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ference operators $D_\alpha$ corresponds to multiplication by $\left(e^{-\alpha} - 1 \right)$. Therefore, the left-hand side of is identified with $\left( \prod_{\alpha \in R_{G,+}} - D_\alpha \right) m_{T_G,V_{G,\lambda}}$.
Now consider the right-hand side of . Since $\lambda + \rho$ is a strictly dominant weight, it is sent by any Weyl group element $w \neq 1$ to the interior of another Weyl chamber. That is, there exists a positive root $\alpha \in R_{G,+}$ such that $\langle \alpha, w(\lambda + \rho) \rangle < 0$. In particular, $w(\lambda + \rho) - \rho$ is never dominant unless $w = 1$. It follows that the restriction of $\left( \prod_{\alpha \in R_{G,+}} - D_\alpha \right) m_{T_G,V_{G,\lambda}}$ to $\Lambda^*_{G,+}$ is equal to the indicator function of $\{\lambda\}$, i.e., equal to the highest weight multiplicity function of $V_{G,\lambda}$.
The idea of using for determining multiplicities of irreducible representations goes back at least to Steinberg [@steinberg61], who proved a formula for the multiplicity $c_{\lambda,\mu}^\nu$ of an irreducible representation $V_{G,\nu}$ in the tensor product $V_{G,\lambda} \otimes V_{G,\mu}$. These multiplicities $c_{\lambda,\mu}^\nu$ are called the *Littlewood–Richardson coefficients* for $G$. Steinberg’s formula involves an alternating sum over the Kostant partition function ; it can be evaluated efficiently as described by Cochet [@cochet05]. De Loera and McAllister give another method for computing Littlewood–Richardson coefficients [@deloeramcallister06], which applies Barvinok’s algorithm to results by Berenstein and Zelevinsky [@berensteinzelevinsky01]. Since the tensor products of irreducible $G$-representations are just the irreducible representations of $G \times G$, the problem of computing Littlewood–Richardson coefficients is again a special case of . The following consequence of the proof of will be convenient in the sequel:
\[steinberg corollary\] Write $\prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right) = \sum_{\gamma \in \Gamma_G}
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i) \in {\prod{\Phi}}$. Hence ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$, and ${\mathbf{f}}\,'$ is a frame.
Finally, definition \[D:ITERATIVE\_TRANSFORM\] calls for the succeeding locus as $\lambda' = \Delta(\lambda, \psi)$. Definition \[D:JUMP\_FUNCTION\] specifies the jump function as a mapping $\Delta \colon \Lambda \times {\prod{\Psi}} \to \Lambda$. It is established above that $\lambda \in \Lambda$ and $\psi \in {\prod{\Psi}}$, so $\lambda' = \Delta(\lambda, \psi) \in \Lambda$ is a locus.
With locus $\lambda' \in \Lambda$, functionality ${\mathit{f}}' \in {\mathscr{F}}$, and frame ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$, we then summarize that ${\mathit{s}}' = (\lambda', {\mathit{f}}', (\psi', \phi')) \in {\mathbb{S}}$ is a step, and conclude that transform $T_{{\mathfrak{A}}}$ is an iterative operator ${\mathbb{S}} \to {\mathbb{S}}$.
Application of the iterative operator $T_{{\mathfrak{A}}}$ induced by automaton ${\mathfrak{A}}$ is denoted ${\mathit{s}}' = {\mathfrak{A}}({\mathit{s}})$.
Let ${\mathbb{S}}$ be a step space with automaton ${\mathfrak{A}}$ inducing iterative operator $T_{{\mathfrak{A}}}$. The walk of ${\mathit{s}} \in {\mathbb{S}}$ under $T_{{\mathfrak{A}}}$ assuming sequence of volatile excitation $\lbrace \xi_n \rbrace$ is denoted ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$.
The notation ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$ is a reminder of the important role of the sequence of volatile excitations $\lbrace \xi_n \rbrace$. While each $\xi$ is entirely determined by initial frame $\psi$ within step ${\mathit{s}}$, this notation emphasizes that the volatile excitations are essentially free variables. Persistent variables are bound.
#### Automaton iterative properties
\[T:AUTOMATON\_ITERATE\_CONJOINT\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with persistent-volatile partition $\Psi = \Phi\Xi$. Suppose step ${\mathit{s}} \in {\mathbb{S}}$ an
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o masses are naturally explained without assuming the triplet fields to be heavy. The Yukawa sector is then the same as the one in the HTM, so that its predictions for the LFV processes are not changed. See Refs. [@Babu:2001ex; @Chun:2003ej] for some discussions about two-loop realization of the $\mu$-term[^1].
This paper is organized as follows. In Sec. \[sec:HTM\], we give a quick review for the HTM to define notation. In Sec. \[sec:1-loop\], the model for radiatively generating the $\mu$ parameter with the dark matter candidate is presented. Some phenomenological implications are discussed in Sec. \[sec:pheno\], and the conclusion is given in Sec. \[sec:concl\]. The full expressions of the Higgs potential and mass formulae for scalar bosons in our model are given in Appendix.
Higgs Triplet Model {#sec:HTM}
===================
In the HTM, an $\text{SU}(2)_L$ triplet of complex scalar fields with hyperchage $Y=1$ is introduced to the SM. The triplet $\Delta$ can be expressed as $$\begin{aligned}
\Delta
&=&
\begin{pmatrix}
\Delta^+/\sqrt{2} & \Delta^{++}\\
\Delta^0 & -\Delta^+/\sqrt{2}
\end{pmatrix} ,\end{aligned}$$ where $\Delta^0 = (\Delta^0_r + i \Delta^0_i)/\sqrt{2}$. The triplet has a new Yukawa interaction term with leptons as $$\begin{aligned}
{\mathcal L}_{\text{triplet-Yukawa}}
&=&
h_{\ell{{\ell^\prime}}}\, \overline{L_\ell^c}\, i\sigma_2\, \Delta\, L_{{\ell^\prime}}+ \text{h.c.} ,\end{aligned}$$ where $h_{\ell{{\ell^\prime}}}$ ($\ell, {{\ell^\prime}}= e, \mu, \tau$) are the new Yukawa coupling constants, $L_\ell$ \[$= (\nu_{\ell L}, \ell)^T$\] are lepton doublet fields, a superscript $c$ means the charge conjugation, and $\sigma_i$ ($i = 1\text{-}3$) denote the Pauli matrices. Lepton number ($L\#$) of $\Delta$ is assigned to be $-2$ as a convention such that the Yukawa term does not break the conservation. A vacuum expectation value $v_\Delta^{}$ \[$=\sqrt{2}\,\langle\Delta^0\rangle$\] breaks lepton number conservation by two units. The new Yukawa interaction then yields the Majorana
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insists that this total reshuffling of physical interpretation should leave the basic mathematical building blocks (a certain generating set of algebras and the symmetry group structure) untouched, then there is only one answer: an associated anti De Sitter (AdS) theory [@Wit]. The nontrivial reprocessing leads to a mathematical isomorphism as described in [@Reh1] i.e. it goes far beyond that picture about the AdS-CQFT correspondence which is limited to the (infinitely remote) boundary of AdS (see in particular the remarks at the end of [@Reh2]). The AdS appearance of the AdS structure as a kind of reprocessed CQFT is less surprizing if one recalls the 6-dimensional lightcone formalism which one uses in order to obtain an efficient description of the conformal compactification $\bar{M}$ of Minkowski space $M $ and the construction of its covering $\tilde{M}$ [@Schroer].
In this way one obtains a (perturbative) new constructive non-Lagrangian access to CQFT which opens a new window into the realm of CQFT beyond those few 4-dimensional Lagrangian candidates for which one had to use a combination of gauge theory with supersymmetry. This means that one has no guaranty that the conformal side at all permits a description in terms of an action.
Particle Structure and Triviality
=================================
We start with recalling an old theorem which clarifies the relation between the particle-versus-field content of conformal field theories. To be more precise the following statement is a result of the adaptation of a combination of several theorems [@BF][@Pohl]
The existence of one-particle states in conformally invariant theories forces the associated interpolating fields to be canonical free fields. The only particle-like structures consistent with interactions are hidden in the structure of those interpolating fields which have anomalous dimensions and whose mass spectrum is continuous with an accumulation of weight at $p^{2}=0,\,\,p_{0}>0.$
The easiest way to get a first glimpse at this situation is to
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elta\E}$ at the next enstrophy level for some sufficiently small $\Delta\E > 0$. As will be demonstrated in §\[sec:3D\_InstOpt\_E0to0\], in the limit $\E_0 \rightarrow 0$ optimization problem \[pb:maxdEdt\_E\] admits a discrete family of closed-form solutions and each of these vortex states is the limiting (initial) member $\widetilde{\mathbf{u}}_{0}$ of the corresponding [maximizing]{} branch. As such, these limiting extreme vortex states are used as the initial guesses ${\mathbf{u}}^0$ for the calculation of $\widetilde{\mathbf{u}}_{\Delta\E}$, i.e., they serve as “seeds” for the calculation of an entire [maximizing]{} branch (as discussed in §\[sec:discuss\], while there exist alternatives to the continuation approach, this technique [in fact results]{} in the fastest convergence of iterations and also ensures that all computed extreme vortex states lie on a single branch). The procedure outlined above is summarized as Algorithm \[alg:optimAlg\], [whereas all]{} details are presented below.
