the Lie-operad, $F_{\mathcal Lie}V=\bigoplus_{n\geq 1} (\mathcal Lie(n)\otimes V^{\otimes n})_{S_n}$ denote the free Lie algebra generated by $V$, and $F_{\mathcal Lie,V}W=\bigoplus_{n\geq 1} (\bigoplus_{k+l=n-1}\mathcal Lie(n)\otimes V^{\otimes k}\otimes W\otimes V^{\otimes l})_{S_n}$ the canonical module over $F_{\mathcal Lie}V$. We will see in proposition \[O\_hat\_algebras\] and example \[exa-comm\],
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hen a set of parameters was tested that was very different from the last set, the experiment almost always produced no atoms, meaning we had to assign a default cost that did not provide meaningful gradient information to the learner. Once the next set of parameters is determined they are sent to the experiment to be tested. After the resultant cost is measured this is then added to the observation set $\mathcal{O}$ with $N\rightarrow N+1$ and the entire process is repeated.
We emphasize that fitting of $H$, estimation of $M_{\hat{\mathscr{C}}}(X)$ and minimization of $B_{\hat{\mathscr{C}}}(X)$ is all done online while the experiment is being run. In a single optimization run, the learner typically performs hundreds more hypothetical experiments than the number physically run in the lab. The MLOO algorithm we developed is open source and available online at [@hush_m-loop_2015] (it uses the package scikit-learn [@pedregosa_scikit-learn:_2011] to evaluate the GPs).
As a benchmark for comparison, we also performed OO using a Nelder-Mead solver [@nelder_simplex_1965], which has previously been used to optimize quantum gates [@kelly_optimal_2014].
We demonstrate the performance of machine learning online optimization in comparison to the Nelder-Mead optimizer in Fig. 2. Here we used the complex parameterization for all 3 ramps, and added an extra parameter that controlled the total time of the ramps, resulting in 16 parameters. If we were to perform a brute force search and optimize the parameters to within a $10\%$ accuracy of the parameters maximum-minimum bounds, the number of runs required would be $10^{16}$. The Nelder-Mead algorithm is able to find BEC much faster than this, in only 145 runs. The machine learning algorithm, on the other hand, is much faster. After the first 20 training runs, where the machine learning and Nelder-Mead algorithm use a common set of parameters, the machine learning algorithm converges in only 10 experiments.
![Optimization of evaporation curves to produce a BEC. The first 2N e
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t adjoint as soon as it is preserved by precomposition with $X$ (see for instance [@maclane Theorem X.7.2] or [@maysig:pht 16.4.12]). In our case when $X = (u\times\id)_!\lI_A$, precomposition with $X$ is just left Kan extension along $u$, which by our assumption of $u\op$-stability preserves the right Kan extension $(\id\times u\op)_\ast$. Thus, $(u\times\id)_!\lI_A \otimes_{[A]} (\id\times u\op)_\ast \lI_A \cong (\id\times u\op)_\ast \big((u\times\id)_!\lI_A\big)$, so it has an analogous universal property, as desired.
Now, if \[item:sd1\] holds, then since weighted colimits are contravariantly functorial on profunctors, the adjunction $(u\times\id)_!\lI_A \adj Z$ yields an adjunction $\colim^Z \adj \colim^{(u\times\id)_!\lI_A} = u_!$. This gives \[item:sd2\]. Conversely, if $Z\in{\sV}(A\times B\op)$ is such that $\colim^Z \adj u_! = \colim^{(u\times\id)_!\lI_A}$, then since composition in $\cProf({\sV})$ is a special case of weighted colimits, we have natural adjunctions $(Z\otimes_{[B]} -) \adj ((u\times\id)_!\lI_A \otimes_{[A]} -)$, which by the bicategorical Yoneda lemma induce an adjunction $(u\times\id)_!\lI_A\adj Z$ in $\cProf({\sV})$.
Similarly, \[item:sd2op\] is equivalent to \[item:sd1op\], since $\colim^{(\id\times u)_!\lI_{A\op}} \cong (u\op)^\ast$. Finally, if \[item:sd2op\] holds then $(u\op)_\ast$, being a weighted colimit, commutes with all left Kan extensions, so that ${\sV}$ is right $u\op$-stable.
Note that \[thm:stable-dual\]\[item:sd2op\] is a generalization of \[thm:wcolim\]\[item:wcl3\] and \[item:wcl4\]. This can be regarded as an explanation of “why” $\Phi$-limits in a right $\Phi$-stable derivator commute with all colimits: they are themselves weighted colimits. (If is not symmetric, then arbitrary weighted colimits need not commute with arbitrary other *weighted* colimits. However, left Kan extensions always commute with all weighted colimits, by \[lem:wcolim\]\[item:wc1\]. If we express left Kan extensions as weighted colimits themselves, then they are in the “center
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therefore the group of equivariant automorphisms of $P_{\mathcal{E}}$ living over the identity on $M$. This is known as the gauge group, ${\mathcal{G}}(P_{\mathcal{E}})$.
From this viewpoint it becomes clear that there is a map of fibrations $ P_{\mathcal{E}}^{Ad} \to {\mathcal{G}}L_n({\mathcal{E}})$ over $M$ (after taking appropriate fibrant and cofibrant replacements), which is an equivalence on the fibers. Therefore there is an equivalence of their spaces of sections, as group-like $A_\infty$-spaces. $$\label{autgl}
\phi: {\mathcal{G}}(P_{\mathcal{E}}) = \Gamma_M( P_{\mathcal{E}}^{Ad}) \simeq \Gamma_M({\mathcal{G}}L_1({\mathcal{E}})) = hAut^R_X({\mathcal{E}}).$$ Now a well known theorem of Atiyah and Bott [@atiyahbott] says that the classifying space of the gauge group ${\mathcal{G}}(P)$ of a principal bundle $G \to P \to M$ is equivalent to the mapping space, $B{\mathcal{G}}(P) \simeq Map_P(X, BG)$. Theorem \[main\] now follows by applying this Atiyah-Bott equivalence to the principal bundle $GL_n(R) \to P_{\mathcal{E}}\to X.$
We now consider an application of Theorem \[main\] to an important special case. Let ${\mathcal{L}}\to X$ be an $R$-line bundle, and let $\oplus_n {\mathcal{L}}\to X$ be the Whitney-sum of $n$-copies of ${\mathcal{L}}$. This is an $R$-module bundle of rank $n$, which is defined to be the pullback under the diagonal map $\Delta^n : X \to X^n$ of the exterior $n$-fold product ${\mathcal{L}}^n \to X^n$. Consider the endomorphism spectrum $End^R({\mathcal{L}})$. As noted above, this is a ring spectrum. We first need the following observation:
\[GLN\] There is an equivalence of group-like monoids $$\lambda : {hAut^R}(\oplus_n {\mathcal{L}}) \simeq GL_n(End^R({\mathcal{L}})).$$
Given a ring spectum $S$, recall that $GL_n(S)$ is defined to be the group-like monoid of units in the endomorphism ring, $$GL_n(S) = GL_1(End^S(\vee_n S)).$$ Notice that there is a natural equivalence $End^S(\vee_n S) = \prod_n (\vee_n S)$. Also notice that $\vee_n End^R({\mathcal{L}}) \simeq Mor^R({\mat
| 3,330
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m{OLS}}} \|\hat\beta_{{\widehat{S}}} - \beta_{{\widehat{S}}} \| \leq C
B_n,$$ with probability at last $1 - \frac{2}{n}$.
[**Remarks.**]{}
1. It is worth recalling that, in the result above as well as in all the result of the paper, the probability is with respect to joint distribution of the entire sample and of the splitting process.
2. For simplicity, we have phrased the bound in in an asymptotic manner. The result can be trivially turned into a finite sample statement by appropriately adjusting the value of the constant $C$ depending on how rapidly $\max\{ B_n, u B_n \} \rightarrow 0$ vanishes.
3. The proof of the above theorem relies namely an inequality for matrix norms and the vector and matrix Bernstein concentration inequalities (see below).
4. Theorems \[thm:beta.accuracy\] and \[thm:beta.accuracy2\] can be easily generalized to cover the case in which the model selection and the computation of the projection parameters are performed on the entire dataset and not on separate, independent splits. In this situation, it is necessary to obtain a high probability bound for the quantity $$\max_{S} \| \beta_S - \hat{\beta}_S \|$$ where the maximum is over all non-empty subsets of $\{1,\ldots,d\}$ of size at most $k$ and $\hat{\beta}_S =
\hat{\Sigma}_{S}^{-1}\hat{\alpha}_{S}$ (see Equation \[eq:alpha.beta.hat\]). Since there are less than $ \left( \frac{e
d}{k} \right)^k $ such subsets, an additional union bound argument in each application of the matrix and vector Bernstein’s inequalities (see Lemma \[lem:operator\]) within the proofs of both Theorem \[thm:beta.accuracy\] and \[thm:beta.accuracy2\] will give the desired result. The rates so obtained will be then worse than the ones from Theorems \[thm:beta.accuracy\] and \[thm:beta.accuracy2\] which, because of the sample splitting do not require a union bound. In particular, the scaling of $k$ with respect to $n$ will be worse by a factor of $k \log \frac{d}{k}$. This immediately gives a rate of consistency for the projection parameter
| 3,331
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from HPLC analysis of periorbital skin (*larger right panel*) and tropical mix seeds (*smaller left panel*). *Arrows* indicate lutein esters *1*, *2* and *3*
Effect of hormone treatment on plasma concentrations of steroids and cholesterol {#Sec14}
--------------------------------------------------------------------------------
DHT treatment, irrespectively of sex, influenced DHT plasma concentrations (Table [2](#Tab2){ref-type="table"}a). Post hoc tests revealed, as expected, that DHT was elevated not only in the DHT-treated birds but also in the T-treated birds, since T can be converted to DHT and this was the case in both sexes (Fig. [2](#Fig2){ref-type="fig"}a).Table 2Results of statistical tests explaining variation of steroid concentrations in peripheral blood plasma with hormone treatment and sex as predictorsVariables and factorsOriginal modelFinal model*dfFpdfFp*(a) Dihydrotestosterone Sex (S)12.480.12 Treatment (Tr)322.13\<0.0001322.31\<0.0001 S × Tr31.140.34 Error3943(b) Testosterone Sex14.740.0414.670.04 Treatment316.87\<0.0001318.01\<0.0001 S × Tr31.180.33 Error3942(c) Estradiol Sex10.460.50 Treatment36.75\<0.00136.37\<0.001 S × Tr31.080.37 Error3943(a) DHT, (b) T, (c) E2Fig. 2Log steroids concentration (averages and SE in pg mL^−1^) measured on day 20 for 47 birds belonging to 4 steroid treatments (*C*, *E*, *T* and *D*). *Letters* refer to significant differences between hormone treatment groups (*p* \< 0.05) (*filled circle* female, *empty circle* male)
Again as expected, T increased only in the T group, both in males and in females, with males showing higher levels than females (Fig. [2](#Fig2){ref-type="fig"}b). Indeed, there were significant effects of hormone treatment and sex on T levels. Although the figure shows that this sex difference is lacking in the T-treatment group there was no significant interaction effect between sex and treatment (Table [2](#Tab2){ref-type="table"}b).
Treatment also significantly affected E2 levels irrespectively of sex (Table
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i(\lambda([c]), \lambda([b])) \neq 0$ by superinjectivity. Since $i([c], [e]) = 0$ for all $e \in P \setminus \{a, b\}$, we have $i(\lambda([c]), \lambda([e])) = 0$ for all $e \in P \setminus \{a, b\}$. But this is not possible because $\lambda([c])$ has to intersect geometrically essentially with some isotopy class other than $\lambda([a])$ and $\lambda([b])$ in $\lambda([P])$ to be able to make essential intersections with $\lambda([a])$ and $\lambda([b])$ since $\lambda([P])$ is a top dimensional maximal simplex. This gives a contradiction to the assumption that $\lambda([a])$ and $\lambda([b])$ do not have adjacent representatives.
\[nonadjacent\] Suppose that $g + n \geq 4$. Let $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ be a superinjective simplicial map. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is not adjacent to $b$ w.r.t. $P$. There exists $a' \in \lambda([a])$ and $b' \in \lambda([b])$ such that $a'$ is not adjacent to $b'$ w.r.t. $P'$ where $P'$ is a set of pairwise disjoint elements of $\lambda([P])$ containing $a', b'$.
Suppose that $g + n \geq 4$. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is not adjacent to $b$ w.r.t. $P$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$. We can find simple closed curves $c$ and $d$ on $N$ such that $c$ intersects only $a$ nontrivially and is disjoint from all the other curves in $P$, $d$ intersects only $b$ nontrivially and is disjoint from all the other curves in $P$, and $c$ and $d$ are disjoint. Since $\lambda$ is injective by Lemma \[inj\], $\lambda$ sends top dimensional maximal simplices to top dimensional maximal simplices. We have $i(\lambda([a]), \lambda([c]) \neq 0$, $i(\lambda([a]), \lambda([x]) = 0$ for all $x \in P \setminus \{a\}$, $i(\lambda([b]), \lambda([d]) \neq 0$, $i(\lambda([d]), \lambda([x
| 3,333
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====================
As we now have a canonical variational principle for fluid dynamics *via* the inverse map, one may obtain its multisymplectic formulation by extending the phase space so that the Lagrangian is affine in the space and time derivatives. In this section we show how to do this for EPDiff($H^1$) as discussed in the previous section.
Affine Lagrangian for EPDiff($H^1$)
-----------------------------------
After introducing the inverse map constraint, the Lagrangian becomes $$L = \frac{1}{2}u_iu_i +
\frac{\lambda^2}{2}u_{i,j}u_{i,j} +
\pi_k\left(l_{k,t} + u_jl_{k,j}\right).$$ Any high-order derivatives and nonlinear functions of first-order derivatives must now be removed from the Lagrangian to make it affine. We introduce a tensor variable $$W_{ij} = u_{i,j}\,;$$ this relationship may be enforced by using Lagrange multipliers. However, it turns out that the multipliers can be eliminated and the Lagrangian becomes $$\label{epmslag}
L = \frac{1}{2}u_iu_i - \frac{\lambda^2}{2}
W_{ij}W_{ij} +\lambda^2W_{ij}u_{i,j}+ \pi_k\left(l_{k,t} +
u_jl_{k,j}\right),$$ which is now affine in the space and time derivatives of $\MM{u}$, $W$, $\MM{l}$ and $\MM{\pi}$.
Multisymplectic structure
-------------------------
The Euler-Lagrange equations for the affine Lagrangian (\[epmslag\]) are $$\begin{aligned}
\delta u_i:&& u_i - \lambda^2W_{ij,j}+ \pi_k l_{k,i} = 0,
\\
\delta l_k:&&
-\pi_{k,t} - (\pi_k u_j)_{,j} = 0.
\\
\delta \pi_k:&&
l_{k,t} + u_j l_{k,j} = 0,
\\
\delta W_{ij}:&&
-\lambda^2 W_{ij} + \lambda^2 u_{i,j}
= 0 .\end{aligned}$$ These equations possess the following multisymplectic structure as in equation (\[Eul-Lag-eqns\]): $$\begin{pmatrix}
0 & \pi_k\partial_i & & -\lambda^2\partial_j \\
-\pi_k\partial_i & 0 & -\partial_t-u_j\partial_j & 0 \\
0 & \partial_t+u_j\partial_j & 0 & 0 \\
\lambda^2\partial_j & 0 & 0 & 0 \\
\end{pmatrix}
\begin{pmatrix}
u_i \\
l_k \\
\pi_k \\
W_{ij}\\
\end{pmatrix}
= \nabla H,$$ where $\partial_t=\partial/\partial t,\ \partial_i=\partial/\partial x_i$, and $$H = -\left(\
| 3,334
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{ for any } 1\leq k\leq 5 , \mbox{ there exists a semi-infinite path } \gamma_{k} \\
\mbox{ included in the cone } C_{2k\pi/5,\varepsilon,r_{k}} \mbox{ and starting from } \\
\mbox{ a vertex } X_{k} \mbox{ satisfying } r_{k}<|X_{k}|\leq r_{k}+1
\end{array} \right\} ~,$$ we get that for all $\varepsilon>0$, there exist some (deterministic) radii $r_{1},\ldots,r_{5}\in{{\mathbb N}}^*$ such that $A_{\varepsilon}(r_{1},\ldots,r_{5})$ occurs with positive probability.\
Let $R=\max\{r_{k}+1 ; 1\leq k\leq 5\}$ and $V_{\varepsilon}(r_{1},\ldots,r_{5})$ be the complementary set of the five cones in the ball $B(O,R)$: $$V_{\varepsilon}(r_{1},\ldots,r_{5}) = B(O,R) \setminus \Big[\Big( \cup_{k=1}^{5} C_{2k\pi/5,\varepsilon,r_{k}} \Big) \cup \{O\} \Big] ~.$$ Now, we are going to change the configuration of the PPP $N$ in $V_{\varepsilon}(r_{1},\ldots,r_{5})$ in such a way that the $X_k$’s are all of different colors. Let $\widetilde{N}=N\cap V_{\varepsilon}^{c}(r_{1},\ldots,r_{5})$ be the thinned PPP obtained by deleting all the points of $N$ belonging to $V_{\varepsilon}(r_{1},\ldots,r_{5})$ (Jacod and Shiryaev [@jacod], II.4.b). It is crucial to remark that deleting the points of $V_{\varepsilon}(r_{1},\ldots,r_{5})$ does not affect the occurrence of $A_{\varepsilon}(r_{1},\ldots,r_{5})$. In other words, if $N$ satisfies the event $A_{\varepsilon}(r_{1},\ldots,r_{5})$, so does $\widetilde{N}$; $${{\mathbb P}}\left( \widetilde{N} \in A_{\varepsilon}(r_{1},\ldots,r_{5}) \right) \geq {{\mathbb P}}\left( N \in A_{\varepsilon}(r_{1},\ldots,r_{5}) \right) > 0 ~.$$ Now, let us consider a PPP $\hat{N}$ on $V_{\varepsilon}(r_{1},\ldots,r_{5})$ with intensity $1$. Let us denote by $r$ the minimum of the $r_k$’s.\
The event $\hat{N}\in B_{\varepsilon}(r_{1},\ldots,r_{5})$ is defined by the three following conditions.
- For any $k\in\{1,\dots,5\}$, if $r_{k}>r$ then for all integers $r \leq n \leq r_{k}-1$, $$\hat{N} \left( B(n e^{\i 2k\pi/5} , \varepsilon) \right) = 1 ~,$$ else $$\hat{N} \left( B(r e^{\i 2k\pi/5} , \varepsilon
| 3,335
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O(1)$ the above quantity $\gamma$ can be made arbitrarily close to one, for large enough problem size $d$. On the other hand, when $p_{j,\ell_j}$ is close to $\kappa_j$, the accuracy can degrade significantly as stronger alternatives might have small chance of showing up in the rank breaking. The value of $\gamma$ is quite sensitive to $b$. The reason we have such a inferior dependence on $b$ is because we wanted to give a universal bound on the Hessian that is simple. It is not difficult to get a tighter bound with a larger value of $\gamma$, but will inevitably depend on the structure of the data in a complicated fashion. To ensure that the (second) largest eigenvalue of the Hessian is small enough, we need enough samples. This is captured by $\eta$ defined as $$\begin{aligned}
\label{eq:eta_def}
\eta \;\;\equiv\;\; \max_{j \in [n]} \{\eta_j\} \;,\; \;\;\;\;\;\text{where} \;\; \;\;\;\;\; \eta_j \;\; = \;\; \frac{\kappa_j}{\max\{\ell_j, \kappa_j - p_{j,\ell_j}\}}\,.\end{aligned}$$ Note that $1 < \eta_j \leq \kappa_j/\ell_j$. A smaller value of $\eta$ is desired as we require smaller number of samples, as shown in Theorem \[thm:main2\]. This happens, for instance, when all separators are at the top, such that $p_{j,\ell_j}=\ell_j$ and $\eta_j=\kappa_j/(\kappa_j-\ell_j)$, which is close to one for large $\kappa_j$. On the other hand, when all separators are at the bottom of the list, then $\eta$ can be as large as $\kappa_j$.
We discuss the role of the topology of data captures by these parameters in Section \[sec:role\].
Main Results {#sec:main}
============
We present the main theoretical results accompanied by corresponding numerical simulations in this section.
Upper Bound on the Achievable Error
-----------------------------------
We present the main result that provides an upper bound on the resulting error and explicitly shows the dependence on the topology of the data. As explained in Section \[sec:intro\], we assume that each user provides a partial ranking according to his/her position of the sepa
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x_kx_mx'_l=\sum_{\substack{{k,l,m}\\m\neq k}}x_kx_lx'_m
=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,m}\\m\neq k}}x_kx_mx'_m
+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,m}\\k<m}}x_kx_mx'_m
+\sum_{\substack{{k,m}\\m<k}}x_kx_mx'_m
+2\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l}\\k<l}}x_kx_lx'_l
+\sum_{\substack{{k,l}\\k<l}}x_kx_lx'_k
+2\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\k<l}}x_kx_lx'_m
+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m,\end{aligned}$$ therefore it suffices to prove $$\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l\leq\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m.$$ This is trivial if $x'_m=0$ for all $m$. Otherwise $\sum_m x'_m>0$, hence $c_i\geq c_j$ yields $$\lambda:=\biggl(\sum_m x_m\biggr)\biggl(\sum_m x'_m\biggr)^{-1}\geq 1.$$ Clearly, we are done if we can prove $$\lambda^2\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l\leq\lambda\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\lambda\sum_{\substack{{k,l,m}\\k<l\\m\neq
k,l}}x_kx_lx'_m.$$ We introduce $\tilde x_m:=\lambda x'_m$, then $$\sum_m\tilde x_m=\sum_m x_m,$$ and the last inequality reads $$\sum_{\substack{{k,l,m}\\k<l}}x_m\tilde x_k\tilde x_l\leq
\sum_{\substack{{k,m}\\m\neq k}}x_k^2\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_l\tilde x_m.$$ By adding equal sums to both sides this becomes $$\sum_{\substack{{k,l,m}\\k<l}}x_m\tilde x_k\tilde
x_l+\sum_{\substack{{k,l,m}\\k<l}}x_kx_l\tilde x_m\leq
\sum_{\substack{{k,m}\\m\neq k}}x_k^2\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_l\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l}}x_kx_l\tilde x_m,$$ which can also be written as $$\biggl(\sum_m x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}\tilde
| 3,337
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|
hi,\Psi],{\mathcal{S}}\Phi]
\nonumber\\
&
+\frac{1}{2}\{\Psi,\Xi\{\eta\Phi,\Xi {\mathcal{S}}\Phi\}\}
+\frac{1}{2}\{\Psi,\Xi[\Phi,\Xi {\mathcal{S}}\eta\Phi]\}
-[\Xi[\Phi,\Psi],\Xi {\mathcal{S}}\eta\Phi]
\nonumber\\
&
-\frac{1}{2}{\mathcal{S}}\Xi[\Phi,\Xi\{\eta\Phi,\Psi\}]
-\frac{1}{2}{\mathcal{S}}\Xi[\eta\Phi,\Xi[\Phi,\Psi]],\\
\delta^{(2)}_{\mathcal{S}}\Psi\ =&\ \frac{1}{6}X\eta[\Phi,[\Phi,{\mathcal{S}}\Phi]]
+\frac{1}{2}X\eta[\Phi,\Xi[{\mathcal{S}}\Phi,\eta\Phi]]
+\frac{1}{2}X\eta\{\eta\Phi,\Xi[\Phi,\Xi {\mathcal{S}}\eta\Phi]\}
\nonumber\\
&
+\frac{1}{2}X\eta[\Phi,\Xi[\eta\Phi,\Xi {\mathcal{S}}\eta\Phi]]\,,\end{aligned}$$ to cancel them by $\delta_{\mathcal{S}}^{(2)}S^{(0)}$: $\delta_{\mathcal{S}}^{(2)}S^{(0)}+\delta_{\mathcal{S}}^{(1)}S^{(1)}+\delta_{\mathcal{S}}^{(0)}S^{(2)}=0$. The procedure is not terminated, so we suppose a full transformation consistent with these results, and then show that it is in fact a symmetry of the complete action.
Complete space-time supersymmetry transformation
------------------------------------------------
Here we suppose that the complete transformation is given by
\[complete transformation\] $$\begin{aligned}
A_{\delta_{\mathcal{S}}}\ =&\ e^\Phi({\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi))e^{-\Phi}
+ \{F\Psi,F\Xi A_{\mathcal{S}}\},
\label{complete tf ns}\\
\delta_{\mathcal{S}}\Psi\ =&\ X\eta F\Xi D_\eta A_{\mathcal{S}}\ =\ X\eta F\Xi {\mathcal{S}}A_\eta\,,
\label{complete tf r}\end{aligned}$$
and show that the complete action (\[complete action\]) is invariant under this transformation. From the formula of the general variation of the action (\[general variation\]), we have $$\begin{aligned}
\delta_{\mathcal{S}}S\ =&\
-\langle e^\Phi({\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi))e^{-\Phi} ,QA_\eta+(F\Psi)^2\rangle
-\langle \{F\Psi,F\Xi A_{\mathcal{S}}\}, QA_\eta+(F\Psi)^2\rangle
\nonumber\\
&\
- {\langle\!\langle}X\eta F\Xi D_\eta A_{\mathcal{S}}\,,Y(Q\Psi+X\eta F\Psi){\rangle\!\rangle}\,.
\label{var S}\end{aligned}$$ We calculate each of these three terms, which we denote
| 3,338
| 700
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| 3,520
| null | null |
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|
6.0 (6.0--11.9) 142.8 (113.0--142.8)
**Reference interval partitioned based on season**
Blood urea nitrogen mmol/L
Fall sampling period 78 33.7 (11.8--62.8) 13.1 (11.0--15.7) 61.3 (56.5--65.9)
Summer sampling period 113 23.6 (6.1--67.8) 7.2 (6.1--8.6) 63.9 (56.8--71.0)
^a^Sample sizes varied among analytes for reference interval estimation due to missing data for a few hematological analytes.
10.1371/journal.pone.0115739.t002
###### Hematology and plasma biochemical parameters that exhibited significant variation between the fall (October and November) and summer (May through September) sampling periods in juvenile loggerhead sea turtles (*Caretta caretta*) sampled in Core Sound, North Carolina, USA.
{#pone.0115739.t002g}
Parameter Fall Median Summer Median P-value
---------------------------------------------- ------------- --------------- -----------
Estimated white blood cell count (X 10^9^/L) 10 9 \< 0.0001
Heterophils (X 10^9^/L) 6.4 4.1 \< 0.0001
Monocytes (X 10^9^/L) 2.8 1.0 0.0001
Total protein (g/L) 37 33 0.0020
Globulin (g/L) 26 22 0.0040
Blood urea nitrogen (mmol/L) 34 24 0.0001
Packed cell volume (%) 29 31 0.0003
Chloride (mmol/L)
| 3,339
| 6,267
| 1,240
| 1,978
| null | null |
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|
l objective of chaotic dynamics is to generate a topology in $X$ with respect to which elements of the set can be grouped together in as large equivalence classes as possible in the sense that if a net converges simultaneously to points $x\neq y\in X$ then $x\sim y$: $x$ is of course equivalent to itself while $x,y,z$ are equivalent to each other iff they are simultaneously in every open set in which the net may eventually belong. This hall-mark of chaos can be appreciated in terms of a necessary obliteration of any separation property that the space might have originally possessed, see property (H3) in Appendix A3. We reemphasize that a set in this chaotic context is required to act in a dual capacity depending on whether it carries the initial or final topology under $\mathcal{M}$.
This preliminary inquiry into the nature of chaos is concluded in the final section of this work.
**5. Graphical convergence works**
We present in this section some real evidence in support of our hypothesis of graphical convergence of functions in $\textrm{Multi}(X,Y)$. The example is taken from neutron transport theory, and concerns the discretized spectral approximation [@Sengupta1988; @Sengupta1995] of Case’s singular eigenfunction solution of the monoenergetic neutron transport equation, [@Case1967]. The neutron transport equation is a linear form of the Boltzmann equation that is obtained as follows. Consider the neutron-moderator system as a mixture of two species of gases each of which satisfies a Boltzmann equation of the type$$\begin{gathered}
\left(\frac{\partial}{\partial t}+v_{i}.\nabla\right)f_{i}(r,v,t)=\\
=\int dv^{\prime}\int dv_{1}\int dv_{1}^{\prime}\sum_{j}W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})\{ f_{i}(r,v^{\prime},t)f_{j}(r,v_{1}^{\prime},t)--f_{i}(r,v,t)f_{j}(r,v_{1},t)\})\end{gathered}$$
where$$W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})=\mid v-v_{1}\mid\sigma_{ij}(v-v^{\prime},v_{1}-v_{1}^{\prime})$$
$\sigma_{ij}$ being the cross-section of interactio
| 3,340
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|
6--400), 3, 0 0-150 0.065
*Albumin (g/L)* 31 (27--36), 1, 0 29 (26--34), 0, 0 23-35 \<0.001
*Cholesterol (mmol/L)* 5.6 (2.5-9.1), 6, 1 5.0 (1.9-7.7), 4, 2 3.5-7.0 0.040
*Creatinine (μmol/L)* 84 (37--123), 1, 0 80 (30--122), 1, 0 20-110 0.033
*Globulins (g/L)* 31 (21--48), 1, 0 27 (22--41), 0, 0 22-40 0.012
*Glucose (mmol/L)* 5.3 (3.5-8.7), 8, 0 5.2 (3.0-7.4), 6, 0 3.5-5.5 0.210
*Urea (mmol/L)* 4.6 (1.6-8.0), 2, 3 5.6 (3.5-8.9), 3, 0 3.5-7.0 0.047
*Triglycerides (mmol/L)* 1.2 (0.6-5.3), 8, 0 0.9 (0.6-2.1), 3, 0 0.6-1.5 0.015
Data are expressed as median (range), number above reference range, and number below reference range. P values quoted for Wilcoxon signed ranks test.
Changes in plasma nutrient concentrations with weight loss
----------------------------------------------------------
Plasma nutrient concentrations are shown in Table [4](#T4){ref-type="table"}. For the 26 dogs with initial and post sample analysis of choline, betaine, and amino acids, the median time between the two samples (weight loss program duration) was 242 days, with a range of 91--674 days. Plasma amino acid concentrations had wide ranges and no systematic change was noted across all nutrients (i.e. some increased while other decreased with weight loss). Additionally, median concentrations of the amino acid were not outside the reference range of the laboratory (determined by two standard deviations from their median score in 131 healthy animals), with the exception of aspartic acid and glutamic acid, which were more than the reference range before and after weight loss.
######
Median (and range) of selected nutrients in dogs pre and post weight loss
***Nutrient***
| 3,341
| 2,572
| 3,453
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|
[|a |b]{}I \[AdjAdj=I\] \_[|b |a]{} \^[a |a]{} \^[b |b]{} = \^[a b]{}I where $I$ is the identity at least as acting upon the current algebra. One can argue more generically that these bilinears are proportional to the unit operator by using the definition of the primary adjoint in terms of the supertrace, and using completeness of the Lie algebra generators. Remember also that the left and right conformal dimensions of the adjoint operator $\mathcal{A}^{a \bar a}$ vanish since they are proportional to the dual Coxeter number of the Lie superalgebra.
The action of the zero modes of the currents generates the group transformations. Since the structure constants are the generators of the Lie superalgebra in the adjoint representation, the OPE between a current and the primary adjoint operator reads : $$\begin{aligned}
\label{jAdj} & j^a_{L,z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_+}{c_++c_-} \frac{i{f^{ab}}_c \mathcal{A}^{c \bar b}}{z-w} + ... \cr
%
& j^a_{L,\bar z}(z) \mathcal{A}^{b \bar b}(w)
= \frac{c_-}{c_++c_-} \frac{i{f^{ab}}_c \mathcal{A}^{c \bar b}}{\bar z- \bar w} +... \cr
%
& j^{\bar a}_{R,z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_-}{c_++c_-} \frac{i{f^{\bar a \bar b}}_{\bar c} \mathcal{A}^{b \bar c}}{z-w} +... \cr
%
& j^{\bar a}_{R,\bar z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_+}{c_++c_-} \frac{i{f^{\bar a \bar b}}_{\bar c} \mathcal{A}^{b \bar c}}{\bar z- \bar w} +... \end{aligned}$$ In section \[primaries\] the concept of primary field will be defined precisely. The coefficients appearing in the previous OPE will be explained, and we will compute the first subleading terms (see equation ).
Moreover, we propose that the following equations hold in the model under consideration: \[dAdj\] \^[a |a]{} = -:j\^c\_[L,z]{} \^[b |a]{}: = -:j\^[|c]{}\_[R,z]{} \^[a |b]{}: \[dbarAdj\] |\^[a |a]{} = -:j\^c\_[L,|z]{} \^[b |a]{}:= -:j\^[|c]{}\_[R,|z]{} \^[a |b]{}:. One argument for the previous equations is the following. We start with the definition of the adjoint operator in terms of the group element , and compu
| 3,342
| 763
| 1,764
| 3,374
| null | null |
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|
the same purpose). We can use $C_1={\mathrm}{Fix}(C_0)$ to define a partition $\dot{f}$ of $\omega_1$ so that for each $\xi\in\omega_1$ and each $s\in S_{\xi^+(C_1)}$, $s^\smallfrown j$ forces that $\dot{f}(\xi)=j$. Now we choose two (names of) functions $\dot{h}_1$ and $\dot{h}_2$ witnessing normality as follows:
- For each $j\in\omega$ and each $i\in 2$, let $\dot{W}^i_j=\bigcup\{\dot{U}(\xi,h_i(\xi)):\xi\in \dot{f}^{-1}(j)\}$,
- the $\{\dot{W}^1_j:j\in\omega\}$ form a discrete family,
- the closure of $\dot{W}^2_j$ is included in $\dot{W}^1_j$.
Choose any countable elementary submodel $M$ with all the above as members of $M$, such that $\delta = M\cap\omega_1$ is an element of $C_1$. We know that there is a name of an integer $\dot{J}_\delta$ satisfying that it is forced that $\dot{U}(\delta,0)\cap\dot{W}_j$ is empty for all $j\geq\dot{J}_\delta$. Choose any $s\in S$ of height at least $\delta^+(C_1)$ that decides a value $J$ for $\dot{J}_\delta$. Let $\bar{s} = s\upharpoonright\delta^+(C_1)$. Notice that $\bar{s}$ decides the truth value of the equation “$\dot{U}(\delta,0)\cap\dot{B}_\alpha=\emptyset$”, for all $\alpha\in M$. For each $n,j\in\omega$, $s$ and hence $s\upharpoonright\delta$ forces that the closure of $\dot{W}^2_j\cap \dot{B}_n$ is included in $\dot{W}^1_j$. By elementarity and compactness, this implies there is a finite $\dot{F}_{j,n}\subseteq\delta$ such that $s\upharpoonright\delta$ forces that $\dot{W}^2_j\cap \dot{B}_n\subseteq\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\}\subseteq\dot{W}^1_j$. But now $\bar{s}$ forces $\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\})$ is empty for all $n$ and all $j\geq J$.
On the other hand, fix any $j\geq J$ and consider what $\bar{s}^\smallfrown j$ is forcing. This forces that $\dot{f}(\delta)=j$ and that $\delta\in W^2_j$, and so $\delta$ is in the closure of the union of the sequence $\{\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in
F_{j,n}\}):n\in\omega\}$. This is a contradiction.
\[cor410\] In any mod
| 3,343
| 2,555
| 2,738
| 2,912
| null | null |
github_plus_top10pct_by_avg
|
in\mathbb{N}}$ that is not the constant sequence $(x_{0})$ at a fixed point? As $i\in\mathbb{N}$ increases, points are added to $\{ x_{0},f(x_{0}),\cdots,f^{I}(x_{0})\}$ not, as would be the case in a normal sequence, as a piled-up Cauchy tail, but as points generally lying between those already present; recall a typical graph as of Fig. \[Fig: tent4\] for example.]{}
[^26]: \[Foot: gen\_eigen\][The technical definition of a generalized eigenvalue is as follows. Let $\mathcal{L}$ be a linear operator such that there exists in the domain of $\mathcal{L}$ a sequence of elements $(x_{n})$ with $\Vert x_{n}\Vert=1$ for all $n$. If $\lim_{n\rightarrow\infty}\Vert(\mathcal{L}-\lambda)x_{n}\Vert=0$ for some $\lambda\in\mathbb{C}$, then this $\lambda$ is a]{} *generalized eigenvalue* [of $\mathcal{L}$, the corresponding eigenfunction $x_{\infty}$ being a]{} *generalized eigenfunction.*
[^27]: \[Foot: cluster\]This is also known as a *cluster point*; we shall, however, use this new term exclusively in the sense of the elements of a derived set, see Definition 2.3.
[^28]: \[Foot: Filter\_conv\][The restatement $$\mathcal{F}\rightarrow x\Longleftrightarrow\mathcal{N}_{x}\subseteq\mathcal{F}\label{Eqn: Def: LimFilter}$$ of Eq. (\[Eqn: lim filter\]) that follows from (F3), and sometimes taken as the definition of convergence of a filter, is significant as it ties up the algebraic filter with the topological neighbourhood system to produce the filter theory of convergence in topological spaces. From the defining properties of $\mathcal{F}$ it follows that for each $x\in X$, $\mathcal{N}_{x}$ is the coarsest (that is smallest) filter on $X$ that converges to $x$.]{}
[^29]: \[Foot: adh\_seq\][In a first countable space, while the corresponding proof of the first part of the theorem for sequences is essentially the same as in the present case, the more direct proof of the converse illustrates how the convenience of nets and directed sets may require more general arguments. Thus if a sequence $(x_{i})_{i\in\mathbb{N}}$ has a s
| 3,344
| 4,186
| 3,845
| 3,096
| 2,314
| 0.780919
|
github_plus_top10pct_by_avg
|
s rank-breaking estimators. This provides guidelines for choosing the weights in the estimator to achieve optimal performance, and also explicitly shows how the accuracy depends on the topology of the data.
This paper provides the first analytical result in the sample complexity of rank-breaking estimators, and quantifies the price we pay in accuracy for the computational gain. In general, more complex higher-order rank-breaking can also be considered, where instead of breaking a partial ordering into a collection of paired comparisons, we break it into a collection of higher-order comparisons. The resulting higher-order rank-breakings will enable us to traverse the whole spectrum of computational complexity between the pairwise rank-breaking and the MLE. We believe this paper opens an interesting new direction towards understanding the whole spectrum of such approaches. However, analyzing the Hessian of the corresponding objective function is significantly more involved and requires new technical innovations.
Proofs
======
Proof of Theorem \[thm:main2\] {#sec:proof_main2}
------------------------------
We prove a more general result for an arbitrary choice of the parameter $\lambda_{j,a}>0$ for all $j\in[n]$ and $a\in[\ell_j]$. The following theorem proves the (near)-optimality of the choice of $\lambda_{j,a}$’s proposed in , and implies the corresponding error bound as a corollary.
\[thm:main\] Under the hypotheses of Theorem \[thm:main2\] and any $\lambda_{j,a}$’s, the rank-breaking estimator achieves $$\begin{aligned}
\label{eq:main1}
\frac{1}{\sqrt{d}} \big\|\,\widehat{\theta} - \theta^* \,\big\|_2 \; \,\leq\, \; \frac{4\sqrt{2}e^{4b}(1+ e^{2b})^2 \sqrt{d \log d} }{\alpha\, \gamma} \frac{\sqrt{\sum_{j=1}^n \sum_{a=1}^{\ell_j} \big(\lambda_{j,a}\big)^2 \big(\kappa_j - p_{j,a}\big)\big(\kappa_j- p_{j,a}+1\big)}}{ \sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})}\;,
\end{aligned}$$ with probability at least $ 1- 3e^{3}d^{-3}$, if $$\begin{aligned}
\label{eq:mai
| 3,345
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_x[{g}(X_{\tau_D})^2]<\infty$. However, if we consider the computation in (\[worksforsquared\]) of the Appendix, which shows that $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$ when ${g}$ is continuous and in $L^1_{\alpha}(D^\mathrm{c})$, then it is easy to see that the same statement holds replacing ${g}$ by ${g}^2$. Under finiteness of the second moment, the [central limit theorem]{}completes the proof.
We now return to the proof of Theorem \[main\]. Our approach is to break it into several parts. For convenience, we shall henceforth write $X^{(x)}=(X^{(x)}(t)\colon t\geq 0)$ to indicate the dependency of $X$ on its initial position $X_0 = x$ (equivalent to writing $(X, \mathbb{P}_x)$). For any $x = (x_1, \dots, x_d)\in\mathbb{R}^d$ such that $x_1>0$, we have $V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1>0\}$ for the open half-space containing $x$ and denote its boundary $\partial V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1=0\}$. For any Borel set $A\subset\mathbb{R}^d$, we write $
\sigma_A = \inf\{t>0 \colon X_t\not\in A\}.
$ We will typically use in place of $A$ the set $V(x)$ as well as $B(x,1) = \{z\in\mathbb{R}^d\colon |z- x|<1\}$, the unit ball centred at $x\in\mathbb{R}^d$. Finally write ${\rm\bf i} = (1,0, \dots, 0)\in\mathbb{R}^d$.
\[scaled\] Without loss of generality (by appealing to the spatial homogeneity of $X$ which allows us to appropriately choose our coordinate system) suppose that $x= |x|\,{
\rm\bf i}\in D$ is such that $\partial V(x)$ is a tangent hyperplane to both $D$ and $B_1$. Then $X^{(x)}_{\sigma_{B_1}}$ is equal in distribution to $|x|\, X^{(\rm\bf i)}_{\sigma_{B({\rm\bf i},1)}}$ and $X^{(x)}_{\sigma_{V(x)}}$ is equal in distribution to $|x|\,X^{(\rm\bf i)}_{\sigma_{V(\mathbf{i})
}}$.