set $\E_0 = 0$ set ${\widetilde{\mathbf{u}}_{\E_0}}= \widetilde{\mathbf{u}}_{0}$ ${\mathbf{u}}_{\E_0}^{(0)} = {\widetilde{\mathbf{u}}_{\E_0}}$ $\E_0 = \E_0 + \Delta \E$ $n = 0$ compute $\R_0 = \R\left({\mathbf{u}}_{\E_0}^{(0)}\right)$
compute the $L_2$ gradient $\nabla^{L_2}\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)$, see equation
compute the Sobolev gradient $\nabla\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)$, see equation
compute the step size $\tau_n$, see equation
set ${\mathbf{u}}_{\E_0}^{(n+1)} = \mathbb{P}_{\mathcal{S}_{\E_0}}\left(\;{\mathbf{u}}_{\E_0}^{(n)} + \tau_n \nabla\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)\;\right)$, see equations –
set $\R_1 = \R\left({\mathbf{u}}_{\E_0}^{(n+1)}\right)$
compute the `relative error` $ = (\R_1 - \R_0)/\R_0$
set $\R_0 = \R_1$
set $n=n+1$
A key step of Algorithm \[alg:optimAlg\] is the evaluation of the gradient $\nabla\R({\mathbf{u}})$ of the objective functional $\R({\mathbf{u}})$, cf. , representing its (infinite-dimensional) sensitivity to perturbations of the v
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al correlations and fluctuation scaling, let us repeat here Eqs. and : $$\begin{aligned}
\alpha(\Delta t)=\alpha_0^\pm + \gamma^\pm \log \Delta t, \nonumber \\
H_i=H_0^\pm + \gamma^\pm \log \ev{f_i}. \nonumber\end{aligned}$$ Beyond the obvious symmetry of these two logarithmic laws, notice that the prefactors are equal: in both equations $\gamma^- \approx 0$ and $\gamma^+\approx 0.05$.
It is easy to show [@eisler.unified] that none of this is a simple coincidence. If both fluctuation scaling and long range autocorrelations are present in data, there are only two possible ways for their coexistence:
1. Correlations are homogeneous throughout the system, $H_i=H_0$, $\gamma = 0$, and $\alpha$ is independent of $\Delta t$. This is realized for $\Delta < 60$ min. For shuffled time series correlations are absent, thus such data also fall into this category.
2. Both the $H(\ev{f_i})$ and $\alpha(\Delta t)$ are logarithmic functions of their arguments with the same coefficient $\gamma^+$. This is realized for $\Delta > 390$ min.
In other words the coexistence of the two scaling laws is so restrictive, that if the strength of correlations depends on $\ev{f}$ at all, then the realized logarithmic dependence is the only possible scenario.
Conclusions
===========
In this paper, we analyzed the empirical properties of trading activity on the New York Stock Exchange. We showed that, in contrast to earlier findings, the distribution of traded value is not in the Levy stable regime, and is not universal. Traded value is nearly uncorrelated on an intraday time scale, while on daily or longer scales fluctuations show strong persistence, whose strength grows logarithmically with the liquidity of the stock. This effect is in harmony with findings on fluctuation scaling, a general scaling framework for complex systems.
All our results imply, that the notion of universality must be used with extreme care in the context of financial markets, where the concepts and the theoretical background are radically different from
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\right\vert},
\end{aligned}$$ so by and , $$\mu^{(n)} (f) \to \bigl[(1-p) \gamma_{\mathbb{C}}+ p \gamma_\alpha\bigr](f)$$ in $L^2$, and hence in probability.
The proof is analogous to that of Theorem \[T:circular-law-correlated\], setting $\alpha = 0$. In that case $\operatorname{Cov}(\lambda_\chi)$ no longer depends on whether $\chi$ is real-valued, which makes it unnecessary to assume that $p_2^{(n)}$ approaches a limit.
We omit $(n)$ superscripts as before. We will assume that $p<1$; the case $p=1$ (which implies that in fact $p_2 = 1$ for sufficiently large $n$) is similar and slightly simpler. Let $A =
\{ a \in G \mid a = a^{-1} \}$. Since $G$ is abelian, $A$ is a subgroup of $G$. The restriction of a character of $G$ to $A$ is a character on $A$, which is necessarily real-valued on $A$. It follows that for $\chi_1, \chi_2 \in \widehat{G}$, $$\label{E:semicircle-covariance}\begin{split}
{\left\vert G \right\vert} {\mathbb{E}}\lambda_{\chi_1} \lambda_{\chi_2} &=
\sum_{a,b \in G} \chi_1(a) \chi_2(b) {\mathbb{E}}Y_a Y_b \\
& = \sum_{a \in G} \chi_1(a) \bigl[\bigl(\overline{\chi_2(a)} +
\alpha \chi_2(a)\bigr) {\mathbbm{1}_{a \neq a^{-1}}}
+ \beta \chi_2(a) {\mathbbm{1}_{a = a^{-1}}} \bigr] \\
& = \sum_{a \in G \setminus A} \chi_1(a) \overline{\chi_2(a)} + \alpha
\sum_{a \in G \setminus A} \chi_1(a) \chi_2(a)
+ \beta \sum_{a \in A} \chi_1(a) \chi_2(a) \\
& = \sum_{a \in G} \chi_1(a) \overline{\chi_2(a)} + \alpha \sum_{a
\in G} \chi_1(a) \chi_2(a)
+ (\beta - \alpha - 1) \sum_{a \in A} \chi_1(a) \chi_2(a) \\
& = {\left\vert G \right\vert} \bigl( {\mathbbm{1}_{\chi_1 = \chi_2}} + \alpha {\mathbbm{1}_{\chi_1 =
\overline{\chi_2}}}\bigr) + {\left\vert A \right\vert} (\beta - \alpha - 1) {\mathbbm{1}_{\chi_1
\vert_A = \overline{\chi_2} \vert_A}} \\
& = {\left\vert G \right\vert} \bigl( {\mathbbm{1}_{\chi_1 = \chi_2}} + \alpha {\mathbbm{1}_{\chi_1 =
\overline{\chi_2}}} + p_2 (\beta - \alpha - 1) {\mathbbm{1}_{\chi_1
\vert_A = \chi_2 \
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and post-quench Hamiltonians are different in the limit $\eta\to 0$, only when $m=0$. The process becomes then reversible in such a limit, provided $m\neq 0$. Now, we investigate the role of the classical Rabi frequency $\Omega$ on the irreversibility. The result is depicted in Fig. \[fig2L2\], where one can see that the NL increases with $\Omega$. This behavior is expected from the detailed analysis of Eq. (\[partf1\]), and it can be physically understood from the fact that $\Omega$ is the work parameter and quantifies the intensity of the sudden quench.
In order to obtain a better understanding of the problem, it is necessary to go on and investigate the role of temperature. The NL as a function of the mean occupation number of the initial thermal state $\bar n$ in Eq. (\[mocn\]) is presented in Fig. \[fig3L3\]. It is noticeable that the AJC and JC models in the trapped ion system may respond so differently to variations of initial thermal energy of the system. In particular, it can be seen from Fig. \[fig3L3\] that the shown sidebands for the AJC and also for $m=0$ (which can be seen as either belonging to the AJC or JC classes) lead to a divergency in the NL as $\bar n\to 0$ ($\beta \to \infty$). This is not observed for for the JC case.
Although the dependence of $\mathcal L$ on the temperature is a bit more intricate, since all factors in Eq. (\[partf1\]) depend on it, we again succeeded in providing an analytical treatment based on asymptotics that helps us to spot the reasons behind such different behavior found in the JC and AJC models. In the high temperature limit $\beta \to 0$ ($\bar n\to \infty$), a successive application of this limit, first to some exponentials and then to the hyperbolic functions in Eq. (\[partf1\]), results in $$\label{limZ-T6}
\!\!\!\frac{\mathcal Z_{\pm}(\lambda_f)}{\mathcal Z(\lambda_i)} \to
\lim_{\beta\to 0} \left[ (\bar{n} + 1)^{-1}
\sum_{n=0}^{\infty} \!{\rm e}^{-\beta \hbar \nu( n + \frac{m}{2})} \right]= 1,$$ which makes $\mathcal L \to 0$. This shows that
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\lambda_q)) \: = \: {\rm ch}(d( \sum_i (-)^i \wedge^i N^*)
\: = \: (1, \vec{0}, 0; 4, \cdots, 4).$$ In addition, $${\rm ch}^{\rm rep}({\rm Td}(TI_{\mathfrak{X}})) \: = \:
(1, \vec{0}, 0; 1, \cdots, 1),$$ hence $${\rm Td}(\mathfrak{X}) \: = \:
\alpha_{\mathfrak{X}}^{-1} {\rm Td}(T I_{\mathfrak{X}})
\: = \:
(1, \vec{0}, 0; 1/4, \cdots, 1/4).$$
Putting this together, we find $$\begin{aligned}
\chi\left( {\cal O}_{\mathfrak{X}}[0] \right) & = &
\int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}( {\cal O}_{\mathfrak{X}}[0])
{\rm Td}(\mathfrak{X}) \\
& = &
\int_{ [T^4/{\mathbb Z}_2] } (1) (1) \: + \:
16 \int_{ [{\rm pt}/{\mathbb Z}_2] } (1) (1/4),\\
& = & 0 \: + \: 4 \int_{ [ {\rm pt}/{\mathbb Z}_2] } 1, \\
& = & 4 \left(\frac{1}{2}\right) \: = \: 2, \\
\chi\left( {\cal O}_{\mathfrak{X}}[1/2] \right) & = &
\int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}( {\cal O}_{\mathfrak{X}}[1/2])
{\rm Td}(\mathfrak{X}) \\
& = &
\int_{ [T^4/{\mathbb Z}_2] } (1) (1) \: + \:
16 \int_{ [{\rm pt}/{\mathbb Z}_2] } (-1) (1/4), \\
& = & 0 \: - \: 4 \int_{ [{\rm pt}/{\mathbb Z}_2] } 1, \\
& = & -4 \left( \frac{1}{2}\right) \: = \: -2.\end{aligned}$$
Let $Y$ denote a minimal resolution of $T^4/{\mathbb Z}_2$. Applying the McKay correspondence [@bkr], it can be shown [@tonypriv] that the bundle ${\cal O}_{\mathfrak{X}}[0]$ maps to ${\cal O}_Y$, and ${\cal O}_{\mathfrak{X}}[1/2]$ maps to ${\cal O}_Y( - (1/2) \sum E_a)$ where the $E_a$ are the exceptional divisors. Furthermore, it can be shown that on $Y$, $\chi( {\cal O}_Y) = +2$ and $\chi( {\cal O}_Y( - (1/2) \sum E_a )) = -2$, matching the Euler characteristics above.
So far we have discussed the index of the operator $\overline{\partial}$. We are not aware of rigorous results concerning the Dirac index, which would be of direct relevance for physics. That said, it is very natural to conjecture that, by analogy with smooth manifolds, the Dirac index is computed by a closely analogous expression, except that ${\rm Td}(TI_{\mathfrak{X}})$ is replaced by $${\rm Td}(TI_{\mathfrak{X}}) \exp\left( - \f
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.
Let $\Sigma$ and $\Delta$ be two alphabets, and let ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$ be a function. If $C_{{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}}$ is a prefix code, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{h}}\in\Sigma^{\leq{n}}$, then $c\in{{\it AC}(\Sigma,\Delta,n)}$.
**Proof** Let us assume that $C_{{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}}$ is prefix code, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{h}}\in\Sigma^{\leq{n}}$, but $c\notin{{\it AC}(\Sigma,\Delta,n)}$. By Definition 2.1, the unique homomorphic extension of c, denoted by $\overline{c}$, is not injective. This implies that $\exists$ $u\sigma u', u\sigma'u''\in\Sigma^{+}$, with $\sigma,\sigma '\in\Sigma$ and $u,u',u''\in\Sigma^{*}$, such that $\sigma\neq\sigma'$ and
$(*)$ $ $ $\overline{c}(u\sigma u')=\overline{c}(u\sigma'u'')$.
We can rewrite $(*)$ by
$(**)$ $ $ $\overline{c}(u)c(\sigma,P_{n}(u))\overline{c}(u')=$ $\overline{c}(u)c(\sigma',P_{n}(u))\overline{c}(u'')$,
where $P_{n}(u)$ is given by $$P_{n}(u)=
\left\{
\begin{array}{ll}
\lambda & \textrm{if $u=\lambda$,} \\
u_{1}\ldots u_{q} & \textrm{if $u=u_{1}u_{2}\ldots u_{q}$ and $u_{1},u_{2},\ldots,u_{q}\in\Sigma$ and $q\leq{n}$,} \\
u_{q-n+1}\ldots u_{q} & \textrm{if $u=u_{1}u_{2}\ldots u_{q}$ and $u_{1},u_{2},\ldots,u_{q}\in\Sigma$ and $q>n$.}
\end{array}
\right.$$ By hypothesis, $C_{P_{n}(u)}$ is a prefix code and $c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\in{C_{P_{n}(u)}}$. Therefore, the set $\{c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\}$ is a prefix code. But the equality $(**)$ can hold if and only if $\{c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\}$ is not a prefix set. Hence, our assumption leads to a contradiction. $\diamondsuit$
The converse of Theorem 2.1 does not hold. We can prove this by taking a counter-example. Let us consider $\Sigma=\{{\texttt{\textup{a}}},{\texttt{\textup{b}}}\}$ and $\Delta=\{0,1\}$ two alphabets, and ${c:\Sigma\times\Sigma^{\leq{2}}\rightarrow\Delta^{+}}$ a function given by the table below.
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g*, [2014](#grl53099-bib-0038){ref-type="ref"}; *Chung and Soden*, [2015](#grl53099-bib-0005){ref-type="ref"}\], for example, to be decomposed into correctable parameterization errors and legitimate differences in model climatology.
Highly accurate calculation of radiative fluxes is of primary interest to the climate modeling community where long free‐running simulations and highly variable greenhouse gas concentrations are the norm. The range of parameterization errors, especially with respect to forcing, implies that specifying changes in greenhouse gas concentrations (as in CMIP) does not completely determine the instantaneous clear‐sky radiative forcing to which each model is subject. In fairness it is the effective radiative forcing, including model‐specific rapid adjustments \[see, e.g., *Sherwood et al.*, [2015](#grl53099-bib-0032){ref-type="ref"}\] and state‐dependent sensitivities, that is relevant to determining the long‐term climate response. Diversity in adjustments \[*Andrews et al.*, [2012](#grl53099-bib-0002){ref-type="ref"}; *Zelinka et al.*, [2014](#grl53099-bib-0037){ref-type="ref"}\] will partly mask and may well outweigh the range of parameterization error. But rapid adjustments themselves can be influenced by radiation parameterization error \[*Ogura et al.*, [2014](#grl53099-bib-0023){ref-type="ref"}\], and, given the maturity of understanding about radiative transfer, global estimates of parameterization error are a necessary first step in efforts to characterize and assess radiative forcing.
Supporting information
======================
######
Readme
######
Click here for additional data file.
######
Data S1
######
Click here for additional data file.
######
Data S2
######
Click here for additional data file.
######
Data S3
######
Click here for additional data file.
The data on which this paper is based, including atmospheric profiles used in the radiative transfer calculations and the results of the calculations as provided by participants, are
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vec
x)\rangle$, or $L[\langle A_0\rangle]$, is used to determine the phase transition temperature $T_c$ as well as critical exponents. The temperature-dependence of the Polyakov loop two-point function relates to the string tension. In the confining phase, for $T<T_c$, and large separations $|\vec x-\vec y|\to\infty$, the two-point function falls off like $$\label{eq:string}
\lim_{|\vec x-\vec y|\to\infty}
\langle L(\vec x) L^\dagger (\vec y) \rangle_{c} \simeq \exp
\left\{-\beta\, \sigma |\vec x-\vec y|\right\}\,.$$ Here, $\langle\cdots \rangle_c$ stands for the connected part of the related correlation function, i.e. $
\langle L(\vec x) L^\dagger (\vec y) \rangle_{c} =
\langle L(\vec x) L^\dagger (\vec y) \rangle-\langle L(\vec x)\rangle
\langle L(\vec y)\rangle$. In turn, its Fourier transform shows the momentum dependence $$\label{eq:stringmomentum}
\lim_{|p|\to 0}
\langle L(0) L^\dagger (p) \rangle_c \simeq \lim_{|p|\to 0}
\0{1}{\pi^2} \0{\beta\sigma} {
((\beta\sigma)^2+p^2)^2}=\0{1}{\pi^2} \0{1} {
(\beta\sigma)^3} \,.$$ We conclude that the Polyakov loop variable has a massive propagator. This directly relates to a massive propagator of $A_0$ in Polyakov gauge.
The approximation scheme is fully set by specifying the regulators $R_{0,k}$ and $R_{\bot,k}$. Naively one would identify the cut-off parameter $k$ in the regulators with the physical cut-off scale $k_{\rm phys}$. For general regulators this is not possible and one deals with two distinct physical cut-off scales, $k_{0,\rm phys}$ and $k_{\bot,\rm phys}$ related to $R_{0,k}$ and $R_{\bot,k}$ respectively, for a detailed discussion see [@Pawlowski:2005xe]. However, within the approximation it is crucial to have a unique effective cut-off scale $k_{\rm phys}=k_{0,\rm
phys}=k_{\bot,\rm phys}$, as different physical cut-off scales $k_{0,\rm phys}\neq k_{\bot,\rm phys}$ necessarily introduce a momentum transfer into the flow which carries part of the physics. This momentum transfer is only fully captured with a non-local approximation to
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|
appearance of $b_{gz}$ with $g>G$ is in the third sum. Since the $b_{uz}$ are a ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis of ${\mathcal{N}}$, that third term $\sum_{g>G} c'_{gz}b_{gz}$ is actually zero.
Now consider where the specific term $b_{Gw}$ appears on the right hand side of . For $g<G$, implies that $\operatorname{{\mathbf{E}}\text{-deg}}c_{gfh}>m$ for each $f, h$ and so $b_{Gw}$ cannot appear in the first sum. Thus it must appear nontrivially in some term $c_{Gf'h'}d'b_{Gw}$ in the second sum. In this case, implies that $\operatorname{{\mathbf{E}}\text{-deg}}c_{Gf'h'} =m$. Hence $d'\in\mathbb C\smallsetminus\{0\}$ and $$a_{Gf'h'} = d'b_{Gw} +\sum_{(uz)\not= (Gw)}
d_{uz}'' b_{uz}.$$ Thus we can replace $b_{Gw}$ by $a_{Gf'h'}$ in our basis for ${\mathcal{N}}$. By ($\dagger$3), the sets $\{a_{G\ell m} : \ell,m\in {\mathbb{Z}}\}$ and $\{b_{Gu} : u\in {\mathbb{Z}}\}$ have equal finite cardinality. After a finite number of steps we therefore have $\{b_{Gu}\} \subseteq
\{a_{G\ell m}\}$ and hence $\{b_{Gu}\}=
\{a_{G\ell m}\}$. This completes the inductive step and hence the proof of the lemma.
We can now pull everything together and prove both Theorem \[main\] and Proposition \[pre-cohh\].
Proof of Proposition \[pre-cohh\] {#subsec-6.21A}
---------------------------------
Recall from Lemma \[thetainjA\] that $\Theta : eJ^k\delta^{k}\to \operatorname{{\textsf}{ogr}}N(k)$ is the natural inclusion. On the other hand, for any $k \geq 0$, Proposition \[grsameA\] implies that the map $\theta: eJ^k\delta^{k}\to
N(k)$ is an isomorphism. Lemma \[filter-injA\](2) therefore implies that $\operatorname{gr}_{\Lambda} N(k) = \operatorname{{\textsf}{ogr}}\theta(eJ^k\delta^{k}) =
\Theta(eJ^k\delta^k) = eJ^k\delta^k$.