The scaling property of $X$ ensures that we can write $$X^{(x)}_{s} = |x|\hat{X}^{(\mathbf{i})}_{|x|^{-\alpha}s}, \qquad s\geq 0,
\label{scaling}$$ where $\hat{X}^{(x)}$ is equal in law to $X^{(x)}$. Note that $$\begin{aligned}
\sigma_{B_1} & = \inf\Bp{t> 0\colon {
| 3,346
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|
ad T_{3 }= \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array} \right) \ ,$$ and the group element $$g= \exp \left[ x_{1} T_{1} + x_{2}T_{2} + (x_{3}-\frac{1}{2} x_{1}x_{2} ) T_{3}\right] = \left( \begin{array}{ccc}
1& x_{1} &x_{3} \\
0 &1 & x_{2}\\
0 & 0 & 1 \\
\end{array} \right) \ .$$ The left-invariant one-forms $g^{-1}dg= L^{i} T_{i}$ are $$L^{1} = dx_{1} \ , \quad L^{2 } = dx_{2} \ , \quad L^{3} = dx_{3} - x_{1} dx_{2} \ .$$
We consider a $\sigma$-model based on this algebra $$\label{eq:Heisenberg}
{\cal L} = E_{ab} L_+^a L_-^b = f_1 L^{1}_+ L^{1}_- + f_2 L^{2}_+ L^{2}_- + \lambda L^{3}_+ L^{3}_- \ ,$$ i.e. $E = \operatorname{diag} (f_1,f_2, \lambda)$, where we allow $f_{1,2}$ to be functions of any spectator coordinates. In the limit $\lambda \rightarrow 0$ the theory develops a gauge invariance (the coordinate $x_3$ drops out of the action all together) and reduces to the $\sigma$-model whose target space is simply $ds^2 = f_1 dx_1^2 + f_2 dx_2^2$. This Rube Goldberg construction allows us to now go head and perform a non-abelian T-duality on the coset following the techniques of [@Lozano:2011kb].
The resulting dual $\sigma$-model is given by $$\mathcal{L}_{dual} = \partial_+ v_a (M^{-1})^{ab} \partial_- v^b$$ in which $$\begin{split}
M_{ab} = E_{ab} + f_{ab}{}^c v_c & = \left( \begin{array}{ccc}
f_1 & v_3 & 0 \\
-v_3 & f_2 & 0 \\
0 & 0 & \lambda \end{array} \right) \ ,
\\
(M^{-1})^{ab} & = \left( \begin{array}{ccc}
h f_2 & - h v_3 & 0 \\
h v_3 & h f_1 & 0 \\
0 & 0 & \frac{1}{\lambda} \end{array} \right)
\ , \quad h= \frac{1}{f_1 f_2 + v_3^2}\ .
\end{split}$$ The matrix $M^{-1}$ diverges in the limit of interest $\lambda \to 0$. In particular, the coefficient of the kinetic term for $v_3$ becomes infinite in the limit and this can be understood as freezing $v_3$ to a constant value. To see this let us rewrite the dual $\sigma$-model as $$\mathcal{L}_{dual} = \partial_+ v_\alpha (M^{-1})^{\alpha\beta} \partial_- v^\beta + \lambda a_+ a_- + a_+ \partial_-v_3 - a_- \partial_+ v_3 \
| 3,347
| 2,028
| 2,846
| 2,895
| 4,057
| 0.768412
|
github_plus_top10pct_by_avg
|
eGa\]
To the best of our knowledge, the expression for $G_{\bar{R}}$ in cannot be simplified for general values of $\Psi$. Thus, the calculation of the average throughput gain for a general number of reconfiguration states involves an infinite integral of a complicated function. We then further consider two special cases, in which the relatively simple expressions for the average throughput gain are tractable.
### $\Psi\le5$
In practice, the number of distinct reconfiguration states is usually small, due to the complicated hardware design and the limited size of antennas. The following Corollary gives the approximated average throughput gain for the case of $\Psi\le5$.
\[Cor:aveGst5\] The approximated expressions for the average throughput gain, $G_{\bar{R}}$, for the case of $\Psi\le5$ are given by $$\begin{aligned}
\label{eq:th_gain_close_125}
&G_{\bar{R}}(\psi=1)\approx1,~
G_{\bar{R}}(\psi=2)\approx1+\frac{1}{{\bar{R}_{\psi}}}\sqrt{\frac{{\sigma^2_{R_\psi}}}{\pi}},\notag\\
&G_{\bar{R}}(\psi=3)\approx1+\frac{3}{2{\bar{R}_{\psi}}}\sqrt{\frac{{\sigma^2_{R_\psi}}}{\pi}},\notag\\
&G_{\bar{R}}(\psi=4)\approx1+\frac{3}{{\bar{R}_{\psi}}}\sqrt{\frac{{\sigma^2_{R_\psi}}}{\pi^3}}\arccos\left(-\frac{1}{3}\right),\notag\\
&G_{\bar{R}}(\psi=5)\approx1+\frac{5}{2{\bar{R}_{\psi}}}\sqrt{\frac{{\sigma^2_{R_\psi}}}{\pi^3}}\arccos\left(-\frac{23}{27}\right).\end{aligned}$$
See Appendix \[App:proofaveGast5\]
### $\Psi$ Is Large
Although $\Psi$ is relatively small in practice, it is of theoretical importance to study the case of large $\Psi$ to capture the limiting performance gain of reconfigurable antennas. We present the approximated average throughput gain for large $\Psi$ in the corollary below.
\[Cor:aveGlsn\] The approximated average throughput gain, $G_{\bar{R}}$, for the case of large $\Psi$ is given by $$\begin{aligned}
\label{eq:GavelargePsi}
&G_{\bar{R}}\approx 1+\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\notag\\ &\left((1-\beta)\mathrm{erf}^{-1}\left(1-\frac{2}{\Psi}\right)
+\beta \mathrm
| 3,348
| 4,413
| 2,781
| 2,697
| null | null |
github_plus_top10pct_by_avg
|
frac{2}{\pi}\sin(\pi\alpha/2)\left(r^2-r_n^2\right)^{-\alpha/2} r_n^{\alpha}\,\frac{{\rm d}r}{r} \times \frac{{\rm d}\theta}{2\pi}\,,\qquad r>r_n.\label{DISPOL}
\end{aligned}$$ From , we see that the angle $\theta$ is sampled uniformly on $[0,2\pi]$ whereas we can sample the radius $r$ via the inverse-transform sampling method. To this end, noting that $\sin(\pi\alpha/2)B(\alpha/2, 1-\alpha/2)=\pi$, the first factor on the right-hand side of is the density of a distribution with cumulative distribution function $F$. The inverse of $F$ can be identified as follows: For $x\in[0,1]$, $$\begin{aligned}
F^{-1}(x)=r_n\left(I^{-1}( 1-x;\alpha/2,1-\alpha/2))\right)^{-1/2}, \end{aligned}$$ where $I^{-1}(x;z,w)$ is the inverse of the incomplete beta function $$I(x;z,w)\coloneqq\frac{1}{B(z,w)}\int_{0}^{x}u^{z-1}(1-u)^{w-1}\,{\rm d}u, \qquad x\in [0,1],$$ and $B(z,w)\coloneqq \int_{0}^{1}u^{z-1}(1-u)^{w-1}\,{\rm d}u$ is the beta function.
The homogeneous part of the solution to is somewhat easier to compute than the inhomogeneous part, which additionally involves numerical computation of the integral $r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot))$ in . To develop this expression, we use the substitution $u=(1-t)/t$ for the integral in and hence, when $d = 2$, for $|y|<1$, $$V_1(0,{\rm d}y)=c_{2,\alpha}B(1-\alpha/2,\alpha/2)|y|^{\alpha-2}(1-I(|y|^2;1-\alpha/2,\alpha/2))$$ with $c_{2,\alpha}=2^{-\alpha}\pi^{-1}\Gamma(\alpha/2)^{-2}$. Moreover, by converting to polar coordinates $(r,\theta)$, the simulated quantity at step $n$ becomes $$\begin{aligned}
& r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot)) \\
& =r_n^{\alpha}c_{2,\alpha}B(1-\alpha/2,\alpha/2) \int_{|y|<1}
{f}(\rho_n+r_ny)|y|^{\alpha-2}\left(1-I(|y|^2;1-\alpha/2,\alpha/2)\right)\,{\rm d}y\\
& =r_n^{\alpha} c_{2,\alpha}B(1-\alpha/2,\alpha/2)2\pi \alph
| 3,349
| 4,820
| 2,240
| 2,613
| null | null |
github_plus_top10pct_by_avg
|
0$ as $u\geq 0$, it follows from the part A above that $w\geq 0$, where $w$ is the solution of the problem . This allows us to conclude that $\phi=w+u\geq 0$, and so $\psi\geq 0$ as desired.
The same argument is valid for the coupled system considered below.
\[re:general\_positivity\_of\_u\] We sketch here a more general proof that, under the standing assumptions of this section, the solution $u$ of the problem is positive, when ${\bf g}\geq 0$.
We allow $S_0(x,E)$ to depend on both $x$ and $E$, but, in order to make the technicalities easier, we replace the assumption by a slightly stronger one, namely $$\begin{aligned}
\label{eq:strong_S_0_ass}
S_0\in C^2(I,C_b(G)),\end{aligned}$$ where $C_b(G)$ is the set of bounded continuous functions on $G$, equipped with the norm ${\left\Vert \cdot\right\Vert}_{L^\infty(G)}$, making it a closed subspace of $L^\infty(G)$. Clearly, assumption implies . In fact, for the argument below to work, it suffices to assume $S_0\in C^1(I,C_b(G))$.
Since ${\frac{\partial S_0}{\partial E}}\in C(I,C_b(G))$, there is a constant $M>0$ such that ${\left\Vert {\frac{\partial S_0}{\partial E}}(E)\right\Vert}_{L^\infty(G)}\leq M$, and hence for every $E,E'\in I$, it holds $${\left\Vert S_0(E')-S_0(E)\right\Vert}_{L^\infty(G)}\leq M|E-E'|.$$
For a given $(x,\omega)\in G\times S$, letting $$P_{x,\omega}:[0,t(x,\omega)]\times I\to{\mathbb{R}};
\quad P_{x,\omega}(t,E):=S_0(x-t\omega,E),$$ we see that $P$ is continuous, and $$|P_{x,\omega}(t,E')-P_{x,\omega}(t,E)|\leq M|E'-E|,$$ i.e. the map $(t,E)\mapsto P_{x,\omega}(t,E)$ satisfies Lipschitz condition on $I$ uniformly with respect to $[0,t(x,\omega)]$.
Therefore, Cauchy-Lipschitz (or Picard-Lindelöf) theorem (cf. [@lang95 Chapter IV, Proposition 1.1]) implies that for every $(x,\omega,E)\in G\times S\times I$ the problem $$\dot{\gamma}(t)={}&S_0(x-t\omega,\gamma(t)),
\quad t\in [0,t(x,\omega)],\\
\gamma(0)={}&E,$$ has a unique solution $\gamma\in C^1([0,\ol{\tau}])$ defined on the maximal interval $[0,\ol{\tau}]$, such that $\gamma(t)\in I
| 3,350
| 2,134
| 2,570
| 2,981
| null | null |
github_plus_top10pct_by_avg
|
frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{I I}
\left\{ W^{\dagger} A W \right\}_{I J}
\nonumber \\
&-&
(ix) e^{- i \Delta_{J} x} \frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{I J}
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&+&
\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} )^2 } \left\{ W^{\dagger} A W \right\}_{I I}
\left\{ W^{\dagger} A W \right\}_{I J}
\nonumber \\
&-&
\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} )^2 } \left\{ W^{\dagger} A W \right\}_{I J}
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&-&
\sum_{K \neq I, J}
\frac{ 1 }{ ( \Delta_{J} - \Delta_{I} ) ( \Delta_{K} - \Delta_{I} ) ( \Delta_{K} - \Delta_{J} ) }
\nonumber \\
&\times& \left[
\Delta_{J} e^{- i \Delta_{I} x} - \Delta_{I} e^{- i \Delta_{J} x}
+ \left( e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} \right) \Delta_{K}
- ( \Delta_{J} - \Delta_{I} ) e^{- i \Delta_{K} x}
\right]
\nonumber \\
&\times&
\left\{ W^{\dagger} A W \right\}_{I K}
\left\{ W^{\dagger} A W \right\}_{K J}.
\label{hatS-IJ(2+4)}\end{aligned}$$ Under generalized T transformation, the second term goes to the third term (both $ix$ terms), and vice versa, and hence they are invariant. The situation is completely the same as in the fourth and the fifth terms. The first and the last terms are invariant in an analogous way as $\hat{S}_{i j} \vert_{i \neq j} [2]$.
For $\hat{S}_{i J} [2]$ and $\hat{S}_{J i} [2]$ generalized T invariance holds as they are. For $\hat{S}_{i i} [2]$ and $\hat{S}_{I I} [2]$ the matrix factor is self-invariant, and therefore generalized T invariance holds.
What we should do in the rest of appendix is to compute $\hat{S}$ matrix elements perturbatively to fourth order in $H_{1}$. In presenting the results of computation of $\hat{S}$ matrix elements, however, we change strategy of our description. That is,
- We present $\hat{S}$ matrix elements at fixed order in $W$. To do this we, of course, have t
| 3,351
| 4,465
| 2,916
| 2,974
| null | null |
github_plus_top10pct_by_avg
|
ft[dz^2 +d{\bf x}^2\right]
=dy^2+a^2(z)d{\bf x}^2.
\label{rsmetric}$$ Here $a(z)=\ell/z$, where $\ell$ is the AdS radius. The branes are placed at arbitrary locations which we shall denote by $z_+$ and $z_-$, where the positive and negative signs refer to the positive and negative tension branes respectively ($z_+ < z_-$). The “canonically normalized” radion modulus $\phi$ - whose kinetic term contribution to the dimensionally reduced action on the positive tension brane is given by $${1\over 2}\int d^4 x \sqrt{g_+}\, g^{\mu\nu}_+\partial_{\mu}\phi
\,\partial_{\nu}\phi,
\label{kin}$$ is related to the proper distance $d= \Delta y$ between both branes in the following way [@gw1] $$\phi=(3M^3\ell/4\pi)^{1/2} e^{- d/\ell}.$$ Here, $M \sim TeV$ is the fundamental five-dimensional Planck mass. It is usually assumed that $\ell \sim M^{-1}$ . Let us introduce the dimensionless radion $$\lambda \equiv \left({4\pi \over 3M^3\ell}\right)^{1/2} {\phi} =
{z_+ \over z_-} = e^{-d/\ell},$$ which will also be refered to as [*the hierarchy*]{}. The effective four-dimensional Planck mass $m_{pl}$ from the point of view of the negative tension brane is given by $m_{pl}^2 = M^3 \ell
(\lambda^{-2} - 1)$. With $d\sim 37 \ell$, $\lambda$ is the small number responsible for the discrepancy between $m_{pl}$ and $M$.
At the classical level, the radion is massless. However, as we shall see, bulk fields give rise to a Casimir energy which depends on the interbrane separation. This induces an effective potential $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\phi)$ which by convention we take to be the energy density per unit physical volume on the positive tension brane, as a function of $\phi$. This potential must be added to the kinetic term (\[kin\]) in order to obtain the effective action for the radion: $$S_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}[\phi]
=\int d^4x\, a_+^4 \left[{1\over 2}g_+^{\mu\nu}\partial_{\mu}\phi\,
\partial_{\nu}\phi +
V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\
| 3,352
| 3,650
| 3,248
| 2,985
| 3,133
| 0.774625
|
github_plus_top10pct_by_avg
|
at this time the equations do not impose $\partial_{+}f_{--}=0$ and furthermore, the transformation rule of $f_{--}$ also differs from the one found off the chiral point.
Conserved **charges for** $\mu l=1$
-----------------------------------
Evaluating the variation of the surface charges using the expressions of [@Carlip] with the asymptotic conditions (\[Asympt relaxed metric\]) one obtains:$$\delta Q_{+}=\frac{4}{l}\int T^{+}\delta f_{++}d\phi\ ,\text{ and }\delta
Q_{-}=\frac{2}{l}\int T^{-}\delta h_{--}d\phi\ .$$ This implies (up to additive constants) $$Q_{+}=\frac{4}{l}\int T^{+}f_{++}d\phi\ ,\text{ and }Q_{-}=\frac{2}{l}\int
T^{-}h_{--}d\phi\ .$$ The crucial new feature found here, apparently overlooked in the previous literature, is that $Q_{-}[T^{-}]$ does not vanish identically. Rather, the relaxation term $h_{--}$ does contribute to it. This behavior is somehow similar to what occurs for scalar fields that saturates the BF bound. One may verify explicitly that on definite solutions, $Q_{-}[T^{-}]$ is not zero [@HMTfuture]. Indeed, for the metric (\[pp-wave chiral\]), $h_{--}=F(x^{-})$ which in general does not vanish.
From the variations (\[deltah–chiral\]) and (\[deltaf–chiral\]) of $h_{--}$ and $f_{++}$ and the asymptotic field equations, one finds that both $Q_{+}[T^{+}]$ and $Q_{-}[T^{-}]$ fulfill the Virasoro algebra with the central charge $$c_{+}=2\,c\,,\;\;\;\;c_{-}=0\,.$$ Even though $Q_{-}[T^{-}]$ does not vanish, the central charge $c_{-}$ is zero because the inhomogeneous terms $-l^{2}\left( \partial_{-}T^{-}+\partial
_{-}^{3}T^{-}\right) /2$ are absent from $\delta_{\eta}h_{--}$.
In this paper we have exhibited the boundary conditions appropriate to accommodate the solutions of topologically massive gravity found in the literature with a slower decay at infinity than the one for pure standard gravity discussed in [@Brown-Henneaux]. These boundary conditions fulfill the consistency conditions listed in the introduction. The analysis proceeds very much as in the case of anti-de Si
| 3,353
| 1,112
| 2,715
| 3,072
| null | null |
github_plus_top10pct_by_avg
|
\hat{\Gamma}(j,j)} > \sqrt{\underline{\sigma}^2 - C \aleph_n} >0$ and, by Weyl’s inequality, the minimal eigenvalue of $\hat{V}$ is no smaller than $v - C
\daleth_n > 0$. In particular, the error terms $\Delta^*_{n,1}$ and $\Delta^*_{n,2}$ are well-defined (i.e. positive). Thus we have that $$\label{eq:A3}
A_3 \leq C \left( \Delta^*_{n,1} + \Delta^*_{n,2} \right) +
\frac{2}{n},$$ where the lower order term $\frac{1}{n}$ is reported to account for the restriction to the event $\mathcal{E}_n$. The result now follows by combining all the bounds, after noting that $\Delta_{1,n} \leq \Delta^*_{1,n}$ and $\Delta_{2,n} \leq \Delta^*_{2,n}$.