Proof of Theorem \[main\] {#subsec-6.21}
-------------------------
\(1) This is immediate from Corollary \[morrat-cor\](1) and Lemma \[Zalgequiv\].
\(2) Fix $i\geq j\geq 0$. Since $c+j$ still satisfies Hypothesis \[main-hyp\], Proposition \[pre-cohh\] implies that $\operatorname{{\textsf}{ogr}}
| 2,413
| 1,012
| 1,132
| 2,540
| 3,653
| 0.77089
|
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|
em$11$}}/ {\hbox{{\rm Br}\kern 0.1em$10$}}]_{case B}$. We assumed the value $[{\hbox{{\rm Br}\kern 0.1em$11$}}/ {\hbox{{\rm Br}\kern 0.1em$10$}}]_{case B} = 0.75$, which is appropriate for $n = 10^2$ and $T_e = 5,000$ K [@chad; @hs87]. For each galaxy, we computed both these ratios for both realizations of the continuum subtraction (with or without the $20\%$ featureless continuum), and used the mean of these four values as the [ $1.7$ ]{}/ ratio, and the standard deviation as an estimate of the uncertainty associated with the continuum subtraction. We also computed the ratio [ $2.06$ ]{}/. Our measured line ratios are listed in table \[tab-line-rats\], along with values of [ $2.06$ ]{}/ from the literature, and a weighted mean for [ $2.06$ ]{}/ that combines new and literature values. (The quoted uncertainty for the weighted mean is the error in the mean.)
Because of the $1.7$ stellar absorption feature, studies that do not subtract the continuum in galaxies with weak [ $1.7$ ]{} will somewhat underestimate the [ $1.7$ ]{} line strength and therefore underestimate . To test the magnitude of this effect, in table \[tab-line-rats\] we list both continuum–subtracted and raw (un–subtracted) [ $1.7$ ]{}/ ratios. For galaxies NGC 3504 and NGC 4102, the lines are so weak relative to the stellar continuum that the [ $1.7$ ]{}/ cannot be measured without continuum subtraction. For the other galaxies in table \[tab-line-rats\], the raw and continuum–subtracted line ratios are very similar; for these galaxies (He 2–10, NGC 3077, NGC 4214, and NGC 4861), the continuum subtraction is not an important source of uncertainty.
MODELLING LINE RATIO BEHAVIOR {#sec:models}
=============================
Because the emission lines used to diagnose effective temperature have different excitation energies, one cannot verify that a particular diagnostic works by simply testing whether it exhibits a one-to-one correlation with another diagnostic. Instead, one must test diagnostics in light
| 2,414
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|
space ${{{\mathbb P}}}^d, \; d\ge 3.$ Then we have $\;\; \chi(G)\le(\omega(G))^2.$
In both cases, there are polynomial time algorithms that, given the lines corresponding to the vertices of $G$, find complete subgraphs $K\subseteq G$ and proper colorings of $G$ with at most $|V(K)|^3$ and $|V(K)|^2$ colors, respectively.
Note that the difference between the two scenarios comes from the fact that parallel lines in the Euclidean space are disjoint, but the corresponding lines in the projective space intersect.
Most computational problems for geometric intersection and disjointness graphs are hard. It was shown by Kratochvíl and Nešetřil [@KrN90] and by Cabello, Cardinal, and Langerman [@CaCL13] that finding the clique number $\omega(G)$ resp. the independence number $\alpha(G)$ of disjointness graphs of segments in the plane are NP-hard. It is also known that computing the chromatic number $\chi(G)$ of disjointness and intersection graphs of segments in the plane is NP-hard [@EET86]. Our next theorem shows that some of the analogous problems are also NP-hard for disjointness graphs of lines in space, while others are tractable in this case. In particular, according to Theorem 3(i), in a disjointness graph $G$ of lines, it is NP-hard to determine $\omega(G)$ and $\chi(G)$. In view of this, it is interesting that one can design polynomial time algorithms to find proper colorings and complete subgraphs in $G$, where the number of colors is bounded in terms of the size of the complete subgraphs, in the way specified in the closing statements of Theorems 1 and 2.
[**Theorem 3.**]{}
*(i) Computing the clique number $\omega(G)$ and the chromatic number $\chi(G)$ of disjointness graphs of lines in ${{\mathbb R}}^3$ or in ${{\mathbb P}}^3$ are NP-hard problems.*
\(ii) Computing the independence number $\alpha(G)$ of disjointness graphs of lines in ${{\mathbb R}}^3$ or in ${{\mathbb P}}^3$, and deciding for a fixed $k$ whether $\chi(G)\le k$, can be done in polynomial time.
The bounding functions in Theorems
| 2,415
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many cardinals $\d=\d(\Gamma)$ where $\Gamma$ has strong closure properties. In fact, we expect that it can be used to prove Theorem 1.1 of [@KKMW] but we certainly haven’t done so. We now start proving [Theorem \[main theorem\]]{}.\
Let $\kappa=\utilde{\delta}^2_1$. By Martin’s theorem (see Theorem 2.31 and Definition 2.30 of [@Jackson]), it is enough to show that $\k$ is $\k$-reasonable, i.e., there is a non-selfdual pointclass $\utilde{\Gamma}$ closed under $\exists^{\mathbb{R}}$ and a map $\phi$ with domain $\mathbb{R}$ satisfying:
1. $\forall x (\phi(x)\subseteq \k\times \k)$,
2. $\forall F: \k\rightarrow \k$, $\exists x \in \mathbb{R} ( \phi(x)=F)$,
3. $\forall \b<\k$, $\forall {\gamma}<\k$, $R_{\b, {\gamma}}\in \utilde{\Delta}$ where
$x\in R_{\b, {\gamma}} \iff \phi(x)(\b, {\gamma}) \wedge \forall {\gamma}^\prime <\k (\phi(x)(\b, {\gamma}^\prime) \rightarrow {\gamma}^\prime={\gamma})$
4. Suppose $\b<\l$, $A\in \exists^{\mathbb{R}}\utilde{\Delta}$, and $A\subseteq R_\b=\{ x : \exists {\gamma}<\k R_{\b, {\gamma}}(x)\}$. Then $\exists {\gamma}_0<\k$ such that $\forall x\in A\exists {\gamma}<{\gamma}_0 R_{\b, {\gamma}}(x)$.
Let $\Gamma=\Sigma^2_1$. We claim that $\utilde{\Gamma}$ is as desired and spend the rest of the proof to argue for it. In what follows, we will freely use the terminology developed for analyzing $\H$ of models of $AD^+$. This terminology has been exposited in many places including [@GeneralizedHjorth], [@StrengthPFA1], [@ATHM], [@CMI], [@OIMT] and more recently in [@SSW]. In particular, recall the definitions of suitable premouse, short tree, maximal tree, short tree iterable and etc. Given a suitable premouse $\P$, we let $\d_\P$ be its Woodin cardinal and $\l_\P$ be the least cardinal which is $<\d_\P$-strong in $\P$.
Suppose $a\in HC$. We say an $a$-premouse $\Q$ is *good* if
1. $\Q$ is $(\omega, \omega_1)$-iterable,
2. $\Q{\vDash}ZFC-Powerset$+“there are no Woodin cardinals" +“there is a largest cardinal"
3. $\Q$ is full, i.e., for every cutpoint $\xi$ o
| 2,416
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| 2,526
| 2,206
| 4,106
| 0.768077
|
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|
ypes of size $n$. We denote by $m_{d,\lambda}(\omega)$ the multiplicity of $(d,\lambda)$ in $\omega$. As with partitions it is sometimes convenient to consider a type as a collection of integers $m_{d,\lambda}\geq 0 $ indexed by pairs $(d,\lambda)\in \Nstar\times \; \calP^*$. For a type $\omega=(d_1,\omega^1)(d_2,\omega^2)\cdots (d_r,\omega^r)$, we put $n(\omega)=\sum_id_i n(\omega^i)$ and $[\omega]:=\cup_id_i\cdot\omega^i$.
When considering elements $a_\muhat\in \Lambda_k$ indexed by multi-partitions $\muhat=(\mu^1,\ldots,\mu^k)\in \calP^k$, we will always assume that they are homogeneous of degree $(|\mu^1|,\ldots,|\mu^k|)$ in the set of variables $\x_1,\dots,\x_k$.
Let $\{a_\muhat\}_{\muhat\in\calP^k}$ be a family of symmetric functions in $\Lambda_k$ indexed by multi-partitions.
We extend its definition to a *multi-type* $\omhat=(d_1,\omhat^1)\cdots(d_s,\omhat^s)$ with $\omhat^p\in(\calP_{n_p})^k$, by
$$a_\omhat:=\prod_p\psi_{d_p}(A_{\omhat^p}).$$
For a multi-type $\omhat$ as above, we put $$C_\omhat^o:=\begin{cases}\frac{\mu(d)}{d}(-1)^{r-1}\frac{(r-1)!}{\prod_\muhat m_{d,\muhat}(\omhat)!}\,\text{ if } d_1=\cdots=d_r=d.\\0\,\text{ otherwise.}
\end{cases}$$ where $m_{d,\muhat}(\omhat)$ with $\muhat\in\calP^k$ denotes the multiplicity of $(d,\muhat)$ in $\omhat$.
We have the following lemma (see [@hausel-letellier-villegas §2.3.3] for a proof).
Let $\{A_\muhat\}_{\muhat\in\calP^k}$ be a family of symmetric functions in $\Lambda_k$ with $A_0=1$. Then $$\Log\left(\sum_{\muhat\in\calP^k}A_\muhat T^{|\muhat|}\right)=\sum_{\omhat}C_\omhat^oA_\omhat T^{|\omhat|}$$ where $\omhat$ runs over multi-types $(d_1,\omhat^1)\cdots(d_s,\omhat^s)$.\[Log-w\]
The formal power series $\sum_{n\geq 0}a_nT^n$ with $a_n\in \Lambda_k$ that we will consider in what follows will all have $a_n$ homogeneous of degree $n$. Hence we will typically scale the variables of $\Lambda_k$ by $1/T$ and eliminate $T$ altogether.
Given any family $\{a_\mu\}$ of symmetric functions indexed by partitions $\mu\in \P$ and a multi-partition $\mu
| 2,417
| 1,402
| 2,590
| 2,279
| 2,948
| 0.775886
|
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|
s an example of a neighbourhood base at that point,* an observation that has often led, together with Eq. (\[Eqn: TB\]), to the use of the term “neighbourhood” as a synonym for “non-empty open set”. The distinction between the two however is significant as neighbourhoods need not necessarily be open sets; thus while not necessary, it is clearly sufficient for the local basic sets $B$ to be open in Eqs. (\[Eqn: TBx\]) and (\[Eqn: TBx\_nbd\]). If Eq. (\[Eqn: TBx\_nbd\]) holds for every $x\in N$, then the resulting $\mathcal{N}_{x}$ reduces to the topology induced by the open basic neighbourhood system $\mathcal{B}_{x}$ as given by Eq. (\[Eqn: nbd-topology\]).
In order to check if a collection of subsets $_{\textrm{T}}\mathcal{B}$ of $X$ qualifies to be a basis, it is not necessary to verify properties $(\textrm{T}1)-(\textrm{T}3)$ of Tutorial4 for the class (\[Eqn: TB\_topo\]) generated by it because of the properties (TB1) and (TB2) below whose strong affinity to (NB1) and (NB2) is formalized in Theorem A1.1.
**Theorem A1.1.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of subsets of $X$ is a* *topological basis on* $X$ *iff*
(TB1) *$X=\bigcup_{B\in\,_{\textrm{T}}\mathcal{B}}B$. Thus each $x\in X$ must belong to some* $B\in\,_{\textrm{T}}\mathcal{B}$ *which implies the existence of a* *local base* *at each point* *$x\in X$.*
(TB2) *The intersection of any two members $B_{1}$ and $B_{2}$ of* $_{\textrm{T}}\mathcal{B}$ *with $x\in B_{1}\bigcap B_{2}$* ***contains another member of* $_{\textrm{T}}\mathcal{B}$: $(B_{1},B_{2}\in\,_{\textrm{T}}\mathcal{B})\wedge(x\in B_{1}\bigcap B_{2})\Rightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!:x\in B\subseteq B_{1}\bigcap B_{2})$.$\qquad\square$
This theorem, together with Eq. (\[Eqn: TB\_topo\]) ensures that a given collection of subsets of a set $X$ satisfying (TB1) and (TB2) induces *some* topology on $X$; compared to this is the result that *any* collection of subsets of a set $X$ is a *subbasis* for some topology on $X$. If $X$, however, already has a topolog
| 2,418
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| 3,410
| 2,314
| 2,641
| 0.778199
|
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|
defined above is a very general mesh/partition on $\Omega$, as we do not specify the shape and conformal property of $K\in \mathcal{T}_h$. The interior penalty discontinuous Galerkin (IPDG) method can be extended to such a general mesh, without any modification of the formulation. However, to ensure its approximation rate, certain conditions must be imposed on $\mathcal{T}_h$ and the discrete function spaces. In this paper, we are interested in discussing the minimum requirements of such conditions. First, we shall give the formulation of the interior penalty discontinuous Galerkin method.
Let $V_K$ be a finite dimensional space of smooth functions defined on $K \in \mathcal{T}_h$. Define $$V_h=\{v\in L^2(\Omega):v|_K\in V_K,\textrm{ for all } K\in\mathcal{T}_h\},$$ and $$V(h)=V_h+\left( H_0^1(\Omega)\cap \prod_{K\in\mathcal{T}_h} H^{r}(K) \right),\qquad \textrm{where }r>\frac{3}{2}.$$ For any internal interface $e = \bar{K_1}\cap\bar{K_2} \in \mathcal{E}_h$, let $\bn_1$ and $\bn_2$ be the unit outward normal vectors on $e$, associated with ${K_1}$ and ${K_2}$, respectively. For $v\in V(h)$, define the average $\{\nabla v\}$ and jump $[v]$ on $e$ by $$\{\nabla v\} = \frac{1}{2}\left(\nabla v|_{K_1} + \nabla v|_{K_2} \right),\qquad
[v] = v|_{K_1} \bn_1 + v|_{K_2} \bn_2.$$ On any boundary segment $e= \bar{K}\cap\partial\Omega$, the above definitions of average and jump need to be modified: $$\{\nabla v\} = \nabla v|_{K},\qquad [v] = v|_{K} \bn_K,$$ where $\bn_K$ is the unit outward normal vector on $e$ with respect to $K$.
Define a bilinear form on $V(h)\times V(h)$ by $$\begin{aligned}
A(u,v) =& \sum_{K\in \mathcal{T}_h}(\nabla u,\nabla v)_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla u\},\, [v]\rangle_e \\
&\quad -\delta \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [u]\rangle_e + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \langle[u],\, [v]\rangle_e,
\end{aligned}$$ where $\delta = \pm 1,\, 0$ and $\alpha>0$. when $\delta=1$, the bilinear form $A(\cdot,\cdot)$ is symmetric. The constant $\alpha$ is usually
| 2,419
| 1,641
| 2,306
| 2,208
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| 0.770055
|
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|
on the different particles from the general expressions Eqs. (\[eq:fp\_unperturb\])-(\[eq:fp\_perturb\]) and from the approximate expressions of Sects. \[sec:inertial\] and \[sec:drag\].
Numerical evaluation of the inertial force {#sec:numerical-inertial}
------------------------------------------
We consider a test particle traveling at the same speed $V_p$ as the impurity or laser beam producing the flow, but located at a distance of 10 coherence lengths in front of it, and 20 coherence lengths in the $y$ direction apart from it. This distance is sufficient to avoid inclusion of $\mathcal U_p$ or $V_{ext}$ in Eq. (\[eq:psi0lin\]) for the neighborhood of the test particle. Condensate and test particle interact via a coupling constant $g_p'$ sufficiently small so that the full force on the later, Eq. (\[eq:fp2\]), is well approximated by the inertial part Eq. (\[eq:fp\_unperturb\]), being the perturbation the particle induces on the flow, and thus the force (\[eq:fp\_perturb\]) completely negligible.
Figure (\[fig:inertial\_force\]) shows, for different values of $V_p = 0.1, 0.8, 1$ at $\gamma=0$, the $x$ component of the time-dependent force produced by the transient flow inhomogeneities hitting the test particle in the form of sound waves. The size of the test particle, taking several values, is called $a'$ to distinguish it from the size $a$ of the particle producing the flow perturbation. Blue lines are computed from the exact Eq. (\[eq:fp2\]) or equivalently from Eq. (\[eq:fp\_unperturb\]) to which it reduces for sufficiently small $g_p'$. Because of the rather explicit appearance of the interaction potential in this formula, we label the blue lines in Fig. (\[fig:inertial\_force\]) as ‘potential force’. High frequency waves arrive before low-frequency ones, because its larger sound speed. We also see how the frequencies become Doppler-shifted for increasing $V_p$. We have derived in Sect. \[sec:inertial\] several approximate expressions for the inertial force. First, Eq. (\[eq:F0full\]) is obtained
| 2,420
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A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&+&
\sum_{n} \sum_{k} \sum_{K} \sum_{m \neq k}
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( h_{m} - h_{k} ) }
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&-&
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{m} ) }
e^{- i ( \Delta_{K} - h_{n} ) x}
+
\frac{ ( h_{k} + 2 h_{m} - 3 \Delta_{K} ) }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{K} - h_{m} )^2 }
e^{- i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{m} )^2 ( h_{m} - h_{k} )^2 }
e^{- i ( h_{m} - h_{n} ) x}
-
\frac{ \left( \Delta_{K} + 2 h_{m} - 3 h_{k} \right) }{ ( \Delta_{K} - h_{k} )^3 ( h_{m} - h_{k} )^2 }
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&+&
\sum_{n} \sum_{k} \sum_{K \neq L}
\biggl[
- \frac{x^2}{2}
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) }
e^{- i ( h_{k} - h_{n} ) x}
- (ix)
\frac{ \left( \Delta_{K} + \Delta_{L} - 2 h_{k} \right) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 }
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{L} - \Delta_{K} ) }
e^{- i ( \Delta_{K} - h_{n} ) x}
+ \frac{ 1 }{ ( \Delta_{L} - h_{k} )^3 ( \Delta_{L} - \Delta_{K} ) }
e^{- i ( \Delta_{L} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{L} - h_{k} )^3 }
\biggl\{
\Delta_{L}^2 + \Delta_{L} \Delta_{K} + \Delta_{K}^2 - 3 h_{k} ( \Delta_{L} + \Delta_{K} ) + 3 h_{k}^2
\biggr\}
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\lef
| 2,421
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| 2,458
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|
e without the CKM factors, respectively. We emphasize that the SM parametrization in Eqs. (\[eq:decomp-1\])–(\[eq:decomp-4\]) is completely general and independent from U-spin considerations. For example, further same-sign contributions in the CF and DCS decays can be absorbed by a redefinition of $t_0$ and $t_2$, see Ref. [@Brod:2012ud]. The meaning as a U-spin expansion only comes into play if we assume a hierarchy for the parameters according to their subscript.