To show that the same bound holds for the coverage of $\tilde{C}^*_\alpha$ we proceed in a similar manner. Using the triangle inequality, and uniformly over all the distributions in ${\cal P}_n$, $$\begin{aligned}
\mathbb{P}(\theta \in \tilde{C}^*_n) & =
\mathbb{P}(\sqrt{n} |\hat\theta_j - \theta _j| \leq \tilde{t}^*_{j,\alpha},
\forall j)\\
& \geq
\mathbb{P}\left( \sqrt{n} |\hat \theta^*_j - \hat{\theta}_j | \leq
\tilde{t}^*_{j,\alpha}, \forall j \Big| (W_1,\ldots,W_n) \right) - (A_1 + A_2 +
A_3)\\
& \geq (1 - \alpha) - (A_1 + A_2 +
A_3),\end{aligned}$$ where $$\begin{aligned}
A_1 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s} \left| \mathbb{P}\left( \sqrt{n} |
\hat{\theta}_j-
\theta_j |
\leq t_j, \forall j \right) - \mathbb{P}( |Z_{n,j}| \leq t_j, \forall j ) \right|,\\
A_2 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s } \left| \mathbb{P}(
| Z_{n,j} | \leq t_j, \forall j ) - \mathbb{P}( |
\hat{Z}_{n,j} | \leq t_j, \forall j ) \right|,\\
\text{and} & \\
A_3 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s } \Big| \mathbb{P}(
| \hat{Z}_{n,j} | \leq
t_j, \forall j) - \mathbb{P}\left( \sqrt{n} | \hat{\theta}_j^* -
\hat{\theta}_j | \leq t_j, \forall j
\Big| (W_1,\ldots,W_n) \right) \Big|.\end{aligned}$$ The term $A_1$ is bounded by $C (\Delta_{1,n} + \Delta_{2,n})$, as shown in the first part of the proof
| 3,354
| 5,019
| 1,443
| 2,707
| null | null |
github_plus_top10pct_by_avg
|
hin the fiducial volume (two meters inward from the detector walls),]{}
2. [There must be two Cherenkov rings,]{}
3. [Both rings must be showering for the $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$ and $p \rightarrow e^{+}\gamma$ modes; one ring must be showering and one ring must be non-showering for the $pp \rightarrow e^{+}\mu^{+}$, $nn \rightarrow e^{+}\mu^{-}$, $nn \rightarrow e^{-}\mu^{+}$ and $p \rightarrow \mu^{+}\gamma$ modes; both rings must be non-showering for the $pp \rightarrow \mu^{+}\mu^{+}$, $nn \rightarrow \mu^{+}\mu^{-}$ modes (see note in [^1]),]{}
4. [There must be zero Michel electrons for the $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$ and $p \rightarrow e^{+}\gamma$ modes; there must be less than or equal to one Michel electron for the $pp \rightarrow e^{+}\mu^{+}$, $nn \rightarrow e^{+}\mu^{-}$, $nn \rightarrow e^{-}\mu^{+}$ and $p \rightarrow \mu^{+}\gamma$ modes; there is no Michel electron cut for the $pp \rightarrow \mu^{+}\mu^{+}$, $nn \rightarrow \mu^{+}\mu^{-}$ modes (see note in [^2]),]{}
5. [The reconstructed total mass, $M_{tot}$, should be $1600\leq M_{tot}\leq 2050$ MeV/c$^2$ for the dinucleon decay modes; the reconstructed total mass should be $800\leq M_{tot}\leq 1050$ MeV/c$^2$ for the nucleon decay modes,]{}
6. [The reconstructed total momentum, $P_{tot}$, should be $0 \leq P_{tot} \leq 550$ MeV/c for the dinucleon decay modes; for the nucleon decay modes, it should be $100 \leq P_{tot} \leq 250$ MeV/c for the event to be in the “High $P_{\text{tot}}$" signal box and $0 \leq P_{tot} \leq 100$ MeV/c for the event to be in the “Low $P_{\text{tot}}$" signal box,]{}
7. [\[SK-IV nucleon decay searches only\] There must be zero tagged neutrons.]{}
Figure \[fig:mass-momentum\] shows the distributions of signal MC events (left panels), atmospheric neutrino background (middle), and data (right) as a function of $P_{tot}$ versus $M_{tot}$ after cut (A4). The signal sele
| 3,355
| 800
| 3,168
| 3,271
| 3,137
| 0.774598
|
github_plus_top10pct_by_avg
|
ed}
\begin{split}
~& 8 \big(\gamma +2 \delta \big) \Big( -1+A(r)\Big) A(r) B(r)^3+4 r \big(5 \gamma -2 \delta
\big) B(r)^3 A'(r)-18 r^2 \gamma A(r) B(r) B'(r)^2
\\& +8 r \big(2 \gamma
+\delta \big) B(r)^2 \Big(-r A'(r) B'(r)+A(r) \left(B'(r)+r
B''(r)\right)\Big)
\\& +r^3 \big(\gamma -\delta \big) B'(r) \Big(5 A(r)
B'(r)^2+3 B(r) \left(A'(r) B'(r)-2 A(r) B''(r)\right)\Big)=0.
\end{split}\end{aligned}$$ The first one provides solutions for $B(r)$, so the two equations decouple. We have found real exact vacuum solutions for 3 couples $(\gamma \, , \, \delta)$ and have computed their associated Kretschmann scalars $ R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$ to see if they suffer from singularites at $r=0$.
- For $\gamma =1$ and $\delta =0$ : $$\begin{aligned}
ds^2 = -k \, r^2 \; dt^2 + \frac{r^2}{p+r^2} \; dr^2 + r^2 d\Omega^2 \quad\quad \text{and} \quad\quad R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta} \to \frac{p^2}{r^8} ,
\end{aligned}$$
- For $\gamma =0$ and $\delta =1$ : $$\begin{aligned}
ds^2 = -\frac{k}{r^4} \; dt^2 + \frac{3}{1+p \, r^2} \; dr^2 + r^2 d\Omega^2 \quad\quad \text{and} \quad\quad R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta} \to \frac{15 p^2-2 p+3}{r^4} ,
\end{aligned}$$
- For $\gamma =1$ and $\delta =-\frac{1}{2}$ : $$\begin{aligned}
ds^2 = -k \; dt^2 + p \;dr^2 + r^2 d\Omega^2 \quad\quad \text{and} \quad\quad R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta} \to \frac{(p-1)^2}{p^2 \, r^4} .
\end{aligned}$$
where $p$ and $k$ are integration constants. Note that $k$ can be set equal to one by a right choice of the time coordinate.
Recall that the Kretschmann scalar of the Schwarzschild solution of the Einstein equations diverges as $ R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta} \to 1/r^6$. Therefore, we see that our first solution has a milder divergence. In the last case, choosing the initial condition $p=1$ provides $R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}=0$ for all $r$ so there is no singularity of curvature because the space-ti
| 3,356
| 5,414
| 675
| 2,840
| null | null |
github_plus_top10pct_by_avg
|
ended detectors
---
Introduction
============
The Unruh effect [@Fulling:1972md; @Davies:1974th; @Unruh:1976db] states that an observer of negligible spatial size on a worldline of uniform linear acceleration in Minkowski spacetime reacts to the Minkowski vacuum of a relativistic quantum field by thermal excitations and de-excitations, in the Unruh temperature $g/(2\pi)$, where $g$ is the observer’s proper acceleration. The acceleration singles out a distinguished direction in space, and an observer with direction-sensitive equipment will in general see a direction-dependent response; however, for the Lorentz-invariant notion of direction-sensitivity introduced in [@Takagi:1985tf], the associated temperature still turns out to be equal to $g/(2\pi)$, independently of the direction. For textbooks and reviews, see [@Birrell:1982ix; @Crispino:2007eb; @Fulling:2014wzx].
In this paper we address the response of uniformly linearly accelerated observers in Minkowski spacetime, operating direction-sensitive equipment of nonzero spatial size. We ask whether the temperature seen by these observers is still independent of the direction. The question is nontrivial: while a spatially pointlike detector with a monopole coupling is known to be a good approximation for the interaction between the quantum electromagnetic field and electrons on atomic orbitals in processes where the angular momentum interchange is insignificant [@MartinMartinez:2012th; @Alhambra:2013uja], finite size effects can be expected to have a significant role in more general situations [@DeBievre:2006pys; @Hummer:2015xaa; @Pozas-Kerstjens:2015gta; @Pozas-Kerstjens:2016rsh; @Pozas-Kerstjens:2017xjr; @Simidzija:2018ddw]. Also, the notion of a finite size accelerating body has significant subtlety: while a rigid body undergoing uniform linear acceleration in Minkowski spacetime can be defined in terms of the boost Killing vector, different points on the body have differing values of the proper acceleration, and the body as a whole does not have an un
| 3,357
| 2,648
| 1,416
| 3,115
| null | null |
github_plus_top10pct_by_avg
|
)
position = [605, 6];
// OR
/*
var position = ((left >= 0 && left <= 80) ? [20, 1] :
((left >= 81 && left <= 198) ? [137, 1] :
((left >= 199 && left <= 315) ? [254, 3] :
((left >= 316 && left <= 430) ? [371, 4] :
((left >= 431 && left <= 548) ? [488, 5] :
((left >= 549) ? [605, 6] : [] ) ) ) ) ) );
*/
if (position.length) {
$(this).animate({
left : position[0]
}, 200);
$(content).children().fadeOut(300, "linear", function() {
$(content).children(':nth-child(' + position[1] + ')').delay(299).fadeIn(600, "linear");
});
}
}
Q:
Why this loop freezes my browser?
Why browser hangs executing this?
for(var i= 9007199254740993;i<9007199254740994;i++) {
console.log(i);
}
A:
The integers you are trying to use are larger than 2^53. JavaScript cannot represent those integers precisely. Lets have a look at the console:
> var i = 9007199254740993;
undefined
> i++
9007199254740992
> i++
9007199254740992
> i++
9007199254740992
...
As you can see, due to loss of precision, the value of i doesn't change, thus the condition will always be true, resulting in an infinite loop.
Q:
yii framework: how to make search result empty in search form?
Hello i created a separate search form having input box and button.
in my model i want to search products by category wise...
but problem is that when input box is empty and clicking on search buttons it displays all entries from the database table..
controller code is-
class AddController extends Controller
{
public function actionAddsearch()
{
$model_form = new Add("search");
$attributes = Yii::app()->getRequest()->getPost("Add");
if(!is_null($attributes))
{
$model_form->setAttributes(Yii::app()->getRequest()->getPost("Add"));
}
$this->render("searchResults", array(
"model" => $model_form,
"models" => $model_form->searchAdd(),
));
}
model code--
class Add extends CActiveRecord
{
public function searchAdd()
{
$criteria = new CDbCriteria();
| 3,358
| 4,147
| 58
| 2,500
| 10
| 0.842453
|
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|
label="fig:fsf"}](5.eps)
![$e_is_{i}=e_if_ie_{i+1} = s_ie_{i+1}$ ($E4''$)[]{data-label="fig:efe"}](6.eps)
As we noted in the previous paper [@Ko4], these basic relations are invariant under the transpositions of indices $i\leftrightarrow n-i+1$ as well as the $\mathbb{Z}[Q]$-linear involution $*$ defined by $(xy)^{*} = y^*x^*$ ($x, y\in A_{n}(Q)$). This implies that if one local move is allowed then other three moves —obtained by reflections with respect to the vertical and the horizontal lines and their composition— are also allowed.
Further, we note that if we put $$e^{[r]} = f_1f_2\cdots f_{r-1}e_1e_2\cdots e_rf_1f_2\cdots f_{r-1}$$ then we can check that $e^{[r]}$, $f$ and $s_i$ ($1\leq i\leq n-1$) satisfy the defining relations of $P_{n,r}(Q)$, the $r$-modular party algebra, defined in the paper [@Ko4]. This means that the local moves shown in the paper [@Ko4] also hold in $A_n(Q)$ (in fact, these local moves are more easily verified in $A_n(Q)$). Some of them are pictorially expressed in Figures \[fig:dpjrE\],\[fig:roeE\],\[fig:dpsE\] and \[fig:dpeE\].
![Defective part jump rope ($R13'$)[]{data-label="fig:dpjrE"}](7.eps)
![Removal (addition) of excrescences ($R14'$)[]{data-label="fig:roeE"}](8.eps)
![Defective part shift ($R16'$)[]{data-label="fig:dpsE"}](9.eps)
![Defective part exchange ($R17'$)[]{data-label="fig:dpeE"}](10.eps)
Standard expressions of seat-plans
==================================
In this section, for a seat-plan $w$ of $\Sigma_n^1$, we define a [*basic expression*]{}, as a word in the alphabet $\Gamma_n^1$. Then we define more general forms called [*crank form expression*]{}s. As a special type of the crank form expression, we define the [*standard expression*]{}. In the next section, we show that any two crank form expressions of a seat-plan will be moved to each other by using the basic relations $(R0)$-$(R4)$ and $(E1)$-$(E5)$ finite times. Consequently, we find that any seat-plan can be moved to its standard expression. To define these expressions, we introduce so
| 3,359
| 319
| 3,219
| 2,548
| 386
| 0.811329
|
github_plus_top10pct_by_avg
|
the last inequality follows from the size property and the inequality $m{\leqslant}n$. [ Therefore, there are $x$ possibilities for the root and hence there are at most $4^k\cdot \binom{{s}}{d}\cdot n^{c_0+2}$ ordered trees with the specified root. Fix an arbitrary $d$-choice $D_t$ as the root for $T$.]{}
Next we fix an arbitrary function ${\text{col}}:\{2,\ldots,k\}\rightarrow \{\text{blue},\text{red}\}$, that gives a blue-red coloring of $2,\ldots,k$. In what follows we establish an upper bound for the probability that ${\mathcal{C}}_{m}$ contains the blue-red colored tree $T\subset {\mathcal{C}}_m$, (according to Definition \[def:br\]). Let $q_1(t)$ be the probability that the $t$-th ball chooses the root of $T$ [(that is, that the $d$-choice made by the $t$-th ball corresponds to the root of $T$)]{}. Then $$\begin{aligned}
\label{pr:no}
\sum_{t=1}^{m}q_1(t){\leqslant}\sum_{t=1}^{m}\frac{1}{\binom{{s}}{d}}{\leqslant}\frac{n}{\binom{{s}}{d}}
,
\end{aligned}$$ because $H$ contains $\binom{{s}}{d}$ distinct $d$-element sets for for every $H\in {\mathcal{E}}_t$. For every $t=2,\ldots, k$, define $q_i(t,{\text{col}}(i))$ to be the probability that the $t$-th ball chooses the $i$-th vertex of the tree (i.e., $i$) with ${\text{col}}(i)$. If ${\text{col}}(i)$ is red then $D_t$ must share at least two bins with $\cup_{j=1}^{i-1}D_{t_j}$, while if ${\text{col}}(i)$ is blue then $D_t$ only shares one bin with its parent. For every $i=2,\ldots, k$, let us derive an upper bound on $q_i(t, \text{blue})$. Here, the $i$-th vertex share one bin with its parent in $T$, say $D_{t_j}$. Now $D_{t_j}$ has $d$ bins and by the balancedness property we get $${\ensuremath{\operatorname{\mathbf{Pr}}\left[D_{t_j}\cap H_t\neq \emptyset\right]}}{\leqslant}\sum_{i\in D_{t_j}}{\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in H_t\right]}}{\leqslant}\frac{\beta d{s}}{n},$$ where $H_t$ is the edge chosen by ball $t$ from ${\mathcal{H}}^{(t)}$, uniformly at random. Suppose that for some [$a{\geqslant}1$]{} we have $|D_{t_j}\cap H_
| 3,360
| 2,055
| 2,001
| 3,137
| 2,124
| 0.782568
|
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|
r\\
\label{eq:jad}\end{aligned}$$ where $d_{\alpha\beta}=\langle\alpha;R\vert(\partial/\partial R)
\vert\beta;R\rangle$ is the nonadiabatic coupling vector and $$S_{\alpha\beta}=E_{\alpha\beta}d_{\alpha\beta}
\left(\frac{P}{M}\cdot d_{\alpha\beta}\right)^{-1}\;.$$ Using Eqs. (\[eq:ilad\]) and (\[eq:jad\]), the equation of motion for the density matrix in the adiabatic basis can be written explicitly as $$\begin{aligned}
\partial_t\rho_{\alpha\alpha^{\prime}}
&=&
-i\omega_{\alpha\alpha^{\prime}}\rho_{\alpha\alpha^{\prime}}
-\frac{P}{M}\cdot\frac{\partial}{\partial R}\rho_{\alpha\alpha^{\prime}}
\nonumber\\
&& -\frac{1}{2}\left(F_{\alpha}+F_{\alpha^{\prime}}\right)\cdot
\frac{\partial}{\partial P}\rho_{\alpha\alpha^{\prime}}
\nonumber\\
&&-\sum_{\beta}\frac{P}{M}\cdot d_{\alpha\beta}
\left(1+\frac{1}{2}S_{\alpha\beta}\cdot\frac{\partial}{\partial P}\right)
\rho_{\beta\alpha^{\prime}}
\nonumber\\
&&-\sum_{\beta^{\prime}}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^*
\left(1+\frac{1}{2}S_{\alpha^{\prime}\beta^{\prime}}^*
\cdot\frac{\partial}{\partial P}\right)
\rho_{\alpha\beta^{\prime}}
\;.\nonumber\\
\label{eq:rho-eq-ad}\end{aligned}$$ The wave fields $\vert\psi^{\iota}(X)\rangle$ and $\langle\psi^{\iota}(X)\vert$ can be expanded in the adiabatic basis as $$\begin{aligned}
\vert\psi^{\iota}(X)\rangle&=&\sum_{\alpha}\vert\alpha;R\rangle
\langle\alpha;R\vert\psi^{\iota}(X)\rangle
=\sum_{\alpha}C_{\alpha}^{\iota}\vert\alpha;R\rangle\nonumber\\
\langle\psi^{\iota}(X)\vert&=&\sum_{\alpha}\langle\psi^{\iota}\vert\alpha;R\rangle
\langle\alpha;R\vert
=\sum_{\alpha}\langle\alpha;R\vert C_{\alpha}^{\iota *}(X)
\;,\nonumber\\\end{aligned}$$ and the density matrix in Eq. (\[eq:rho-ansatz\]) becomes $$\begin{aligned}
\rho_{\alpha\alpha^{\prime}}(X,t)
&=&\sum_{\iota}w_{\iota}C_{\alpha}^{\iota}(X,t)C_{\alpha^{\prime}}^{\iota *}(X,t)
\;.
\label{eq:rho-ansatz-ad}\end{aligned}$$ In order to find two separate equations for $C_{\alpha}^{\iota}$ and $C_{\alpha^{\prime}}^{\iota *}$, one cannot insert Eq. (\[eq:rho-ansatz-ad\]
| 3,361
| 1,418
| 849
| 3,635
| null | null |
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|
of the corresponding families, and the descriptions given in the statement are immediately verified in these cases.
The second part of Lemma \[PNCtolimits\] may be viewed as the analogue in our context of an observation of Pinkham (‘sweeping out the cone with hyperplane sections’, [@MR0376672], p. 46).
\[eluding\] Denote by $R$ the proper transform in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the set of singular matrices in ${{\mathbb{P}}}^8$. Lemma \[PNCtolimits\] asserts that the set of limits of ${{\mathscr C}}$ is the image of the union of the PNC and $R$. A more explicit description of the image of $R$ has eluded us; for a smooth curve ${{\mathscr C}}$ of degree $\ge 5$ these ‘star limits’ have two moduli. It would be interesting to obtain a classification of curves ${{\mathscr C}}$ with smaller ‘star-moduli’.
The image of the [*intersection*]{} of $R$ and the PNC will play an important role in this paper. Curves in the image of this locus will be called ‘rank-$2$ limits’; we note that the set of rank-$2$ limits has dimension $\le 6$.
Lemma \[PNCtolimits\] translates the problem of finding the limits for families of plane curves ${{\mathscr C}}\circ\alpha(t)$ into the problem of describing the PNC for the curve ${{\mathscr C}}$. Each component of the PNC is a $7$-dimensional irreducible subvariety of ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N$. We will describe it by listing representative points of the component. More precisely, note that ${\text{\rm PGL}}(3)$ acts on ${{\mathbb{P}}}^8$ by right multiplication, and that this action lifts to a right action of ${\text{\rm PGL}}(3)$ on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$. Each component of the PNC is a union of orbits of this action. For each component, we will list germs $\alpha(t)$ lifting on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ to germs $\tilde\alpha(t)$ so that the union of the orbits of the centers $\tilde\alpha(0)$ is dense in that component.
Marker germs {#germlist}
------------
| 3,362
| 1,553
| 2,286
| 3,063
| 3,807
| 0.769994
|
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|
reen’s formula (\[green-ex\]), which in combination with the fact that $$P'(x,\omega,E,D)\phi=-P(x,\omega,E,D)\phi-{\frac{\partial S_0}{\partial E}}\phi,$$ leads us to $${\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
={}&
-{\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
-{\left\langle}{\frac{\partial S_0}{\partial E}}\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} \\
{}&
+{\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2-{\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}^2 \\
{}&
+{\left\langle}S_0(\cdot,0)\gamma_0(\phi),\gamma_0(\phi){\right\rangle}_{L^2(G\times S)}
-{\left\langle}S_0(\cdot,E_m)\gamma_m(\phi),\gamma_m(\phi){\right\rangle}_{L^2(G\times S)}.$$ Using this equation, and performing estimations as in the proof of Theorem \[csdath1\], allows us to deduce the inequality $$\begin{aligned}
{\left\langle}{\bf f},\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g}, \gamma_-(\phi){\right\rangle}_{T^2(\Gamma_-)}
\geq
c'{\left\Vert \phi\right\Vert}_{H_1}^2,\end{aligned}$$ from which the desired estimate , and therefore uniqueness of solutions, follow.
\[ttc\] Suppose that the assumption ${\bf TC}$ is valid and that $\phi\in {{{\mathcal{}}}H}_P$ such that \[assttc\] \_[|\_-]{}T\^2(\_-) (,,E\_[m]{})L\^2(GS). Then at least in some cases one is able to show that (cf. [@cessenat85]) \[asscl-a\] \_[|\_+]{}T\^2(\_+) (,,0)L\^2(GS). This would make the assumption of part (iii) of Theorem \[csdath3\] superfluous. We omit further considerations of this issue here.
Similarly as above we see that the variational equation corresponding to the original problem (\[se1\]), (\[se2\]), (\[se3\]) is $$\tilde{B}_0(\tilde{\psi},v)=F_0(v)\quad \forall v\in H_2,$$ where $\tilde\psi \in H_1$ and $\tilde{B}_0(\cdot,\cdot)$ is the continuous extension onto $H_1\times H_2$ of the bilinear form $B_0(\cdot,\cdot):C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)\to{\mathbb{R}}$ defined by (that is, the bilinear form (\[csda27\]) with
| 3,363
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| 1,735
| 3,322
| null | null |
github_plus_top10pct_by_avg
|
nario is if the adversary creates malicious examples when noise removing operations are turned on in all possible locations. It is possible that such adversarial examples would also fool the classifier when the defense is only applied in a subset of the layers. Fortunately, we note that for FGS, IGS, and CW2, transferability of attacks between defense arrangements is limited as seen in Table \[table:transferability\_layer\_lenet\]. The column headers are binary strings indicating presence or absence of defense at the 3 relevant points in LeNet: input layer, after the first convolutional layer, after the second convolutional layer. Column \[0, 0, 0\] shows the attack success rate against an undefended model, and column \[1, 1, 1\] shows the attack success rate against a fully deterministically-defended model. The remaining columns show the transfer attack success rate of the perturbed images created for the \[1, 1 ,1\] defense arrangement. The most surprising result is that defense arrangements using 2 autoencoders are more susceptible to a transfer attack than defense arrangements using a single autoencoder. Specifically, \[1, 0, 0\] is more robust than \[1, 1, 0\] which does not make sense intuitively. Although the perturbed images are engineered to fool a classifier with VAEs in all convolutional layers in addition to the input layer, it is possible that the gradient used to generate such images is orthogonal to the gradient required to fool just a single VAE in the convolutional layers.
------------------------------ ------------- ------------- ------------- ------------- ------------- ------------- ------------- -------------
(r)[2-3]{} (r)[4-9]{} Attack \[0, 0, 0\] \[1, 1, 1\] \[0, 0, 1\] \[0, 1, 0\] \[1, 0, 0\] \[0, 1, 1\] \[1, 0, 1\] \[1, 1, 0\]
FGS 0.176 0.1451 0.035 0.019 0.117 0.01
| 3,364
| 1,456
| 2,939
| 3,109
| null | null |
github_plus_top10pct_by_avg
|
ation method to find the coefficient vectors, and calculate the corresponding average computation rate.
We first show that as stated in Remark \[remark:KPractical\], for high dimension and large power, the number of real-valued approximations $K$ can be set to a rather small value without degrading the rate apparently. As shown in Figure \[figure:K\_L4\], for dimension $L=4$ and power $P$ from 0dB to 20dB, the average computation rate quickly converges as $K$ increases from 1 to 4. Further increasing $K$ up to 10 incurs additional computational cost with little improvement in the average computation rate.
With the above observation, it is reasonable to introduce the upper bound $K_u$ for $K$, and adopt the criterion in to determine $K$. $K_u$ can be calculated off-line by simulations prior to applying the method, which incurs no additional processing complexity in real-time. The values of $K_u$ according to the simulation results are listed in Table \[table:Ku\].
[c|\*[15]{}[r]{}]{} $L$ &2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\
$K_u$ &2 & 3 & 4 & 5 & 5 & 5 & 6 & 6 & 6 & 6 & 7 & 6 & 6 & 6 & 4\
\
We then show the effectiveness of our method by comparing the average computation rate with those of other existing methods. The methods covered include the following.
- Our QP relaxation (QPR) method that gives the suboptimal solution.
- The branch-and-bound (BnB) method proposed by Richter [*et al.*]{} in [@Richter2012] that provides the optimal solution.
- The method developed by Sahraei and Gastpar in [@Sahraei2014] that finds the optimal solution with an average-case complexity of $O(P^{0.5}L^{2.5})$ for i.i.d. Gaussian channel entries. We refer to this method as the “SG" method for short.
- The LLL method proposed by Sakzad [*et al.*]{} in [@Sakzad2012], which is based on the LLL lattice reduction (LR) algorithm. The parameter $\delta$ in the LLL LR algorithm is set as 0.75 since further increasing $\delta$ towards 1 achieves little gain in the computation rate but requi
| 3,365
| 1,447
| 1,915
| 3,043
| null | null |
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|
th slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(1:0:0)$.
The limit curves are then obtained by choosing a side of the polygon with slope strictly between $-1$ and $0$, and setting to $0$ the coefficients of the monomials in $F$ [*not*]{} on that side. These curves are studied in [@MR2002d:14083]; typically, they consist of a union of cuspidal curves. The kernel line is part of the distinguished triangle of such a curve, and in fact it must be one of the distinguished tangents.
Here is the Newton polygon for the curve of Example \[exone\], with respect to the point $(1:0:0)$ and the line $z=0$:
Setting to zero the coefficient of $z^3$ produces the limit $y(y^2+xz)$.
[*Type V:*]{} Let $p=\operatorname{im}\alpha(0)$ be a singular point of the support of ${{\mathscr C}}$, and let $m$ be the multiplicity of ${{\mathscr C}}$ at $p$. Again choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ so that $p=(1:0:0)$ and $z=0$ is the selected line.
We may describe ${{\mathscr C}}$ near $p$ as the union of $m$ ‘formal branches’, cf. §\[formalbranches\]; those that are tangent to the line $z=0$ (but not equal to it) may be written $$z=f(y)=\sum_{i\ge 0} \gamma_{\lambda_i} y^{\lambda_i}$$ with $\lambda_i\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_0}\ne 0$.
The choices made above determine a finite set of rational numbers, which we call the ‘characteristics’ for ${{\mathscr C}}$ (w.r.t. $p$ and the line $z=0$): these are the numbers $C$ for which there exist two branches ${{\mathscr B}}$, ${{\mathscr B}}'$ tangent to $z=0$ that agree modulo $y^C$, differ at $y^C$, and have $\lambda_0<C$. (Formal branches are called ‘pro-branches’ in [@MR2107253], Chapter 4; the numbers $C$ are ‘exponents of contact’.)
Let $S$ be the number of branches that agree with ${{\mathscr B}}$ (and ${{\mathscr B}}'$) modulo $y^C$. The initial exponents $\lambda_0$ and
| 3,366
| 1,717
| 2,920
| 3,098
| 1,453
| 0.789361
|
github_plus_top10pct_by_avg
|
1}^{-1})h_{1}$$
Thus $g \rhd h_{1}^{-1} = h_{1}^{-1}$, i.e. $h_{1}^{-1} \in {\rm
Fix}(H)$. In a similar way we can show that ${\rm Fix}(G)$ is a subgroup of $G$. Using the compatibility condition [(\[eq:2\])]{} we obtain that the map given by: $$\varphi_{\rhd}: {\rm Fix}(G) \rightarrow {\rm Aut}(H), \quad
\varphi_{\rhd}(g)(h) := g \rhd h$$ for all $g \in {\rm Fix}(G)$, $h\in H$ is a morphism of groups. Thus we can construct the left version of the semidirect product associated to the triple $(H, \,
{\rm Fix}(G), \, \varphi_{\rhd})$: that is $H
{}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) : = H\times {\rm
Fix}(G)$ with the multiplication: $$(h, g) (h', g') = \bigl(h(g \rhd h'), \, gg'\bigl)$$ for all $h$, $h' \in H$ and $g$, $g' \in {\rm Fix}(G)$. Similarly, using [(\[eq:3\])]{} we obtain that the map given by: $$\psi_{\lhd}: {\rm Fix}(H) \rightarrow {\rm Aut}(G), \quad \psi_{\lhd}(h)(g):= g \lhd h$$ for all $h \in {\rm Fix}(H)$, $g\in G$ is a morphism of groups and we can construct the right version of the semidirect product associated to the triple $(G, \, {\rm Fix}(H), \, \psi_{\lhd})$: i.e. ${\rm Fix}(H) \rtimes_{\psi_{\lhd}} G := {\rm Fix}(H) \times
G$ with the multiplication: $$(h, g)(h', g') = \bigl(hh', \, (g \lhd h')g'\bigl)$$ for all $h$, $h' \in {\rm Fix}(H)$ and $g$, $g' \in G$. Moreover, the inclusion maps $$\overline{i}: {\rm Fix}(H) \times {\rm Fix}(G) \hookrightarrow H
{}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) \quad {\rm and} \quad
\overline{j} : {\rm Fix}(H) \times {\rm Fix}(G) \hookrightarrow
{\rm Fix}(H) \rtimes_{\psi_{\lhd}} G$$ are morphisms of groups by straightforward verifications.
On the other hand we can easily prove that the canonical inclusions $$i: H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G)
\hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad
i(h,g) = (h,g)$$ and $$j: {\rm Fix}(H) \rtimes_{\psi_{\lhd}} G
\hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad
j(h,g) = (h,g)$$ are morphisms of groups. Indeed for $h$, $h'\in
H$ and $g$, $g' \in {\rm Fix}(G)$ w
| 3,367
| 2,324
| 2,496
| 3,097
| null | null |
github_plus_top10pct_by_avg
|
group.
Moreover, $\langle y \rangle$ acts transitively on $\{\Delta_1, \Delta_2, \ldots, \Delta_k\}$.
- The length of an orbit $\nabla$ of $\langle y \rangle$ on $\Omega$ is $k=p$, and there are $m$ orbits $\nabla_1:=\nabla$, $\nabla_2\dots, \nabla_m$. Hence there is a partition $$\Omega=\nabla_1\cup \cdots \cup \nabla_m$$ and both ${{\operatorname}{O}_{p'}(A)}$ and $B$ act transitively on the set $\{\nabla_1, \nabla_2, \dots, \nabla_m\}$. In particular, for each $1\leq i\leq m$, there exists $a_i\in{{\operatorname}{O}_{p'}(A)}$ such that $\nabla_1^{a_i}=\nabla_i$; $a_1=1$.
Since the lenght of an orbit of $\langle y \rangle$ on $\Omega$ divides $p$ and $\langle y \rangle$ does not normalise any $N_i$, by Lemma \[new\], the first assertion follows. Now, the fact that $G=\langle y \rangle {{\operatorname}{O}_{p'}(A)}N=\langle y\rangle BN$ gives the last assertion.
- It follows from (ii) and (iii) that $r=pm$, with $1=(m,p)$.
- Without loss of generality, we may consider $\Delta=\{N_1,\ldots, N_m\}$, and we set $M_{\Delta}:=N_1\times \cdots \times N_m$. Then $M_{\Delta}$ is a minimal normal subgroup of $NC$.
Moreover, if $1\neq R\leq N$ and $R\unlhd NC$, then there exist $\{x_1, \ldots, x_d\}\subseteq \langle y \rangle$ such that $R=M_{\Delta}^{x_1} \times \cdots \times M_{\Delta}^{x_d}$.
- Since $r > 1$, then $m>1$.
Recall that $r >1$, by Lemma \[new\]. If $m=1$, then $\langle y \rangle$ has only one orbit on $\Omega$, i.e. $\langle y \rangle$ acts transitively on $\Omega=\{N_1, \ldots, N_k\}$. Suppose, for instance, that $N_i=N_1^y$, for $i > 1$. If there exists a non-trivial element $x\in{{\operatorname}{C}_{N_1}(y)}$, then $x=x^y\in N_1\cap N_1^y=N_1 \cap N_i$, a contradiction. Hence ${{\operatorname}{C}_{N_1}(y)}=1$, and it follows that ${{\operatorname}{C}_{N}(y)}=1$. But since $N\cap A\neq 1$, by Lemma \[ncapa\], we can choose an element of prime powe order $1 \neq x \in N \cap A$ such that $x \in C_N(y)$ because the hypotheses on the indices (recall that $N$ is a $p'$-group),
| 3,368
| 2,075
| 2,252
| 2,986
| null | null |
github_plus_top10pct_by_avg
|
scriptstyle}(1)}}(v_1,v'_1){\nonumber}\\
=\sum_{u',u'',v'}\bigg(&2\psi_\Lambda(v_1,u')\,\tilde G_\Lambda(u',u'')
\Big({{\langle \varphi_{u'}\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,u'')+\tilde
G_\Lambda(u',u'')\,\delta_{u,u''}\Big)\,\psi_\Lambda(u'',v'_1){\nonumber}\\
&\times{{\langle \varphi_{v_1}\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}\varphi_{
v'_1} \rangle}}_\Lambda\psi_\Lambda(v',v)+(\text{permutation of $u$ and }v')
\bigg),\end{aligned}$$ where the permutation term corresponds to the second term for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)$ in Figure \[fig:P-def\].
In addition to the above quantities, we define (see the third line in Figure \[fig:P-def\]) $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}}_\Lambda^2
{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda,
{\label{eq:P'0-def}}\\[5pt]
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}}
_\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}
_\Lambda\sum_{v'}{{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}
\varphi_x \rangle}}_\Lambda\,\psi_\Lambda(v',v),{\label{eq:P''0-def}}\end{aligned}$$ and let $$\begin{aligned}
{\label{eq:P'P''-def}}
P'_{\Lambda;u}(y,x)=\sum_{j\ge0}P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x),&&
P''_{\Lambda;u,v}(y,x)&=\sum_{j\ge0}P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x),\end{aligned}$$ where $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)$ are the leading contributions to $P'_{\Lambda;u}(y,x)$ and $P''_{\Lambda;u,v}(y,x)$, respectively.
Finally, we define $$\begin{aligned}
Q'_{\Lambda;u}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big)
P'_{\Lambda;u}(z,x),{\label{eq:Q'-def}}\\
Q''_{\Lambda;u,v}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)
\big)P''_{\
| 3,369
| 3,733
| 3,041
| 2,994
| null | null |
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|
t normal in $G$. Note that $G$ acts on $\Omega:=\{H^g\: : \: g\in G\}$ transitively. If $H<G$, then certainly $|\Omega|>1$ and, by Lemma \[FKS\], there exists a prime power order element $x\in G$ acting fixed-point-freely on $\Omega$. But the hypotheses imply that $x\in H^z$ for some $z\in G$, so $H^{zx}=H^z$ and this is a contradiction.
Preliminaries on (almost) simple groups and their prime graphs
==============================================================
We begin this section with a useful result on the centralisers of automorphisms of simple groups, which is a refinement of [@DPSS Lemma 2.6]. In fact, the own proof of that lemma provides this stronger result:
\[aut\] Let $N$ be a simple group. Then there exists $r\in \pi(N) \setminus\pi({\operatorname}{Out}(N))$ such that $(r, {\ensuremath{\left| {{\operatorname}{C}_{N}(\alpha)} \right|}})=1$ for every non-trivial $\alpha \in {\operatorname}{Out}(N)$ of order coprime to ${\ensuremath{\left| N \right|}}$.
Following the proof of [@DPSS Lemma 2.6], we can assume that $N=G(q)$ is a simple group of Lie type, with $q=p^{e}$, $p$ a prime and $e \geq 3$ a positive integer. In that proof it is shown that the prime $r$ is in fact a primitive prime divisor of $p^{em}-1$ for some integer $m \geq 2$, and that such $r$ always exist under the given assumptions. Now having in mind the orders of the outer automorphisms of the simple groups of Lie type (see for instance [@LPS Table 2.1]) and applying Lemma \[Zsi\] we can deduce that $r \not\in \pi({\operatorname}{Out}(N))$ (see also [@LPS 2.4. Proposition B]).
We will denote the *prime graph* of a group $G$, also called, the *Grünberg-Kegel* graph, by $\Gamma(G)$. The set of vertices of such graph is the set $\pi(G)$ of prime divisors of $|G|$, and two vertices $r,s$ are adjacent in $\Gamma(G)$ if there exists an element of order $rs$ in $G$. The connected components of the prime graph of a simple group are known from [@Wil] and [@Kon]. We will denote by $\mathcal{Z}( \Gamma(G))$, the center of the graph, i.e. $\math
| 3,370
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in{aligned}
cv_1 &\Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]-\tr\big[ \Er_\La(\La) \Er_\La(v_1I_r+(1-v_1)\La)^{-1}\big]\Big]
\\
&\leq cv_1 \Big\{ 2\tr \Er_\La(\La) - {1\over v_1}\tr \Er_\La(\La)\Big\}
= c (2v_1-1) \tr \Er_\La(\La),\end{aligned}$$ which implies that $$\De(W;\pi_{GB}^J)/m_{GB}
\leq \{ - (q-r-1) + c(2v_w/v_0-1)\}\tr \Er_\La(\La).$$ Hence, it holds true that $ - (q-r-1) + c(2v_w/v_0-1)\leq 0$ if $$\min \{0, - (q-r-1) + r+1\} \leq c \leq 0.
\label{eqn:sc2}$$ Combining (\[eqn:sc1\]) and (\[eqn:sc2\]) yields the condition $- (q-r-1) + r+1 \leq c \leq (q-r-1)/(2-v_w/v_0)$, namely, $-q+4\leq a+b\leq (q-r-1)/(2-v_w/v_0)-2r+2$. From (\[eqn:bound-cond\]), the sufficient conditions on $(a,b)$ for minimaxity can be written as $a>2-q$, $b>2$ and $a+b\leq (q-r-1)/(2-v_w/v_0)-2r+2$ if $$\begin{aligned}
&\{(q-r-1)/(2-v_w/v_0)-2r+2\}-\{-q+4\}\\
&=\{q-r-1+(2-v_w/v_0)(q-2r-2)\}/(2-v_w/v_0)>0.\end{aligned}$$ Thus the proof is complete. $\Box$
Take $v_x=v_w=v_0=1$. Let $X|\Th\sim\Nc_{r\times q}(\Th,I_r\otimes I_q)$. Consider the problem of estimating the mean matrix $\Th$ under the squared Frobenius norm loss $\Vert\Thh-\Th\Vert^2$. The Bayesian estimator with respect to (\[eqn:pr\_Th\]) and (\[eqn:pr\_GB\]) is expressed as $$\Thh_{GB}=\bigg[I_r-\frac{\int_{\Rc_r}\Om|\Om|^{(q+a)/2-1}|I_r-\Om|^{b/2-1}\exp[-\tr(\Om XX^\top)/2]\dd\Om}{\int_{\Rc_r}|\Om|^{(q+a)/2-1}|I_r-\Om|^{b/2-1}\exp[-\tr(\Om XX^\top)/2]\dd\Om}\bigg]X.$$ Then the same arguments as in this section yield that $\Thh_{GB}$ is proper Bayes and minimax if $a>0$, $b>2$, $q>3r+1$ and $ 2 < a+b \leq q-3r+1$. $\Box$
Superharmonic priors for minimaxity {#sec:superharmonic}
===================================
In estimation of the normal mean vector, Stein (1973, 1981) discovered an interesting relationship between superharmonicity of prior density and minimaxity of the resulting generalized Bayes estimator. The relationship is very important and useful in Bayesian predictive density estimation. In this section we derive some Bayesian minimax pre
| 3,371
| 1,228
| 1,595
| 3,367
| 2,196
| 0.781873
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0.003 $-0.081$
(k) $\nu[321]3/2$ $\nu[211]1/2$ 7.14 0.001 0.106
(l) $\pi[220]1/2$ $\pi[101]1/2$ 7.96 0.037 0.0095
(m) $\pi[211]3/2$ $\pi[101]1/2$ 7.95 0.015 $-0.011$
(n) $\pi[321]3/2$ $\pi[220]1/2$ 14.0 0.004 0.313
(o) $\pi[312]5/2$ $\pi[211]3/2$ 14.7 0.002 $-0.338$
(p) $\pi[211]3/2$ $\pi[110]1/2$ 14.1 0.002 0.280
(q) $\pi[211]1/2$ $\pi[101]1/2$ 11.5 0.002 $-0.256$
----- --------------- --------------- ------------------------ -------------------------------------------- ----------------------
: Same as Table \[26Ne\_0-\] but for the $K^{\pi}=1^{-}$ state in $^{26}$Ne at 8.76 MeV. This mode has $B(E1)=1.65 \times10^{-2}~e^{2}$fm$^{2}$, $B(Q^{\nu}1)=3.58 \times10^{-2} e^{2}$fm$^{2}$, $B(Q^{\mathrm{IV}}1)=1.00 \times10^{-1} e^{2}$fm$^{2}$, and $\sum|Y_{\alpha\beta}|^{2}=2.93\times 10^{-3}$. []{data-label="26Ne_1-"}
In Fig. \[strength\], we show the transition strengths in the low-energy region. The neutron emission threshold is 6.35 MeV, and the resonance which is composed of several discrete states appears just above the threshold. In contrast to the low-lying quadrupole state in $^{22}$O, the transition strengths for the dipole states in this region converge at the cutoff energy of about 40 MeV. We made a detailed analysis of the QRPA eigenmodes and show in Tables \[26Ne\_0-\],\[26Ne\_1-\] the microscopic structures of the $K^{\pi}=0^{-}$ state at 8.25 MeV and the $K^{\pi}=1^{-}$ state at 8.76 MeV, which have the largest transition strength for each sec
| 3,372
| 2,033
| 3,828
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0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Hence $\ker {\hat{T}}^{\chi }_{p,\Lambda }=U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{\Lambda '}$ and ${\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda } = U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $.