The letters used to denote the amplitudes should not be confused with any ideas about the diagrams that generate them. That is, the use of $p_0$ and $t_0$ is there since in some limit $p_0$ is dominated by penguin diagrams and $t_0$ by tree diagrams. Yet, this is not always the case, and thus it is important to keep in mind that all that we do know at this stage is that the above is a general reparametrization of the decay amplitudes, and that each amplitude arises at a given order in the U-spin expansion. In the topological interpretation of the appearing parameters, $t_0$ includes both tree and exchange diagrams, which are absorbed [@Muller:2015lua]. Moreover, $s_1$ contains the broken penguin and $p_0$ includes contributions from tree, exchange, penguin and penguin annihilation diagrams [@Muller:2015lua; @Brod:2012ud].
We note that the U-spin parametrization is completely general when we assume no CPV in the CF and DCS decays, which is also the case to a very good approximation in the SM. Beyond the SM, there can be additional amplitude contributions to the $\overline{D}^0\rightarrow K^+\pi^-$ and $\overline{D}^0\rightarrow \pi^+K^-$ decays which come with a relative weak phase from CP violating new physics. We do not discuss this case any further here.
In terms of the above amplitudes, the branching ratios are given as $$\begin{aligned}
\mathcal{BR}(D\rightarrow P_1P_2 ) &= \vert \mathcal{A}\vert^2 \times
\mathcal{P}(D,P_1,P_2)\,,
\nonumber \\
\mathcal{P}(D,P_1,P_2)
| 2,422
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| 2,682
| 2,252
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|
sed $\hat{A}$ as \_[ab]{} = \_[ab]{}/2 + \_[ab]{}, where $\hat{h}_{ab}$ is symmetric and $\hat{B}_{ab}$ is anti-symmetric. The trace part of $\hat{h}_{ab}$ is denoted [^4] \_a\^a.
The effective metric (\[metric\]) is \_ = \_\^a\_[ab]{}\_\^b \_ + \_[ab]{} + 2 \_[ab]{} + [O]{}(A\^2). We also have \_\^[a]{}\_a\^ = D (1 + + [O]{}(A\^2)), and thus = 1 + D + [O]{}(A\^2). One can ignore the integration measure $\det\hat{e}$ when the 3-point vertex under consideration does not involve $\hat{\phi}$ as an external leg.
Expanding the field strength in powers of $\hat{A}$, we have F\_[abc]{} &=& - \_a\^\_b\^(\_\_[c]{} - \_\_[c]{}),
Then, F\_[abc]{}F\^[abc]{} &=& \^[(0)]{}\_[abc]{}\^[(0)abc]{} - 2 \^[(0)abc]{}\_a\^[d]{}\^[(0)]{}\_[dbc]{} + [O]{}(A\^4), \[F2\] where \^[(0)]{}\_[a]{} \_\_[a]{} - \_\_[a]{}.
Similar to the perturbative expansion in terms of $A$, the second and third terms in the action (\[general-action\]) do not contribute to 3-point interactions of the traceless part of $h_{ab}$.
It is remarkable that for this action (\[general-action\]) there is a single 3-point vertex for graviton interaction (the second term in (\[F2\])) \^[(3)]{} = - 2 \^[(0)abc]{}\_a\^[d]{}\^[(0)]{}\_[dbc]{}. This is precisely of the form $F_{(0)}AF_{(0)}$ needed to explain the double-copy procedure at tree level as we discussed in Sec.\[heuristic\]. This is of course also a significant simplification compared with the usual expression of GR.
Higher-Form Gauge Symmetries {#HigherGauge}
============================
In the above we have focused on the gauge symmetries with 1-form potentials and 0-form gauge parameters. We can also apply the same notion of generalization to gauge symmetries with higher-form gauge potentials. The basic ideas of our generalization are the following:
1. \[g1\] The symmetry generators do not have to be factorizable in the form T({f, a}) = \_n f\_n(x) T\_[a\_n]{}, \[factorizable\] where $f_n(x)$’s are functions on the base space and $T_a$’s are elements of a finite-dimensional Lie algebra.
2. \[g2\]
| 2,423
| 971
| 2,906
| 2,483
| null | null |
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|
o, there is no way to rank the relative importance of hazards of mixed type. Loss of the ability to compare risks of all hazards is a flagrant omission. Under correct physics, risks of multiple hazards are additive. This is not the case under MIL-STD-882.
The formal sense of *assurance* is lost by these definitional variants. Being quantitatively assured requires a limit value on proportion or mean deviation and a statement of statistical confidence for this limit; the Standard clearly lacks this characteristic. Properly assurance is a numeric quantity associated with statistical control of risk, not an engineering activity to further psychological confidence. Despite that its developers may express great confidence in the methodology, software built under the Standard is not quantitatively assured.
Repair of MIL-STD-882 {#S:REPAIR}
---------------------
Rehabilitation of MIL-STD-882 is straightforward. It must be amended to contain an engineering introduction to statistical risk for software, including allied procedures. This subject matter is covered here in mathematical language, but should be presented differently for engineers’ consumption. The revised Standard should distinguish between formal assurance and design confidence, and classify what procedures support either. Generally, the concerns associated with design risk align with developmental software engineering, while those of statistical risk align with responsibilities of system safety engineering, part of systems engineering. Software and Systems Engineering should not duplicate each other’s efforts.
Groundwork {#Ch:GROUNDWORK}
==========
This appendix examines ensembles, a fundamental structure in the theory of systems which formalizes the notion of stimulus and response. From this start, discussion proceeds into the Cartesian product, choice spaces and subspaces, and partitions of choice spaces into persistent and volatile components. Dyadic notation is introduced.
Ensemble {#S:ENSEMBLE}
--------
An ensemble is a special form of a more
| 2,424
| 321
| 3,234
| 2,655
| null | null |
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|
}_{t}f(x)={\mathbb{E}}(P_{t}^{N,Z}f(x))=\sum_{m=0}^{\infty
}I_{m}(f)(x)$$with$$I_{0}(f)(x)={\mathbb{E}}(1_{\{N(t)=0\}}P_tf(x)) =e^{-\rho t}P_tf(x)$$ and for $m\geq 1$, $$\begin{aligned}
I_{m}(f)(x) &={\mathbb{E}}\Big(1_{\{N(t)=m\}}\frac{m!}{t^m}%
\int_{0<t_{1}<...<t_{m-1}<t_{m}\leq
t}P_{t-t_{m}}\prod_{i=0}^{m-1}U_{Z_{m-i}}P_{t_{m-i}-t_{m-i-1}}f(x)dt_{1}%
\ldots dt_{m}\Big) \\
&=\rho^me^{-\rho t}{\mathbb{E}}\Big(\int_{0<t_{1}<...<t_{m-1}<t_{m}\leq t}%
\Big(\prod_{i=0}^{m-1}P_{t_{m-i+1}-t_{m-i}}U_{Z_{m-i}}\Big)%
P_{t_1}f(x)dt_{1}\ldots dt_{m}\Big),\end{aligned}$$ in which $t_0=0$ and $t_{m+1}=t$. We come now to the hypothesis on $\phi .$ We assume that for every $z\in E,$ $\phi _{z}\in C^{\infty }({\mathbb{R}}%
^{d})$ and $\nabla \phi _{z}\in C_{b}^{\infty }({\mathbb{R}}^{d})$ and that for every $q\in {\mathbb{N}}$$$\begin{aligned}
\left\Vert \phi \right\Vert _{1,q,\infty }& :=\sup_{z\in E}\left\Vert \phi
_{z}\right\Vert _{1,q,\infty }=\sum_{1\leq \left\vert \alpha \right\vert
\leq q}\sup_{z\in E}\sup_{x\in {\mathbb{R}}^{d}}\left\vert \partial
_{x}^{\alpha }\phi (z,x)\right\vert <\infty, \label{J4'} \\
\widehat{\phi }& :=\sup_{z\in E}\left\vert \phi _{z}(0)\right\vert <\infty .