For all $w\in {\mathrm{Aut}}({\mathbb{Z}}^I)$ and ${\alpha }\in {\mathbb{Z}}^I$ let $w(e^{\alpha })=e^{w({\alpha })}$, and extend this definition linearly on formal characters. We investigate the effect of the maps ${\hat{T}}_p,{\hat{T}}^-_p$ on formal characters. For all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$ with $\bfun{\chi '}({\alpha }_i)<\infty $ let ${\dot{\sigma }}^{\chi '}_i:{\mathbb{Z}}^I\to {\mathbb{Z}}^I$ be the affine transformation $$\begin{aligned}
{\dot{\sigma }}_i^{\chi '}({\alpha })={\sigma }_i^{\chi '}({\alpha })
+(1-\bfun{\chi '}({\alpha }_i)){\alpha }_i.
\label{eq:sdot}\end{aligned}$$ Note that then ${\dot{\sigma }}_i^{r_i(\chi ')}{\dot{\sigma }}_i^{\chi '}({\alpha })={\alpha }$ for all ${\alpha }\in {\mathbb{Z}}^I$.
\[le:Tpfch\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and ${\alpha }\in {\mathbb{Z}}^I$. Then $$\begin{aligned}
\label{eq:Tpweight+}
{\hat{T}}_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha })
\subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })},\\
\label{eq:Tpweight-}
{\hat{T}}^-_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha })
\subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })}.
\end{aligned}$$ In particular, $$\begin{aligned}
\fch{M^\chi (\Lambda )} =
{\dot{\sigma }}_p^{r_p(\chi )}
(\fch{M^{r_p(\chi )}({t}_p^\chi (\Lambda ))}).
\label{eq:fchM}
\end{aligned}$$
Let $\Lambda '={t}^\chi _p(\Lambda )$ and $u\in U(r_p(\chi ))_{\alpha }$. Then $${\hat{T}}_p(uv_{\Lambda '})={T}_p(u)F_p^{{b}-1}v_\Lambda
\in U(\chi )_{{\sigma }_p^{r_p(\chi )}({\alpha })}U(\chi )_{(1-{b}){\alpha }_p}v_\Lambda$$ by Prop. \[pr:LTdeg\]. This proves Eq. , since $v_\Lambda \in M^\chi (\Lambda )_0$, The proof of Eq. is similar. B
| 3,373
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| 2,625
| 3,080
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t explicitly discuss in this article) and Bayesians. The categorization used here is motivated by a short blog post (see <http://labstats.net/articles/overview.html>). ]{}]{}
[^4]: [[Note that this method also utilizes mechanisms usually known from the frequentist school; i.e., hypothesis testing.]{}]{}
[^5]: [[We could tackle this problem by e.g., applying the *Bonferroni correction* which we leave open for future work]{}]{}
[^6]: [[In order to confirm our findings we also applied an additional way of determining the accuracy which is motivated by a typical evaluation technique known from link predictors [@liben]. Concretely, it counts how frequently the true next click is present in the TopK (k=5) states determined by the probabilities of the transition matrix. In case of ties in the TopK elements we randomly draw from the ties. By applying this method to our data we can mirror the evaluation results obtained by using the described and used ranking technique. Note that we do not explicitly report the additional results of this evaluation method throughout the paper.]{}]{}
[^7]: <http://thewikigame.com/>
[^8]: <http://en.wikipedia.org/wiki/Category:Main_topic_classifications>
[^9]: <http://www.cs.mcgill.ca/~rwest/wikispeedia/>
[^10]: <http://schools-wikipedia.org/>
[^11]: <http://kdd.ics.uci.edu/databases/msnbc/msnbc.html>
[^12]: <http://msnbc.com>
[^13]: Except for teleportation which we do not model in this work.
[^14]: Both priors agree throughout all our investigations in this article.
[^15]: [[Consequently, we might get better representations of the data by using Markov chain models that, instead modeling state transitions in equal time steps, additionally stochastically model the duration times in states (e.g., semi Markov or Markov renewal models). However, we leave these investigations open for future work.]{}]{}
[^16]: <https://github.com/psinger/PathTools>
---
author:
- 'Quanshi Zhang, Ruiming Cao, Ying Nian Wu, and Song-Chun Zhu *Fellow, IEEE*'
bibliography:
- 'TheBib.bib'
title: Mining
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\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1$, $J^1= {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]A^1$, ,
$L_c(\mu)$, simple factor of $\Delta_c(\mu)$,
${\mathcal{L}}_1 ={\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}(1)$, ${\mathcal{L}}={\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$, ,
$\mathcal{O}_c$, category $\mathcal{O}$ for $H_c$,
$\widetilde{{\mathcal{O}}}_c$, graded category $\mathcal{O}$ for $H_c$,
$[n]_v! = (1-v)^{-n}\prod_{i=1}^n (1-v^i)$,
$N(k)=B_{k0}eH_c$,
$\overline{N(k)} = {\mathbb{C}}\otimes N(k)$, $\underline{N(k)} = N(k)\otimes {\mathbb{C}}$,
$\operatorname{{\textsf}{ord}},\operatorname{{\textsf}{ogr}}$, order filtration and order gradation,
${\mathcal{P}}_1$, ${\mathcal{P}}$, the rank $n!$ Procesi bundles, ,
$p(M,v)$, Poincaré series,
$p(V,s,t)$, bigraded Poincaré series,
$p(M,v,{{W}})$, ${{W}}$-graded Poincaré series,
$\operatorname{{\textsf}{qgr}}$, $\operatorname{{\textsf}{Qgr}}$, quotient categories,
$Q^{c+1}_{c} = eH_{c+1}e_-\delta =e H_{c+1}\delta e$,
$\mathbb{R}(n,l)= {\rm H}^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1\otimes {\mathcal{B}}_1^{l})$,
$\rho_1: {\mathbb{X}}_n\to \operatorname{Hilb^n{\mathbb{C}}^2}$, $\rho: X_n\to \operatorname{Hilb(n)}$, ,
$\mathbb{S} = \oplus \mathbb{J}^i$, $S=\oplus {J}^i$, ,
$S_q = S_q(n,n)$, $q$-Schur algebra,
$\mathcal{S}$, the reflections in ${{W}}$,
$\sigma(r)$, the principal symbol of $r$,
$\operatorname{{\textsf}{sign}}$, the sign representation of ${{W}}$,
Specht module $Sp_q(\mu)$,
$ \tau: \operatorname{Hilb^n{\mathbb{C}}^2}\rightarrow {\mathbb{C}}^{2n}/{{W}}$,
$ \tau: \operatorname{Hilb(n)}\rightarrow {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$,
$\operatorname{{\textsf}{triv}}$, the trivial representation of ${{W}}$,
$U_c=eH_{c}e$, the spherical subalgebra,
$U_c^-=e_-H_ce_-$, the anti-spherical subalgebra,
${{W}}=\mathfrak{S}_n$, the symmetric group,
$\mathbb{X}_n$, $X_n$, isospectral Hilbert schemes, ,
[GGOR]{}
E. Backelin and K. Kremnitzer, Quantum flag varieties, equivariant quantum $\mat
| 3,375
| 2,419
| 1,113
| 3,333
| 3,593
| 0.771324
|
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|
given in (\[eqn:JS\]), when $\al_1=\cdots=\al_r=0$ and $\be=\be^{JS}$, and to $\Thh_{EM}$, given in (\[eqn:EM\]), when $\al_1=\cdots=\al_r=\al^{EM}$ and $\be=0$. In estimation of the normal mean matrix relative to the quadratic loss $L_Q$, $\Thh_{JS}$ and $\Thh_{EM}$ are minimax.
If $\Thh_{MS}$ with certain specified $\al_1,\ldots,\al_r$ and $\be$ has good performance, the prior density $\pi_{SH}$ with the same $\al_1,\ldots,\al_r$ and $\be$ would produce a good Bayesian predictive density. From Tsukuma (2008), $\Thh_{MS}$ is a minimax estimator dominating $\Thh_{EM}$ when $\al_i=\al_i^{ST}$ for $i=1,\ldots,r$ and $0\leq \be\leq 4(r-1)$. A reasonable choice for $\be$ is $\be^{MS}=2(r-1)$ and this suggests that we should consider a prior density of the form $$\label{eqn:pr_MS2}
\pi_{MS2}(\Th)
=\{\tr(\Th\Th^\top)\}^{-\be^{MS}/2}\prod_{i=1}^r \la_i^{-\al_i^{ST}/2}
=\pi_{MS1}(\Th)\prod_{i=1}^r \la_i^{-\al_i^{ST}/4}.$$ The prior density $\pi_{MS2}(\Th)$ is not superharmonic, and it is not known whether the resulting Bayesian predictive density is minimax or not. In the next section, we verify risk behavior of the Bayesian predictive density with respect to $\pi_{MS2}(\Th)$ through Monte Carlo simulations.
Monte Carlo studies {#sec:MCstudies}
===================
This section briefly reports some numerical results so as to compare performance in risk of some Bayesian predictive densities for $r=2$ and $q=15$.
First we investigate risk behavior of generalized Bayes predictive densities $\ph_{GB}(Y|X)$ with $v_0=1$ in the following six cases: $$(a,\, b)=(-11,\, 3),\ (-11,\, 9),\ (-11,\, 15),\ (-5,\, 3),\ (-5,\, 9),\ (1,\, 3)$$ for the second-stage prior (\[eqn:pr\_GB\]). When $r=2$ and $q=15$, $\ph_{GB}(Y|X)$ with the above six cases are minimax and, in particular, $\ph_{GB}(Y|X)$ with $(a,\, b)=(1,\,3)$ is proper Bayes for any $v_x$ and $v_y$ (see Corollary \[cor:proper2\]).
The risk has been simulated by 100,000 independent replications of $X$ and $Y$, where $X|\Th\sim\Nc_{r\times q}(\Th, v_xI_r\otimes I_q)$ and $Y
| 3,376
| 2,106
| 1,061
| 3,286
| 3,359
| 0.772882
|
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|
uations should be compared with the corresponding exact ones of Appendix A4. We shall see that the net of functions (\[Eqn: phieps\]) converges graphically to the multifunction Eq. (\[Eqn: singular\_eigen\]) as $\varepsilon\rightarrow0$.
In the discretized spectral approximation., the singular eigenfunction $\phi(\mu,\nu)$ is replaced by $\phi_{\varepsilon}(\mu,\nu)$, $\varepsilon\rightarrow0$, with the integral in $\nu$ being replaced by an appropriate sum. The solution Eq. (\[Eqn: CaseSolution\_HR\]) of the physically interesting half-space $x\geq0$ problem then reduces to [@Sengupta1988; @Sengupta1995] $$\Phi_{\varepsilon}(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})e^{-x/\nu_{i}}\phi_{\varepsilon}(\mu,\nu_{i})\qquad\mu\in[0,1]\label{Eqn: DiscSpect_HR}$$
where the nodes $\{\nu_{i}\}_{i=1}^{N}$ are chosen suitably. This discretized spectral approximation to Case’s solution has given surprisingly accurate numerical results for a set of properly chosen nodes when compared with exact calculations. Because of its involved nature [@Case1967], the exact calculations are basically numerical which leads to nonlinear integral equations as part of the solution procedure. To appreciate the enormous complexity of the exact treatment of the half-space problem, we recall that the complete set of eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ are orthogonal with respect to the half-range weight function $W(\mu)$ of half-range theory, Eq. (\[Eqn: W(mu)\]), that is expressed only in terms of solution of the nonlinear integral equation Eq. (\[Eqn: Omega(-mu)\]). The solution of a half-space problem then evaluates the coefficients $\{ a(\nu_{0}),a(\nu)_{\nu\in[0,1]}\}$ from the appropriate half range (that is $0\leq\mu\leq1$) orthogonality integrals satisfied by the eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ with respect to the weight $W(\mu)$, see Appendix A4 for the necessary details of the half-space problem in neutron transport theory.
As ma
| 3,377
| 2,951
| 3,715
| 3,285
| 3,851
| 0.769696
|
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|
of $E_{2}$ converge with the initiator approximation.
In Fig.\[fig1:compinit-alpha\] we plotted the second order energy for the Carbon dimer with the cc-pVQZ basis computed with the QMC-LCC framework. The black line correspond to the simulations presented in fig.\[fig1:swalknum\] i.e. 50k walkers and a initiator threshold of 3 in replica 0 and an initiator criterion of 1, 3 and no initiator approximation for the first order response function. It can be seen that without approximation, the computed energy is in agreement with the one computed using MPS-LCC, shown in blue for reference. When the initiator approximation is used, we obtain a second order energy that is slightly higher than the correct one. However the estimation remains quite good since the error in the energy is lower than 1 $mE_{H}$ with the two criteria used here.
![Comparison of the E2 energy with respect to the number of walkers. The black curve is obtained by setting the initiator threshold to different values, respectively 3,1 and no initiator threshold. The blue curve is obtained by setting the initiator threshold to 1 and controlling the number of walkers on $\protect\Ket{\Psi_{1}}$ by using the $\alpha$ controlling parameter. For comparison purpose, the value obtained using DMRG is in red.\[fig1:compinit-alpha\]](fig3){width="80.00000%"}
In Fig.\[fig1:compinit-alpha\], the blue curve is obtained with an initiator threshold of 1 and different values of $\alpha$ to constrained the number of walkers. The most interesting feature is that, for the same number of walkers on $\Ket{\Psi_{1}}$ it seems to be better to actually set a smaller initiator threshold and to use the $\alpha$ trick than increasing the initiator threshold. For instance for roughly 1M walkers the value obtained with initiator threshold of 1 and an $\alpha$ value of $\approx0.73$ is 0.16 $mE_{H}$ lower than the one obtained by using an initiator of 3.
In order to improve further the efficiency of our implementation, we notice that the ${\cal H}_{1}$ space can be split as t
| 3,378
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| 3,002
| null | null |
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is in the ballpark of the redshift range where the expansion of the Universe apparently made a transition from deceleration to acceleration [@decel].
The result (\[total3\]) was derived in an oversimplified case, where the possible effects of other branes and orientifolds were not taken into account. However, as we now show, the ideas emerging from this simple example persist in more realistic structures. As argued in [@Dfoam], the presence of orientifolds, whose reflecting properties cause a D-brane on one side of the orientifold plane to interact non-trivially with its image on the other side, and the appropriate stacks of D8-branes are such that the *long-range contributions* of the D-particles to the D8-brane energy density *vanish*. What remain are the short-range D0-D8 brane contributions and the contributions from the other D8-branes and O8 orientifold planes. The latter are proportional to the fourth power of the relative velocity of the moving D8-brane world: $$\mathcal{V}_{\rm{long} \,, \,D8-D8, , D8-O8} = V_8\frac{\left(aR_0 - b R_2(t)\right)v^4}{2^{13}\pi^9 {\alpha '}^5}~,
\label{vlongo8d8}$$ where $V_8$ is the eight-brane volume, $R_0$ is the size of the orientifold-compactified ninth dimension in the arrangement shown in Fig. \[fig:inhom\], and the numerical coefficients in (\[vlongo8d8\]) result from the relevant factors in the appropriate amplitudes of strings in a nine-dimensional space-time. The constants $a > 0$ and $b >0$ depend on the number of moving D8-branes in the configuration shown in Fig. \[fig:inhom\]. If there are just two moving D8-branes that have collided in the past, as in the model we consider here, then $a=30$ and $b = 64$ [@Dfoam]. The potential (\[vlongo8d8\]) is positive during late eras of the Universe as long as $R_2(t) < 15R_0/32$. One must add to (\[vlongo8d8\]) the negative contributions due to the D-particles near the D8-brane world, so the total energy density on the 8-brane world becomes: $$\mathcal{\rho}_{8} \equiv \frac{\left(aR_0 - b R_2(t)\right)v^4}{2^{1
| 3,379
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| 4,068
| 3,250
| null | null |
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C^c$ OPE will look like: $$\begin{aligned}
\dots \kappa^{ac} :j_{e\bar z} j_{z}^e :
+
\dots (:j^a_{\bar z} j^c_{ z} :
+
(-1)^{ac} :j^c_{\bar z} j^a_{ z} :).\end{aligned}$$ The first term is proportional to a component of the energy-momentum tensor (and to the kinetic term in the Lagrangian). The other term indicates that at higher order, we need a new four-tensor index structure in the current-current operator product expansion. At the same time, the special properties of the linear operator given above show that only few four-tensors will appear. It is certainly feasible to push the above calculation, and therefore the other calculations in the bulk of the paper to higher order.
The Virasoro algebra from the current algebra {#TjandTT}
---------------------------------------------
In [@Ashok:2009xx] it was shown that the Virasoro algebra emerges from the current algebra via the Sugawara construction. More precisely it was argued that the normal ordered classical expression for the stress tensor : T = :j\_[L,zb]{} j\^[b]{}\_[L,z]{}: satisfies the OPEs : \[TjApp\] T(z) j\^a\_[L,z]{}(w) = + + (z-w)\^0 \[TTApp\] T(z) T(w) = + + + (z-w)\^0. In this section, we fill a gap in the demonstration of equation . We reconsider the OPE between a current and the bilinear operator $:j_{L,zb} j^{b}_{L,z}:$. To perform this computation in [@Ashok:2009xx] we truncated the current algebra at the order of the poles. We obtained : \[j:jj:\] j\_[L,z]{}\^a(z) :j\_[L,zb]{} j\^b\_[L,z]{}:(w) &=& 2c\_1 + c\_2 ( (-1)\^[bc]{} :j\^b\_[L,z]{} j\^c\_[L,z]{}:+ :j\^c\_[L,z]{} j\^b\_[L,z]{}:(w)) &&+ (c\_2-g) [f\^a]{}\_[bc]{} ( (-1)\^[bc]{}:j\^b\_[L,z]{} j\^c\_[L,|z]{}:+: j\^c\_[L,|z]{} j\^b\_[L,z]{}:(w) )\
& & +... where the ellipses contain terms of order zero in the distance between the insertion points $z$ and $w$. We will now show that the subleading terms in the current algebra do not modify this result. Let us divide these terms into two sets. First we have the regular terms and the terms that multiply an $n^{th}$-derivative of a single c
| 3,380
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| 3,521
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ction of $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ to $\Lambda_{\leq 2}
\subset \Lambda$. In other words, we do not need to compute the full $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ and to construct a full resolution $P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ of the cyclic $A$-bimodule $A_\#$; it suffices to construct $P^i_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}= P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}([i])$ for $i=1,2$ (and then apply the functor ${\operatorname{\sf tr}}$).
With the choices made above, we set $P^1_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}= P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, and we let $P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ be the cone of the map $$\begin{CD}
P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}@>{(\tau \boxtimes {\operatorname{\sf id}})\oplus({\operatorname{\sf id}}\boxtimes \tau)}>> (A \boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}})\oplus(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\boxtimes
A).
\end{CD}$$ The involution $\sigma:[2] \to [2]$ acts on $P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ in the obvious way. We also need to define the transition maps $\iota_f$ for the two injections $d,d':[1] \to [2]$ and the two surjections $s,s':[2] \to [1]$. For $d_1$, the transition map $\iota_d:A
\boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ is the obvious embedding, and so is the transition map $\iota_{d'}$. For the surjection $s$, we need a map $\iota_s$ from the cone of the map $$\begin{CD}
P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}@>{(\tau \otimes {\operatorname{\sf id}})\oplus({\operatorname{\sf id}}\otimes \tau)}>> P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\oplus P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\end{CD}$$ to $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$. On $P_{{\:\raisebox{1pt
| 3,381
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he remaining variables becomes H\_0= \_[ij]{} K\_[ab]{} \_[ij]{} \_a \_[ij]{} \_b - \_i u(\_2-\_1), \[2N0\] where the matrix $K_{ab}, \, a,b =1,2$ is related to the original parameters as K\_[11]{}&=& , K\_[22]{}= ,\
K\_[12]{} &=& - . \[Ktg\] . The stability of $H_0$ is guaranteed by K\_[11]{}K\_[22]{} - K\_[12]{}\^2= >0. \[stab\] The condition for SP can be obtained along the lines of the logic [@Wen; @Lubensky; @Toner; @Sondhi; @Kane_2001; @Ashwin_2001] which ignores the compactness of $\phi_{1,2}$. Specifically, introducing the variables $\varphi =\phi_1 + \phi_2$ and $\theta= \phi_2 -\phi_1$ and, then, integrating out $\varphi$, the resulting partition function becomes Z\_0=D\^[-H\_]{}, H\_ = d\^2 x , \[Z\_0\] where the notation K= \[K\] is introduced and the long wave limit is considered – so that the summation along the layers is replaced by the integration $\int d^2 x ...$.
As long as the compactness of $\theta$ is ignored, Eqs. (\[Z\_0\]) represent the standard Sine-Gordon model in 2D. The RG analysis predicts (see in, e.g., [@Lubensky_book]) that at $K<K_d= 1/(8\pi)$ the renormalization renders the Josephson coupling $u$ irrelevant in the thermo-limit $L\to \infty$. More specifically, the renormalized $u$ should flow to zero as $u_r \sim u L^{b}\to 0,\, b= 2(1- K_d/K)<0$. Such a behavior is supposed to occur together with the persistence of the algebraic order along the planes. This requirement imposes further restrictions on the values of $K_{ab}$.
Without loss of generality let’s assume $K_{11} <K_{22}$ and introduce the notations: $T=1/K_{11}$ as a measure of temperature, and $Y=K_{22}/K_{11} >1,\, X=K_{12}/K_{11} $. Then, the condition $K<1/(8\pi)$ for SP becomes T>T\_d=. \[SSP\]
In order to guarantee the algebraic order in each layer no BKT transition should occur in the layers. In order to determine possible types of vortices responsible for the transition, we examine the form (\[2N0\]) by “reinstating” the compactness of the variables in the limit $u=0$ (which is supposed to renor
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care whether the number of points in a neighborhood is finite or infinite.
Previous construction
=====================
We recall the construction from [@MMS]. Fix a basepoint $x_0\in X$. Given $N>0$, an $N$-sequence in $X$ based at $x_0$ is an infinite list $x_0,x_1,\ldots$ of points in $X$ with $d(x_i,x_{i+1})\leq N$ for each $i\geq 0$. Since we are interested in the large scale structure of $X$, we are only interested in sequences that go to infinity. An $N$-sequence $x_0,x_1,\ldots$ goes to infinity if $d(x_0,x_i)\to\infty$. Let ${\text{S}}_N(X,x_0)$ be the set of all $N$-sequences in $X$ based at $x_0$ that go to infinity.
We call two sequences $s,t\in {\text{S}}_N(X,x_0)$ equivalent if there is a finite list $s_0,
\ldots,s_n\in {\text{S}}_N(X,x_0)$ with $s_0=s$, $s_n=t$, and for each $i\geq 0$, $s_{i+1}$ is either a subsequence of $s_i$ or $s_i$ is a subsequence of $s_{i+1}$. If $s_i$ is a subsequence of $s_{i+1}$ we say $s_{i+1}$ is a supersequence of $s_i$. Let $[s]_N$ denote the equivalence class of $s$ in ${\text{S}}_N(X,x_0)$ and let $\sigma_N(X,x_0)$ be the set of equivalence classes.
The cardinality of the set $\sigma_N(X,x_0)$ is the desired invariant. It essentially determines the number of different ways of going to infinity in $X$. Since this cardinality depends on $N$, we have the following definition. For each integer $N>0$ there is a function $\phi_N:\sigma_N(X,x_0)\to\sigma_{N+1}(X,x_0)$ that sends the equivalence class $[s]_N$ to the equivalence class $[s]_{N+1}$. $X$ is said to be $\sigma$-stable if there is a $K>0$ for which $\sigma_N$ is a bijection for each integer $N\geq K$. If $X$ is $\sigma$-stable let $\sigma(X,x_0)$ denote the cardinality of $\sigma_K(X,x_0)$.
It would be better to call $X$ “$\sigma$-stable with respect to $x_0$” since apparently this definition depends on basepoint. In fact it does not; this issue is addressed in the next section.
The following is the main theorem of [@MMS]. It is the theorem that we wish to extend to coarse equivalences.
Suppose $f:X\to
| 3,383
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|
a_1\eta_2+h.c.,\end{aligned}$$ where $$\begin{aligned}
y_{N_R} &=
\left[\begin{array}{ccc}
\alpha_\nu &0 & 0 \\
0 & \beta_\nu &0 \\
0& 0 & \gamma_\nu \\
\end{array}\right]
\left[\begin{array}{ccc}
y_{1} & y_{3} &y_{2} \\
y_{3} & y_{2} &y_{1} \\
y_{2} & y_{1} & y_{3} \\
\end{array}\right],\\
y_{N_L} &=
\left[\begin{array}{ccc}
y_{6,1} & y_{6,3} &y_{6,2} \\
y_{6,3} & y_{6,2} &y_{6,1} \\
y_{6,2} & y_{6,1} & y_{6,3} \\
\end{array}\right]^*
\left[\begin{array}{ccc}
a_\nu &0 & 0 \\
0 & 0 & c_\nu \\
0& b_\nu & 0 \\
\end{array}\right]
+
\left[\begin{array}{ccc}
y'_{6,1} & y'_{6,3} &y'_{6,2} \\
y'_{6,3} & y'_{6,2} &y'_{6,1} \\
y'_{6,2} & y'_{6,1} & y'_{6,3} \\
\end{array}\right]^*
\left[\begin{array}{ccc}
a'_\nu &0 & 0 \\
0 & 0 & c'_\nu \\
0& b'_\nu & 0 \\
\end{array}\right],
\label{eq:mn}\end{aligned}$$ where ${\bf Y^{(6)}_3}\equiv[ y_{6,1},y_{6,2},y_{6,3}]^T$, ${\bf
Y'^{(6)}_3}\equiv[ y'_{6,1},y'_{6,2},y'_{6,3}]^T$, and we impose the perturbativity limit, ${\rm Max}[y_{N_{R,L}}]\lesssim\sqrt{4\pi}$, in the numerical analysis.
Then, the neutrino mass matrix is calculated as $$\begin{aligned}
&m_{\nu}=\kappa [y_{N_R} y_{N_L} +(y_{N_R} y_{N_L})^T]\equiv \kappa \tilde m_\nu,\\
&\kappa=\frac{\lambda_0^* v_{SM}^2}{32\pi^2M_D} F(M_D^2,\eta_1^2,\eta_2^2),\\
& F(a,b,c)=-a\frac{(b-c)a\ln a+(c-a)b\ln b+(a-b)c\ln c}{(a-b)(b-c)(c-a)},\end{aligned}$$ where $v_{SM}\simeq$ 246 GeV is the VEV of the SM Higgs boson. Notice that $\kappa$ does not depend on the flavor structure due to the modular symmetry.
Thus, we determine $\kappa$ by $$\begin{aligned}
(NH):\ \kappa^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_3}^2-\tilde D_{\nu_1}^2},
\quad
(IH):\ \kappa^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_2}^2-\tilde D_{\nu_3}^2},
\end{aligned}$$ where $\tilde m_\nu$ is diagonalized by $V^\dag_\nu (\tilde m_\nu^\dag
\tilde m_\nu)V_\nu=(\tilde D_{\nu_1}^2,\tilde D_{\nu_2}^2,\tilde
D_{\nu_3}^2)$ and $\Delta m_{\rm atm}^2$ is the atmospheric neutrino mass-squared difference. Here, NH and IH stand for the n
| 3,384
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|
g\}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
{\label{eq:Theta[1]-rewr}}
&\Theta_{y,x;{{\cal A}}}-\Theta'_{y,x;{{\cal A}}}{\nonumber}\\
&=\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}\}$}}}.\end{aligned}$$
It remains to bound the right-hand side of [(\[eq:Theta\[1\]-rewr\])]{}, which is nonzero only if $m_b$ is even and $n_b$ is odd, due to the source constraints and the conditions in the indicators. First, as in [(\[eq:2nd-ind-fact\])]{}, we alternate the parity of $n_b$ by changing the source constraint into ${\partial}{{\bf n}}=y{\vartriangle}b{\vartriangle}x$ and multiplying by $\tau_b$. Then, by conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)$ as in [(\[eq:3rd-ind-prefact\])]{} (i.e., conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)={{\cal B}}$, letting ${{\bf m}}'={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf m}}''={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf n}}'={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\bf n}}''={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, and then summing over ${{\cal B}}\subset\Lambda$), we obtain $$\begin{aligned}
{\label{eq:2ndind-contr}}
\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}'={\varnothing}\\ {\partial}{{\bf n}}'=
{\overline{b}}{\vartriangle}x}}\frac{\tilde w_{{{\cal A}}{^{
| 3,385
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|
{i} ) ( \Delta_{K} - h_{i} ) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{i} \right) e^{- i \Delta_{J} x}
- \left( \Delta_{J} - h_{i} \right) e^{- i \Delta_{K} x}
- \left( \Delta_{K} - \Delta_{J} \right) e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W^{\dagger} A W \right\}_{K J}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{J} - h_{i} )^2 }
\biggl[
(ix) \left( e^{- i h_{i} x} + e^{- i \Delta_{J} x} \right)
+ 2 \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\nonumber \\
&+&
\sum_{k \neq i}
\biggl[
- \frac{ (ix) e^{- i \Delta_{J} x} }{ ( \Delta_{J} - h_{i} )( \Delta_{J} - h_{k} ) }
+ \frac{ 1 }{ ( h_{i} - h_{k} ) (\Delta_{J} - h_{i} )^2 (\Delta_{J} - h_{k} )^2 }
\nonumber \\
&\times&
\biggl\{
(\Delta_{J} - h_{k} )^2 e^{- i h_{i} x}
- (\Delta_{J} - h_{i} )^2 e^{- i h_{k} x}
+ ( h_{i} - h_{k} )( h_{i} + h_{k} - 2 \Delta_{J} ) e^{- i \Delta_{J} x}
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\left\{ W^{\dagger} A (UX) \right\}_{J k}
\left\{ (UX)^{\dagger} A W \right\}_{k J}
\nonumber \\
&+&
\sum_{K \neq J}
\biggl[
- \frac{ (ix) e^{- i h_{i} x} }{ (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) }
+ \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) (\Delta_{J} - h_{i})^2 ( \Delta_{K} - h_{i} )^2 }
\nonumber \\
&\times&
\biggl\{
( \Delta_{K} - h_{i} )^2 e^{- i \Delta_{J} x}
- (\Delta_{J} - h_{i})^2 e^{- i \Delta_{K} x}
+ e^{- i h_{i} x} (\Delta_{J} - \Delta_{K}) (\Delta_{J} + \Delta_{K} - 2 h_{i} )
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W^{\dagger} A (UX) \right\}_{K i}
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\nonumber \\
&+&
\sum_{K \neq J}
\sum_{k \neq i}
\frac{ 1 }{ (\Delta_{J} - \Delta_{K}) ( h_{k} - h_{i} ) (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) (\Delta_{K} - h_{
| 3,386
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\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=
\oint_{C(t)}\frac{1}{\rho}{\frac{\partial \ell}{\partial a}}\diamond a\cdot {\mathrm{d}}\MM{x}.$$ For the incompressible Euler equations, $a$ is the relative density $\rho$, so $${\frac{\partial \ell}{\partial a}}\diamond a = \rho\nabla{\frac{\partial \ell}{\partial \rho}},$$ which leads to the circulation theorem $${\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=\oint_{C(t)}\nabla
{\frac{\partial \ell}{\partial \rho}}\cdot{\mathrm{d}}\MM{x}=0.$$
A note on multisymplectic integrators {#numerics}
=====================================
In this section we discuss briefly how to produce multisymplectic numerical integrators, using the inverse map formulation given in this paper. We note in particular that the multisymplectic method will satisfy a discrete form of the particle-relabelling symmetry and hence we will obtain a method that has discrete conservation laws for $-\MM{\pi}\cdot\nabla\MM{l}$.
Variational integrators
-----------------------
A multisymplectic integrator for a PDE is a numerical method which preserves a discrete conservation law for the two-form $\kappa$ given in equation (\[kappa\]) (Bridges & Reich, 2001). As described in (Hydon, 2005), a discrete variational principle with a Lagrangian that is affine in first-order differences automatically leads to a set of difference equations which are multisymplectic. This now makes it very simple to construct multisymplectic integrators for fluid dynamics using the inverse map formulation: one simply replaces the spatial and time integrals in the action with numerical quadratures, replaces the first-order derivatives by differences, and takes variations following the standard variational integrator approach (Lew *et al.*, 2003). Whilst the method will preserve the discrete conservation law for the two-form $\kappa$, the one-form quasi-conservation law will not be preserved in general, and hence the other conservation laws will not be exactly preserved.
Discrete relabelling s
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4]. The character $\calU_\tau$ of $L_\lambda$ decomposes as,
$$\calU_\tau=|W_\lambda|^{-1}\sum_{w\in W_\lambda}\chi^\tau_w\cdot R_{T_w}^{L_\lambda}(1)$$where $\chi^\tau_w$ denotes the value of $\chi^\tau$ at $w$. Applying the Harish-Chandra induction $R_{L_\lambda}^G$ to both side and using the transitivity of induction we find that
$$R_{L_\lambda}^G(\calU_\tau)=|W_\lambda|^{-1}\sum_{w\in W_\lambda}\chi^\tau_w\cdot R_{T_w}^G(1).$$We are now in position to use the calculation in [@hausel-letellier-villegas]. Notice that the right handside of the above formula is the right hand side of the first formula displayed in the proof of [@hausel-letellier-villegas Theorem 4.3.1] with $(M,\theta^{T_w},\tilde{\varphi})=(L_\lambda,1,\chi^\tau)$ and so the same calculation to get [@hausel-letellier-villegas (4.3.2)] together with [@hausel-letellier-villegas (4.3.3)] gives in our case
$$R_{L_\lambda}^G(\calU_\tau)(C)=\sum_\alpha z_\alpha^{-1}\chi^\tau_\alpha\sum_{\{\beta\,|\,[\beta]=[\alpha]\}}Q_\beta^\omega(q)z_{[\alpha]}z_\beta^{-1}$$where the notation are those of [@hausel-letellier-villegas § 4.3]. We now apply [@hausel-letellier-villegas Lemma 2.3.5] to get
$$R_{L_\lambda}^G(\calU_\tau)(C)=\left\langle \tilde{H}_\omega(\x;q),s_\tau(\x)\right\rangle.$$
If $\alpha$ is the type $(1,(\lambda_1))\cdots(1,(\lambda_r))$, then $s_\alpha(\x)=h_\lambda(\x)$. Hence we have:
If $C$ is a conjugacy class of $G$ type $\omega$, then
$$R_{L_\lambda}^G(1)(C)=\left\langle \tilde{H}_\omega(\x,q),h_\lambda(\x)\right\rangle.$$ \[R\]
Put $\calF^\#_{\lambda,\omega}(q):=\#\{X\in\calF_\lambda\,|\, g\cdot X=X\}$ where $g\in G$ is an element in a conjugacy class of type $\omega$. Then
$$\tilde{H}_\omega(\x,q)=\sum_\lambda \calF^\#_{\lambda,\omega}(q)m_\lambda(\x).$$
It follows from Lemma \[R=F\] and Corollary \[R\].
We now recall how to construct from a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ of $n$ a certain family of irreducible characters of $G$. Choose $r$ distinct linear character $\alpha_1,\dots,\alpha_r$ of $\F_q^{\times}$.
| 3,388
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M\geq 0,\ c>0$, \[csda38\] |B(u,v)|M[u]{}\_[X]{}[v]{}\_[Y]{}uX, vY ([boundedness]{}) and \[csda37a\] B(v,v)c[v]{}\_[X]{}\^2vY ([coercivity]{}). Suppose that $F:X\to{\mathbb{R}}$ is a bounded linear form. Then there exists $u\in X$ (possibly non-unique) such that \[csda38-a\] B(u,v)=F(v)vY.
See e.g. [@treves p. 403] or [@grisvard p. 234].
Let $$P(x,\omega,E,D)\phi:= -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi$$ The space $$\begin{gathered}
{{{\mathcal{}}}H}_P(G\times S\times I^\circ):=\{\phi\in L^2(G\times S\times I)\ | \\
P(x,\omega,E,D)\phi\in L^2(G\times S\times I)\ {\rm in\ the \ weak\ sense}\}\end{gathered}$$ is a Hilbert space when equipped with the inner product (cf. section \[m-d\]) $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}_P(G\times S\times I^\circ)}={\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}P(x,\omega,E,D)\phi,P(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}.$$
With this notation, the equations and can be written as $$\begin{aligned}
P(x,\omega,E,D)\psi+\Sigma\psi - K\psi = f,\end{aligned}$$ and $$\begin{aligned}
P(x,\omega,E,D)\phi+C S_0\phi+\Sigma\phi-K_C\phi=e^{CE}f,\end{aligned}$$ respectively.
In the context if Lions-Lax-Milgram Theorem we shall make us of the following assumption which we call as ${\bf TC}$:
\[as:TC\] Let $\gamma_{\pm}(\phi)=\phi_{|\Gamma_{\pm}}$ and $\gamma_{\rm m}(\phi):=\phi(\cdot,\cdot,E_{\rm m}),\ \gamma_{0}(\phi):=\phi(\cdot,\cdot,0)$. The assumption is that the linear maps $$\begin{aligned}
\gamma_{\pm}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(\Gamma_{\pm},|\omega\cdot\nu| d\sigma d\omega dE), \\
\gamma_{\rm m}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(G\times S), \\
\gamma_{0}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(G\times S),\end{aligned}$$ are well-defined and continuous.
In the case where $S=S(E)$ is independent of $x$ and $S\in C(I)$ one can show that the assumption ${\bf TC}$ holds. The proof can be based on the
| 3,389
| 1,135
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entrally-extended two-dimensional abelian algebra, demonstrating that this is equivalent to a TsT transformation of the full supergravity background. There are additional classes of deformations that can be constructed as non-abelian T-duals. These come from considering particular non-semisimple subalgebras of $\mathfrak{su}(2,2) \oplus \mathfrak{su}(4)$, whose existence relies on the non-compactness of $\mathfrak{su}(2,2)$. There are a number of such algebras that are non-abelian and admit central extensions [@Borsato:2016ose], such that when we T-dualise the metric with respect to this centrally-extended subalgebra we find a deformation of the original metric [@Hoare:2016wsk; @Borsato:2016pas] that coincides with a certain Yang-Baxter deformation. To illustrate this richer story we present a summary of two examples showing how the techniques described in this paper also apply, i.e. the R-R fluxes following from non-abelian T-duality agree with those of the Yang-Baxter $\sigma$-model.
An $r$-matrix $$r = r^{ab} T_a \wedge T_b \ ,$$ is said to be non-abelian if $[T_a , T_b]\neq 0$ for at least some of the generators. An $r$-matrix is said to be unimodular if $$r^{ab} [T_a, T_b] = 0 \ .$$ For a solution of the classical Yang-Baxter equation the unimodularity of the $r$-matrix is equivalent to the unimodularity ($f_{ab}{}^b = 0$) of the corresponding subalgebra. In [@Borsato:2016ose] it was shown that the background defined by a Yang-Baxter $\sigma$-model based on a non-unimodular non-abelian $r$-matrix is not a supergravity background, but rather solves the modified supergravity described above. The first example we discuss corresponds to a non-abelian but unimodular $r$-matrix, while the second is a non-unimodular $r$-matrix.
Unimodular r-matrix {#sapp:340}
-------------------
The first example corresponds to an $r$-matrix considered in [@Borsato:2016ose] $$r = \eta~ \mathfrak{M}_{23}\wedge \mathfrak{P}_1 + \zeta~ \mathfrak{P}_2\wedge \mathfrak{P}_3 \ .$$ This is non-abelian e.g. $[\mathfrak{M}_{23}, \mathfrak
| 3,390
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| 2,018
| 3,088
| null | null |
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|
6mr\right) {a}^{4}\alpha \left(\cos\left(\theta \right)\right)^{5} \nonumber\\
& +\left(15{a}^{4}{m}^{2}{r}^{2}-20{a}^{4}{\alpha}^{2}{e}^{2}m{r}^{3}-15{a}^{6}{\alpha}^{2}{m}^{2}{r}^{2}-{\frac {17{a}^{2}{\alpha}^{2}{e}^{4}{r}^{4}}{3}}-10{a}^{4}{e}^{2}mr+7/6{a}^{4}{e}^{4}\right) \left( \cos \left( \theta \right) \right) ^{4} \nonumber\\
&-20 \left(-{e}^{2}m{r}^{2}+ \left( -2{a}^{2}{m}^{2}+{\frac {19{e}^{4}}{30}}\right) r+{a}^{2}{e}^{2}m \right) {r}^{2}{a}^{2}\alpha \left( \cos\left( \theta \right) \right) ^{3}+ \nonumber\\
& \left( 7/6{\alpha}^{2}{e}^{4}{r}^{6}-{\frac {17{a}^{2}{e}^{4}{r}^{2}}{3}}-15{a}^{2}{m}^{2}{r}^{4
}+10{a}^{2}{\alpha}^{2}{e}^{2}m{r}^{5}+15{a}^{4}{\alpha}^{2}{m}^{2}{r}^{4}+20{a}^{2}{e}^{2}m{r}^{3} \right) \left( \cos \left( \theta\right) \right) ^{2} \nonumber\\
&+ 10 \left( {e}^{2}-6/5mr \right) {r}^{4}\alpha \left( {a}^{2}m+1/6{e}^{2}r \right) \cos \left( \theta
\right) + \left( -{a}^{2}{\alpha}^{2}{m}^{2}+{m}^{2} \right) {r}^{6}-2{e}^{2}m{r}^{5}+7/6{e}^{4}{r}^{4} ).\end{aligned}$$
From the above scalars we can calculate the ratio $ P=\sqrt{\dfrac{W}{K}} $, defined in [@entropy1], which serves as the measure of gravitational entropy, $S_{grav}$ of black holes. The four-dimensional determinant of the metric is $$g=-{\frac { \left( \sin \left( \theta \right)\right) ^{2} \left( {r}^{2}+{a}^{2} \left( \cos \left( \theta
\right) \right) ^{2} \right) ^{2}}{ \left( \alpha\,r\cos \left( \theta \right) -1 \right) ^{8}}}.$$ Here again the axisymmetric metric denies us the calculation of the spatial metric due to the nonzero metric component $ g_{t\phi} $. Therefore as in the previous calculation for rotating black holes, the entropy density is calculated by using the metric determinant $ g $ in the covariant derivative. We thus have $$\label{enden2}
s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left(\dfrac{\partial}{\partial r}\sqrt{-g}P\right).$$ Here we have intentionally avoided writing the exact expression of entropy density as it is lengthy and too much complicated, bu
| 3,391
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RRT\
226/3511 (6.4%)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
*Abbreviations.* AKI: acute kidney injury; AKIN: Acute Kidney Injury Network; CKD: chronic kidney disease; F: female; ICD: International Classification of Diseases; ICM: ischemic cardiomyopathy; KDIGO: Kidney Disease: Improving Global Outcomes; LVAD: left ventricular assist device; RVAD: right ventricular assist device; RRT: renal replacement therapy; SCr: serum creatinine; USA: Unites States of America; RIFLE: Risk, Injury, Failure, Loss of kidney function, and End-stage kidney disease.