\label{J4''}\end{aligned}$$Moreover we define $\sigma (\phi _{z})=\nabla \phi _{z}(\nabla \phi
_{z})^{\ast }$ and we assume that there exists a constant $\varepsilon (\phi
)>0$ such that for every $z\in E$ and $x\in {\mathbb{R}}^{d}$$$\det \sigma (\phi _{z})(x)\geq \varepsilon (\phi ). \label{J3}$$
\[rem-J\] We recall that in Appendix \[app:ibp\] we have denoted $%
V_{\phi _{z}}f(x)=f(\phi _{z}(x))=U_{z}f(x).$ With this notation, under ([J4’]{}), (\[J4”\]), (\[J3\]) we have proved in (\[ip10\]) and ([ip12]{}) that, for every $z\in E$, $$\begin{aligned}
\left\Vert U_{z}f\right\Vert _{q,-\kappa ,\infty }& \leq C1\vee \widehat{%
\phi }^{2\kappa }\left\Vert \phi \right\Vert _{1,q,\infty }^{q+2\kappa
}\left\Vert f\right\Vert _{q,-\kappa ,\infty }, \label{J5} \\
\left\Vert U_{z}^{\ast }f\right\Vert _{q,\kappa ,p}& \leq C\,\frac{1\vee
\wid
| 2,425
| 1,061
| 1,095
| 2,368
| null | null |
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|
enstrophy in finite time might be to increase its kinetic energy by allowing for a smaller instantaneous rate of growth (i.e., with an exponent $2 < \alpha \le 3$ instead of $\alpha = 3$). This can be achieved by prescribing an additional constraint in the formulation of the variational optimization problem, resulting in
\[pb:maxdEdt\_KE\] $$\begin{aligned}
{\widetilde{\mathbf{u}}_{\K_0,\E_0}}& = & \mathop{\arg\max}_{{\mathbf{u}}\in{\mathcal{S}_{\K_0,\E_0}}} \, \R({\mathbf{u}}) \\
{\mathcal{S}_{\K_0,\E_0}} & = & \left\{{\mathbf{u}}\in H_0^2(\Omega)\,\colon\,\nabla\cdot{\mathbf{u}}= 0, \; \K({\mathbf{u}}) = \K_0, \; \E({\mathbf{u}}) = \E_0 \right\}.\end{aligned}$$
It differs from problem \[pb:maxdEdt\_E\] in that the maximizers are sought at the intersection of the original constraint manifold ${\mathcal{S}_{\E_0}}$ and the manifold defined by the condition $\K({\mathbf{u}})
= \K_0$, where $\K_0 \le (2\pi)^2 \E_0$ is the prescribed energy. While computation of such maximizers is more complicated, robust techniques for the solution of optimization problems of this type have been developed and were successfully used in the 2D setting by [@ap13a]. Preliminary results obtained in the present setting by solving problem \[pb:maxdEdt\_KE\] for $\K_0 =
1$ are indicated in figure \[fig:K0E0\], where we see that the flow evolutions starting from ${\widetilde{\mathbf{u}}_{\K_0,\E_0}}$ do not in fact produce a significant growth of enstrophy either. An alternative, and arguably more flexible approach, is to formulate this problem in terms of multiobjective optimization [@k99] in which the objective function $\R({\mathbf{u}})$ in problem \[pb:maxdEdt\_E\] would be replaced with $$\R_{\eta}({\mathbf{u}}) := \eta\, \R({\mathbf{u}}) + (1-\eta)\, \K({\mathbf{u}}),
\label{eq:Rm}$$ where $\eta \in [0,1]$. Solution of such a multiobjective optimization problem has the form of a “Pareto front” parameterized by $\eta$. Clearly, the limits $\eta \rightarrow 1$ and $\eta \rightarrow 0$ correspond, respectively, to the extreme vo
| 2,426
| 1,038
| 2,391
| 2,310
| 3,951
| 0.769035
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|
bda}\,, &
\phi &= \Phi + \frac{T}{\lambda}\,, &
r &= 1 + \lambda R\,.\end{aligned}$$ We also introduce a new coordinate $u$ for the polar angle via $u=\cos\theta$. The NHEK limit is then obtained by taking the $(a\to M, \lambda \to 0)$ limit of the Kerr metric in these coordinates, which yields the line element $$\begin{aligned}
\text{d}s^2 =
2 \Gamma(u)\bigg[ &-R^2\,\text{d}T^2 + \frac{\text{d}R^2}{R^2}
+ \frac{\text{d}u^2}{1 - u^2} \\ \nonumber
&+ \Lambda(u)^2(\text{d}\Phi + R\,\text{d}T)^2 \bigg], \end{aligned}$$ where $\Gamma(u) = (1 + u^2)/2$ and $\Lambda(u) = 2 \sqrt{1-u^2}/ (1 + u^2)$. This metric is interpreted on the region $T\in(-\infty,+\infty)$, $\Phi\in[0,2\pi)$, $R\in(0,+\infty)$, $u\in[-1,1]$.
From now on we will refer to $(T,\Phi,R,u)$ as *Poincaré coordinates*. The $T,R$-coordinates of NHEK are similar to the Poincaré coordinates on the two-dimensional anti-de Sitter space $AdS_2$, which only cover a subspace of the global spacetime called the *Poincaré patch*. In particular, the $u=\pm1$ submanifolds are both precisely $AdS_2$. We can make this metric geodesically complete by defining the *global coordinates* $(\tau,\varphi,\psi,u)$ according to [@Bardeen:1999px] $$\begin{aligned}
T &=\frac{\sin\tau}{\cos\tau - \cos \psi}, \quad
R = \frac{\cos\tau - \cos \psi}{\sin \psi},\\ \nonumber
\Phi &= \varphi + \ln\left|\frac{\cos\tau - \sin\tau \cot\psi}{1+\sin\tau \csc\psi}\right|,\end{aligned}$$ where $\tau \in (-\infty,+\infty)$, $\psi \in [0, \pi]$, $\varphi \sim \varphi + 2\pi$. The NHEK metric in global coordinates is $$\begin{aligned}
\text{d}s^2 = 2 \Gamma(u)\bigg[& (-\text{d}\tau^2 + \text{d}\psi^2)\csc^2\psi
+ \frac{\text{d}u^2}{1 - u^2} + \\ \nonumber
&+\Lambda(u)^2(\text{d}\varphi - \cot\psi\,\text{d}\tau)^2 \bigg].\end{aligned}$$
The NHEK spacetime has four Killing vector fields (KVFs), which generate the isometry group $G\equiv {\ensuremath{SL(2,\mathbb{R})\times U(1)}}$. The four generators in Poincaré coordinates are given by $$\begin{a
| 2,427
| 5,076
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|
olume $\text{(fm}^4\text{)}$ Configurations
---- ----------------- --------- ---------- ------------------------------- ----------------
1w $16^3\times 32$ 5.70 0.179 $2.87^3 \times 5.73$ 100
1i $16^3\times 32$ 4.38 0.166 $2.64^3 \times 5.28$ 100
2 $10^3\times 20$ 3.92 0.353 $3.53^3 \times 7.06$ 100
3 $8^3 \times 16$ 3.75 0.413 $3.30^3 \times 6.60$ 100
4 $16^3\times 32$ 3.92 0.353 $5.65^3 \times 11.30$ 100
5 $12^3\times 24$ 4.10 0.270 $3.24^3 \times 6.48$ 100
6 $32^3\times 64$ 6.00 0.099 $3.18^3 \times 6.34$ 75
: Details of the lattices used to calculate the gluon propagator. Lattices 1w and 1i have the same dimensions and approximately the same lattice spacing, but were generated with the Wilson and improved actions respectively. Lattice 6 was generated with the Wilson action.[]{data-label="table:latlist"}
Gauge fixing on the lattice is achieved by maximizing a functional, the extremum of which implies the gauge fixing condition. The usual Landau gauge fixing functional implies that $\sum_\mu {\partial}_\mu A_\mu
= 0$ up to [${\cal O}(a^{2})$]{}. To ensure that gauge dependent quantities are also [${\cal O}(a^{2})$]{} improved, we implement the analogous [${\cal O}(a^{2})$]{} improved gauge fixing.[@LandauGaugeDE] The dimensionless lattice gluon field $A_{\mu}(x)$ is calculated from the link variables in the usual way, which agrees with the continuum to ${\cal O}(a^2)$. We then calculate the scalar part of the propagator $$D(x-y) = \sum_\mu \langle A_\mu(y) A_\mu(x) \rangle \, .$$ To isolate the nonperturbative behavior of the gluon propagator, we can divide the propagator by its lattice tree level form (i.e., that of lattice perturbation theory).[@long_glu] For the momentum space gluon propagator $D(q^2)$, we see that in the continuum $q^2D(q^2)$ will approach a constant
| 2,428
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|
X\_a=\_[i\_a]{}-\_[i\_a]{}\^A\_[i\_A]{}-|\_[i\_a]{}\^A\_[i\_A]{},A=1,…,k.
Substituting expressions (\[eq:3.28\]) into (\[eq:3.31\]) or (\[eq:3.32\]) and simplifying gives the unified form
\[eq:3.34\] [( \^[( + )]{}\_[(a\_1.]{}[( + )]{}…\_[[a\_s]{})]{}[( + )]{})]{}=0.
The index $s$ is either $\max(s)=q-1$ or $\max(s)=2k-1$, we choose the one which is more convenient from the computational point of view. In this case, for $k\geq 1$ the two approaches, CSM and GMC, become essentially different and, as we demonstrate in the following example, the CSM can provide rank-$2k$ solutions which are not Riemann $2k$-waves as defined by the GMC since we weaken the integrability conditions (\[eq:2.1\]) for the wave vectors $\lambda$ and $\bar{\lambda}$.
####
A change of variable on $X\times U$ allows us to rectify the vector fields $X_a$ and considerably simplify the structure of the overdetermined system (\[eq:3.21\]) which classifies the preceding construction of multimodes solutions. For this system, in the new coordinates, we derive the necessary and sufficient conditions for the existence of rank-$2k$ solutions of the form (\[eq:3.5\]). Suppose that there exists an invertible $2k\times 2k$ subblock matrix
\[eq:3.35\] H=(\^s\_t), 1s,t2k,
of the larger matrix $\Lambda\in {\mathbb{C}}^{2k\times p}$, then the independent vector fields $X_a$ can be written as
\[eq:3.36\] X\_a=\_[x\^[a+2k]{}]{}-(H\^[-1]{})\^s\_t\^t\_[a+2k]{}\_[x\^A]{},
which have the required form (\[eq:3.15i\]) and for which the orthogonality conditions (\[eq:3.14\]) are fulfilled. We introduce new coordinate functions
\[eq:3.37\] z\^1=r\^1(x,u),…, z\^[k]{}=r\^[k]{}(x,u),z\^[k+1]{}=|[r]{}\^1(x,u),…, z\^[2k]{}=|[r]{}\^[k]{}(x,u),\
z\^[2k+1]{}=x\^[2k+1]{},…,z\^p=x\^p,v\^1=u\^1,…, v\^q=u\^q,
on $X\times U$ space which allow us to rectify the vector fields (\[eq:3.36\]). As a result, we get
\[eq:3.38\] X\_1=,…, X\_[p-2k]{}=.