Incidence of AKI in LVAD patients {#s0007}
---------------------------------
Fifty-six studies \[[@CIT0015]\] evaluated AKI incidence in LVAD patients. The pooled incidence of reported AKI was 24.9% (95%CI: 20.1%--30.4%, *I*^2^ = 99
| 3,392
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|
------------------------------------------
We now apply the highest-/lowest-weight formalism to NHEK. In the Schwarzschild spacetime, the orbit space of the isometry $SO(3)$ is $S^2$, therefore we expect a $2+2$ decomposition of the whole manifold. Similarly, in the NHEK spacetime, the isometry group ${\ensuremath{SL(2,\mathbb{R})}}\times\nobreak{}U(1)$ acts on the three-dimensional hypersurfaces $\Sigma_u$ of constant polar angle $\theta$ (or $u$). This enables us to perform a $3+\nobreak 1$ decomposition of the spacetime. In both cases, we can simultaneously diagonalize some algebra elements, including the Casimir, in various tensor spaces.
However there is an important difference between the two spacetimes. In the NHEK case, we encounter the non-compact group ${\ensuremath{SL(2,\mathbb{R})}}$. It is known that for non-compact simple Lie groups like ${\ensuremath{SL(2,\mathbb{R})}}$, the only irreducible unitary finite-dimensional representation is the trivial representation [@Barut:1986dd]. As a result, one can find two distinct unitary representations of ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$: the *highest-weight module* or the *lowest-weight module*. Both of them are infinite-dimensional representations in the NHEK case. For compact groups like $SO(3)$, these two modules coincide.
Our method to find the general (scalar, vector, and symmetric tensor) basis functions $\xi$ associated with the highest-weight module of NHEK’s isometry can be summarized into four steps. Notice that the method presented here is not restricted to NHEK spacetime. For instance it can also be applied to finding the basis functions in near-horizon near-extremal Kerr (near-NHEK) which has the same isometry group as NHEK’s [@Bredberg:2009pv]. This will be left for future work. For readers who are more interested in what the bases of NHEK’s isometry look like either in Poincaré or global coordinates, the explicit expressions are given in App. \[app:S-V-T-Basis\].
#### Orbit space.
For each point $p\in\mathcal{M}$, the
| 3,393
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\beta (\gamma_3+ i \gamma_4) \big) \big] \ ,$$ where $$r = 1+ |\alpha|^2 + |\beta|^2 \ , \quad s^2 = \frac{1}{2\sqrt{r} }(1+ \sqrt{r}) \ , \quad t^2 = \frac{1}{2\sqrt{r} (1+ \sqrt{r})} \ .$$ These coordinates give a metric on $S^5$ that makes manifest the structure of $S^5$ as a $U(1)$ fibration over $\mathbb{C}\mathbf{P}^2$ $$\label{eq:CPform}
\begin{aligned}
&ds^2_{S^5} = ds^{2}_{\mathbb{C}\mathbf{P}^2} + \Psi^{2}\ , \quad ds^{2}_{\mathbb{C}\mathbf{P}^2} = \frac{1}{r}( |d\alpha|^2 + |d\beta|^2 ) - \frac{1}{r^2}| \omega |^2 \ , \\
& \Psi= d\phi + \frac{1}{r} \Im( \omega) \ , \quad \omega = \bar\alpha d \alpha + \bar\beta d \beta \ .
\end{aligned}$$ The global one-form $\Psi = \sum_{i=1\dots 3} x_i dy_i - y_i d x_i$ where $z_i = x_i + i y_i$ are coordinates on $\mathbb{C}^3$ given by $z_1 = \frac{1}{\sqrt{r}} e^{i \phi}$, $z_2 = \frac{\alpha}{\sqrt{r}} e^{i \phi}$, $z_3 = \frac{\beta}{\sqrt{r}} e^{i \phi}$. One can think of $\Psi$ as a contact form whose corresponding Reeb vector has orbits which are the $S^1$ fibres. For computational purposes we note that frame fields for $\mathbb{C}\mathbf{P}^2$ can be found in e.g. [@Eguchi:1980jx].
When dealing with the dipole deformation in section \[ssec:app3\] we will need the full ten-dimensional space-time. This is readily achieved by taking a block diagonal decomposition, i.e. $g = g_{AdS_{5}} \oplus g_{S^5}$, with the generators of $\mathfrak{su}(2,2) \oplus \mathfrak{su}(4)$ given by $8 \times 8$ matrices, with the $\mathfrak{su}(2,2)$ and $\mathfrak{su}(4)$ generators in the upper left and lower right $4 \times 4$ blocks respectively. Traces are then replaced with “supertrace” (the bosonic restriction of the supertrace on $\mathfrak{psu}(2,2|4)$) given by the matrix trace of the upper $\mathfrak{su}(2,2)$ block minus the matrix trace of the lower $\mathfrak{su}(4)$ block.
Further Examples of Deformations in AdS5 {#app:furtherexamples}
========================================
In section \[sec:examples\] we considered non-abelian T-dualities with respect to a c
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name{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
Naoya Yamaguchi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
yamaguchi@nagasaki-u.ac.jp
Yuka Yamaguchi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
yamaguchiyuka@nagasaki-u.ac.jp
Ryuei Nishi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
nishii.ryuei@nagasaki-u.ac.jp
---
abstract: |
We examine the issue of the cosmological constant in the $many$ $inflating$ branes scenario, extending on two recent models by I.Oda and Randall-Sundrum.The exact solution in a closed form is found in the slow roll approximation of the radion. Defining an effective expansion rate, which depends on the location of each brane in the fifth dimension and demanding stability for this case we show that each positive tension brane has a localized, decaying cosmological constant (the opposite process applies to the negative energy branes \[4\]) .\
The reason is that the square of the effective expansion rate enters as a source term in the Einstein equations for the branes.Thus the brane has two scale factors depending on time and the fifth dimnesion respectively .The brane will roll along the fifth dimension in order to readjust its effective expansion rate in such a way that it compensates for its internal energy changes due to inflation and possible phase transitions.
author:
- |
Laura Mersini\
Department of Physics\
University of Wisconsin-Milwaukee\
Milwaukee, WI 53201\
lmersini@uwm.edu\
Wisc-Milw-99Th-14
date: 'September 20, 1999'
title: 'Decaying Cosmological Constant of the Inflating Branes in the Randall-Sundrum -Oda Model'
---
Setup
=====
In this paper we consider and find solutions to a five dimensional m
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{} (1426, 14)[$w$]{} (376,-1186)[$0$]{} (3226,-1186)[$1$]{} (676,314)[Im]{} (2251,-1186)[$Q^2$]{} (4576,-811)[$\theta_{i_5}$]{} (5401,-1186)[$\theta_{i_1}$]{} (5701,-886)[$\theta_{i_2}$]{} (4651,-136)[$\theta_{i_4}$]{} (3526,-1186)[Re]{} (5626,-211)[$\theta_{i_3}$]{} (751,-586)[$e_0$]{} (1276,-736)[$e_P$]{} (1876,-736)[$e_Q$]{} (301,-961)[( 1, 0)[3300]{}]{}
Now consider the product map $\Psi_{\langle 12345\rangle} \times \Psi_{\langle 21435\rangle}:
\Theta_5 \to {\cal T}\times {\cal T}$ which sends $\theta$ to $(\Delta_{\langle 12345\rangle,\theta},\Delta_{\langle 21435\rangle,\theta})$. Then we claim that
\[Lem:5injective\] The map $\Psi_{\langle 12345\rangle} \times \Psi_{\langle 21435\rangle}$ is injective.
Let $((P_1, Q_1), (P_2, Q_2))$ be any element of the image of $\Psi_{\langle 12345\rangle} \times \Psi_{\langle 21435\rangle}$, that is, $P_1$, $Q_1$, $P_2$, $Q_2$ are the lengths of the sides $(12)345$, $12(34)5$, $(21)435$, $21(43)5$, respectively. Suppose that $\theta = (\theta_1,\ldots,\theta_5)$ be any element in the inverse image of $((P_1, Q_1), (P_2, Q_2))$ by $\Psi_{\langle 12345\rangle} \times \Psi_{\langle 21435\rangle}$. Then $\theta$ lies in the intersection $\Psi_{\langle 12345\rangle}^{-1}((P_1,Q_1)) \cap \Psi_{\langle 21435\rangle}^{-1}((P_2,Q_2))$. Let $w_1$ be the point in ${{\bold H}}^2 \cong \Psi_{\langle 12345\rangle}^{-1}((P_1,Q_1))$ corresponding to $\theta$ under the identification in Lemma \[Lem:5fibration\], and similarly $w_2$ the point in ${{\bold H}}^2 \cong \Psi_{\langle 21435\rangle}^{-1}((P_2,Q_2))$ corresponding to $\theta$. These can be obtained by the inverse procedure of Fig. \[Fig:5\]. Then we see that the triangles $\triangle w_101$ and $\triangle w_201$ are congruent since both have the external angles $\theta_5$, $\theta_1+\theta_2$, $\theta_3+\theta_4$. Thus $w_1$ and $w_2$ must be identical in the upper half plane ${{\bold H}}^2$. Let $w=w_1=w_2$.
The angle at $P_1^2$ in the triangle $\triangle w0P_1^2$ is $\theta_2$, and the angle at $P_2^2$ in the triangle
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|
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fields and the scalar fields of the model, their representations under the $A_4\times Z_3$ symmetry and their modular weights are given in Tab. \[tab:fields\]. We also show the representations of the Yukawa couplings in Tab. \[tab:couplings\]. Under these symmetries, we write the renormalizable Lagrangian for the lepton sector as follows: $$\begin{aligned}
-{\cal L}_L &=
\sum_{\ell=e,\mu,\tau}y_\ell \bar L_{L_\ell} H_{SM} e_{R_\ell}{\nonumber}\\
&\hspace{3ex}+\alpha_\nu \bar L_{L_e} (Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1}\tilde H_1
+\beta_\nu \bar L_{L_\mu}(Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1''}\tilde H_1
+\gamma_\nu \bar L_{L_\tau}(Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1'}\tilde H_1{\nonumber}\\
&\hspace{3ex}+a_\nu (\bar N_{L_e} \otimes Y^{(6)*}_{\bf 3})_{\bf1} {L^C_{L_e}}\tilde H_2
+b_\nu (\bar N_{L_\mu}\otimes Y^{(6)*}_{\bf 3})_{\bf1'} {L^C_{L_\mu}}\tilde H_2
+c_\nu (\bar N_{L_\tau}\otimes Y^{(6)*}_{\bf 3})_{\bf1''} {L^C_{L_\tau}}\tilde H_2{\nonumber}\\
&\hspace{3ex}+a'_\nu (\bar N_{L_e}\otimes Y'^{(6)*}_{\bf 3})_{\bf1} {L^C_{L_e}}\tilde H_2
+b'_\nu (\bar N_{L_\mu}\otimes Y'^{(6)*}_{\bf 3})_{\bf1'} {L^C_{L_\mu}}\tilde H_2
+c'_\nu (\bar N_{L_\tau}\otimes Y'^{(6)*}_{\bf 3})_{\bf1''} {L^C_{L_\tau}}\tilde H_2{\nonumber}\\
&\hspace{3ex}+ {M_D} (\bar N_{L}\otimes N_{R})_{\bf1}
+ {\rm h.c.},
\label{eq:lag-lep}\end{aligned}$$ where $\tilde H\equiv i\sigma_2 H^*$ with $\sigma_2$ being the second Pauli matrix and $(A\otimes B)_{\bf R}$ indicates that the representation, $\bf R$, is contracted from $A$ and $B$. Here, $M_D$ includes a modular invariant coefficient, $1/(i\tau-i\tau^*)$, and the charged-lepton matrix is diagonal thanks to the $A_4$ symmetry.
----------- --------------------------------------------------- ----------------------------------------- ---------- ------------ ------------ ------------
$(\bar L_{L_e},\bar L_{L_\mu},\bar L_{L_\tau})$ $(e_{R_e},e_{R_{\mu}},e_{R_{\tau}})$ $N_{}$ $H_{SM}$ $H_1^*$ $H_2$
$SU(2)_L$ $\bm{2}$
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ft(\chi(a)^2 Y_a^2 + {\left\vert \chi(a) \right\vert}^2 {\left\vert Y_a \right\vert}^2 \right)\right] \\
&= \frac{1}{2}\left( {\left\vert G \right\vert} + \alpha \sum_{a \in G} \chi^2(a) \right)
= \frac{{\left\vert G \right\vert}}{2}\bigl( 1 + \alpha {\mathbbm{1}_{\chi = \overline{\chi}}}\bigr).
\end{aligned}$$ In the last step we have used that unless $\chi$ is real-valued, $\chi$ and $\overline{\chi}$ are distinct characters, and hence orthogonal in $\ell^2(G)$. In similar fashion, we find that $$\operatorname{Cov}\bigl(\lambda_\chi\bigr)
= \frac{1}{{\left\vert G \right\vert}} \sum_{a \in G}
\operatorname{Cov}\bigl(\chi(a) Y_a\bigr) = \frac{1}{2} \bigl( I_2 +
{\mathbbm{1}_{\chi = \overline{\chi}}} \bigl[\begin{smallmatrix} \alpha & 0 \\
0 & - \alpha \end{smallmatrix}\bigr]\bigr).$$ Observe in particular that if $\alpha = 1$ and $\chi$ is real-valued, then $\lambda_\chi$ is almost surely real, with variance $1$; in that case we treat $\lambda_\chi$ as a random variable in ${\mathbb{R}}$, as opposed to a random vector in ${\mathbb{R}}^2$. Proposition \[T:Lindeberg\] and (recalling that ${\left\vert \chi(a) \right\vert} = 1$ always) now imply that there is a sequence $\delta_n$ decreasing to $0$ such that for each $\chi \in \widehat{G}$, $${\left\vert {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}}\sum_{a \in G} \chi(a) Y_a\right)
- \gamma_\alpha(f) \right\vert} \le \delta_n$$ if $\chi$ is real-valued, and $${\left\vert {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}}\sum_{a \in G} \chi(a) Y_a\right)
- \gamma_{\mathbb{C}}(f) \right\vert} \le \delta_n$$ otherwise. Writing $\nu^{(n)} = (1-p_2^{(n)}) \gamma_{\mathbb{C}}+ p_2^{(n)}
\gamma_\alpha$, by it follows that $$\label{E:mean-bound} \begin{split}
{\left\vert {\mathbb{E}}\mu(f) - \nu(f) \right\vert} & = \Biggl\vert \frac{1}{{\left\vert G \right\vert}}
\sum_{\chi = \overline{\chi}} {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}}
\sum_{a \in G} \chi(a) Y_a\right) - p_2
| 3,398
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}V\_[ab]{}(z-z’) J\^[(a)]{}\_[z,ij]{}J\^[(b)]{}\_[z’,ij’]{}\
&+& \_[i,z]{} J\^2\_[z,i]{}, \[Hdual\] where the matrix $V_{ab}(z-z')$ is defined in terms of the matrix $K_{ab}$. It reflects the asymmetry between odd and even layers. Explicitly, $V_{11} (z)=YV_{22}(z)$, for $z=z-z'$ being even, describes the interaction between odd layers, and $V_{22}(z)$ is defined between even layers; $V_{12}(z)= - X [V_{22}(z+1) +V_{22}(z-1)]$ refers to the interaction between odd and even layers (that is, $z$ is odd), and V\_[22]{}(z)= \_[q\_m]{} , \[Vzz22\] with $ z=0,\pm 2, \pm 4, ...$ and the summation running over $q_m=4\pi m/N, m=0,1,..., N/2 -1$.
The dual formulation for $N_z$ layers allows obtaining the asymptotic expression for $u_r$ within the same logic used for deriving Eq.(\[UR3\]). We will repeat it here. The loop formation can be viewed as a process of random walks of two ends of a broken loop – exactly along the line of the Worm Algorithm [@WA]. Such a walk of each end is controlled by energetics of creating one bond element $|J|=1$ in a randomly chosen direction – either along a given plane or perpendicular to it. Very similar to the case of the two layers, the energy to create such an element alone along the odd layer costs energy $>>T \sim 1$, while the same element along an even layer costs energy $\sim 1$. The only option for creating a loop in an odd layer is if its energy is compensated by parallel elements in the even plane. This feature is caused by the strong current-current interaction $\sim X$. Thus, if the walk occurs along $z$-direction from some even layer $z$ toward the neighboring odd layer $z+1$, the subsequent move along the odd layer will be too energetically costly so that the walker would either move further toward $z+2$ layer or will go back to the original layer $z$. Thus, the inter-layer elements are characterized by either $J_{i,z}=J_{i,z+1}=\pm 1$ or $J_{i,z}=J_{i,z+1}=0$. The weight of such a process is either $\exp(-1/u_V)$ or $1$, respectively. Even if the walker makes a st
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dots \le m_{d-1},$
3. $m_{i+(q-1)} > m_i$ for all $1 \le i \le d-q.$
Clearly this implies that $N=\sum_{i=1}^{d} \rho_q(i,m_i),$ but it is not clear a priori that $m_1,\dots,m_d$ satisfy conditions 2 and 3 as well. Conditions 2 and 3 would follow once we show that $m_d \ge m_{d-1}$ and either $m_d > m_{d-q+1}$ or $m_d=m_{d-q+1}=-1$. First of all, if $m_d=-1$, then $N=0$ and $(m_d,\dots,m_1)=(-1,\dots,-1)$. Hence there is nothing to prove in that case. Assume $m_d \ge 0$. From Equation and Lemma \[lem:help\] we see that $$\label{eq:existence1}
N-\rho_q(d,m_d)<\rho_q(d,m_d+1)-\rho_q(d,m_d)=\sum_{i=1}^{\min\{d,q-1\}}\rho_q(d-i,m_d).$$ First suppose that $d \le q-1$. First of all, Condition 3 is empty in that setting. Further, Equation implies $$N-\rho_q(d,m_d) < \sum_{i=1}^{d}\rho_q(d-i,m_d) = \sum_{i=1}^{d-1}\rho_q(d-i,m_d) +1$$ and hence $$N-\rho_q(d,m_d) \le \sum_{i=1}^{d-1}\rho_q(d-i,m_d)= \sum_{j=0}^{d-2}\rho_q(d-1-j,m_d) < \rho_q(d-1,m_d+1).$$ This shows that $m_{d-1} \le m_d$ as desired.
Now suppose that $d \ge q$. In this situation Equation implies $$N-\rho_q(d,m_d) < \sum_{i=1}^{q-1}\rho_q(d-i,m_d) = \sum_{j=0}^{q-2}\rho_q(d-1-j,m_d)< \rho_q(d-1,m_d+1).$$ Hence $m_{d-1} \le m_d$ as before. Finally assume that $m_d \le m_{d-q+1}$. Then by the previous and Condition 2, we have $m_{d}= m_{d-1}=\cdots = m_{d-q+1}$. Hence $N \ge \sum_{i=0}^{q-1}\rho_q(d-i,m_d)=\rho_q(d,m_d+1)$ which is in contradiction with Equation . This concludes the induction step and hence the proof of existence.
We call the representation of $N$ in the above theorem the $d$-th Macaulay representation of $N$ with respect to $q$. One retrieves the usual $d$-th Macaulay representation letting $q$ tend to infinity. We refer to $(m_d,\dots,m_1)$ as the coefficient tuple of this representation. A direct corollary of the above is the following.
\[cor:greedy\] The coefficient tuple $(m_d,\dots,m_1)$ of the $d$-th Macaulay representation with respect to $q$ of a nonnegative integer $N$ can be computed using the following greedy algorithm: The
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