The $p$-dimensional submanifold invariant under $X_{1},\ldots,X_{p-2k}$, is defined by equations of the form
\[
| 2,429
| 3,183
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|
}{2} + l) \hbar ) \right. \nonumber\\
& & ~~~~ + \hat{a}^i_\pm (u \mp \frac{1}{4} k \hbar ) \nonumber \\
& &~~~~ + \left. \sum_{l=i+1}^N \hat{b}^{il}_{\pm}
( u \pm \frac{1}{2} (\frac{k}{2} + l) \hbar )
- \sum_{l=i+2}^N \hat{b}^{i+1,l}_{\pm}
( u \pm \frac{1}{2} (\frac{k}{2} + l -1) \hbar ) \right\}:~, \label{hpn}\end{aligned}$$
$$\begin{aligned}
& & E^+_i(u) = - \frac{1}{\hbar}
\sum_{m=1}^i :\mbox{exp} \left\{(b+c)^{mi}
( u + \frac{1}{2} (m-1) \hbar ) \right\} \nonumber\\
& & ~~~~\times \left[
\mbox{exp} \left( \hat{b}^{m,i+1}_+
( u + \frac{1}{2} (m-1) \hbar )
-(b+c)^{m,i+1} ( u + \frac{1}{2} m \hbar ) \right) \right. \nonumber \\
& &~~~~~~~~ - \left. \mbox{exp} \left( \hat{b}^{m,i+1}_-
( u + \frac{1}{2} (m-1) \hbar )
-(b+c)^{m,i+1}
( u + \frac{1}{2} (m-2) \hbar ) \right) \right] \nonumber\\
& &~~~~\times
\mbox{exp} \left\{ \sum_{l=1}^{m-1} \left[
\hat{b}^{l,i+1}_+ ( u + \frac{1}{2} (l-1) \hbar )
-\hat{b}^{li}_+ ( u + \frac{1}{2} l \hbar ) \right] \right\} :~, \label{ep}\end{aligned}$$
$$\begin{aligned}
& & E^-_i(u) = - \frac{1}{\hbar} \left\{
\sum_{m=1}^{i-1} :\mbox{exp} \left( (b+c)^{m,i+1}
( u - \frac{1}{2} (k+m) \hbar ) \right) \right. \nonumber\\
& &~~~~\times \left[
\mbox{exp} \left( -\hat{b}^{mi}_- ( u - \frac{1}{2} (k+m) \hbar )
-(b+c)^{mi} ( u - \frac{1}{2} (k+m-1) \hbar ) \right) \right. \nonumber\\
& &~~~~~~~~ - \left.
\mbox{exp} \left( -\hat{b}^{mi}_+ ( u - \frac{1}{2} (k+m) \hbar )
-(b+c)^{mi} ( u - \frac{1}{2} (k+m+1) \hbar ) \right) \right] \nonumber \\
& &~~~~\times \mbox{exp} \left(
\sum_{l=m+1}^i \hat{b}^{l,i+1}_{-}
( u - \frac{1}{2} (k+l-1) \hbar )
- \sum_{l=m+1}^{i-1} \hat{b}^{li}_{-}
( u- \frac{1}{2} (k+l) \hbar) \right. \nonumber\\
& &~~~~~~~~ + \left.\hat{a}^i_- (u) + \sum_{l=i+1}^N \hat{b}^{il}_{-}
( u - \frac{1}{2} (k+l) \hbar)
- \sum_{l=i+2}^N \hat{b}^{i+1,l}_{\pm}
( u- \frac{1}{2} (k + l -1) \hbar ) \right): \nonumber\\
& & ~~~~+ :\mbox{exp} \left( (b+c)^{i,i+1}
( u - \frac{1}{2} (k+i) \hbar ) \right) \nonumber\\
& & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_- (u)
+ \sum_{l=i+1}^N
| 2,430
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|
=(z_x,z_y)\in {{\mathbb Z}}^2$ and $r$ sufficiently large, we have that $\card(N\cap \Lambda(z(X),r))\leq \card(N\cap B(O,|X|)\cap B(X,|X-\mathcal{A}(X)|)),$ the latter quantity being 0 since $[\mathcal{A}(X),X]$ is an edge of the RST, implying that there is no point of $N$ in $\Lambda(z(X),r)$. Thus, for $r\geq \sqrt{2}$: $$\check{X}^{>c}_r\leq Y_r:= \sum_{\substack{z=(z_x,z_y)\in {{\mathbb Z}}^2 \\ z_x\geq r}} {{\bf 1}}_{\{\Lambda(z,r)\cap N=\emptyset\}}, ~ \mbox{ a.s.}\label{2.7}$$Notice that if $r$ and $r'$ are such that $z_x\geq r'\geq r$, then $\Lambda(z,r)\subset \Lambda(z,r')$. This implies that $r\mapsto Y_r$ is almost surely a decreasing function of $r$. Then, if we fix $r_0 \geq \sqrt{2}$, $\forall r\geq r_0$, $\check{X}^{>c}_r\leq Y_{r_0}$. The volume of $\Lambda(z,r_0)$ is of the order of $|z|^2$ and for a given integer $\rho\geq r_0^2$, the number of points $z$ such that $|z|^2=\rho$ is of the order of $\sqrt{\rho}$. Thus, for two positive constants $C$ and $C'$: $$\begin{aligned}
{{\mathbb E}}\big(Y_{r_0}\big)=\sum_{\substack{z=(z_x,z_y)\in {{\mathbb Z}}^2 \\ z_x\geq r_0}} {{\mathbb P}}\big(\Lambda(z,r_0)\cap N=\emptyset\big)\leq C\sum_{\rho\geq r_0^2} \sqrt{\rho} e^{-C'\rho}<+\infty.\label{etape3}\end{aligned}$$It now remains to prove that ${{\mathbb E}}(Y_{r_0}^2)<+\infty$. For this, we compute $$\begin{gathered}
{{\mathbb E}}\Big(Y_{r_0}(Y_{r_0}-1)\Big)= \sum_{\substack{z=(z_x,z_y)\in {{\mathbb Z}}^2,\\ z'=(z'_x,z'_y)\in {{\mathbb Z}}^2 \\ z_x\geq r_0, \ z'_x\geq r_0\\ z_x\not=z'_x}} {{\mathbb E}}\left({{\bf 1}}_{\{\Lambda(z,r_0)\cap N=\emptyset\}}{{\bf 1}}_{\{\Lambda(z',r_0)\cap N=\emptyset\}}\right)\\
\leq 2 \sum_{\rho\geq r_0^2} \sum_{\substack{z=(z_x,z_y)\in {{\mathbb Z}}^2,\\ |z_x|^2=\rho }} \sum_{\substack{z'=(z'_x,z'_y)\in {{\mathbb Z}}^2,\\ z'_x\geq r_0 \\ |z'|\leq |z|}} {{\mathbb P}}\Big(\Lambda(z,r_0)\cap N=\emptyset\Big)
\leq C \sum_{\rho\geq r_0^2} \rho^{3/2}e^{-C'\rho}<+\infty\label{etape4}\end{gathered}$$for two positive constants $C$ and $C'$. and show that ${{\mathbb E}}(Y_{
| 2,431
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|
y_{d-1}-\frac{\epsilon}{\triangle},\
y_{d}+\epsilon+\delta,\
y_{d+1},\ \cdots, \\ \\
y_{n-m-1},\ y_{n-m}-\epsilon-\delta,\ \displaystyle
y_{n-m+1}+\frac{\epsilon}{m}, \ \cdots,\ y_n+\frac{\epsilon}{m}).
\end{array}$$ Now define $x$ as the direct sum of the vectors $(y_1,\dots,y_{t})$ and $\bar{x}'^{\da}$, that is $$x= (y_1,\cdots, y_{t}, \bar{x}'^{\da}).$$ By Lemma 1 we have $x\in T(y)$. On the other hand, $$\sum_{i=1}^{d} x_i = \sum_{i=1}^{d} y_i + \delta> \sum_{i=1}^{d} y_i,$$ $$\sum_{i=1}^{l} x_i \leq \sum_{i=1}^{l} y_i {\ \ \ \rm for \ \ \ \
}1\leq l<d,$$ and $$\begin{array}{rcl}
x_1x_{d}&=&\displaystyle y_1(y_{d}+\epsilon+\delta)\\ \\
&>&y_1y_{d}\geq y_{t+1}^2\\
\\&>&\displaystyle (y_{t+1}-\frac{\epsilon}{\triangle})^2=x_{t+1}^2,
\end{array}$$ so we have $x\not \in M(y)$ from Lemma 2. That completes our proof.$\Box$
Suppose $t$ and $m$ denote the numbers of the components which are equal to $y_n$ and which are equal to $y_1$, respectively. Let $d$ be the maximal index of the components which are greater than $y_{n-t}$. If $d>m+1$ and $y_ny_{d}\leq y_{n-t}^2$ then we can also deduce that $T(y)\not\subseteq M(y)$.
An interesting special case of Theorem \[thm:mrdeterm\] is when $n>4$, if $y_1>y_2>y_3>y_{n-1}>y_n$ and $y_1y_3\geq y_2^2$ then $T(y)\not\subseteq M(y)$.
Theorem \[thm:mrdeterm\] in fact gives us a method to construct a vector $y$ for which $T(y)\not\subseteq M(y)$. To be more specific, given a vector $\bar{y}$ such that $T(\bar{y})\neq S(\bar{y})$, we can derive a desired $y$ by the following two steps. First, add a sufficiently large component to $\bar{y}$ such that the conditions presented in Theorem \[thm:mrdeterm\] are satisfied for the new vector (notice that from Theorem 6 of [@DK01], when $T(\bar{y})\neq S(\bar{y})$, the condition $d<n-m$ in Theorem \[thm:mrdeterm\] holds automatically); second, normalize the vector to $y$ such that it is a probability vector. For example, given $\bar{y}=(0.5,0.25,0.25,0)$, we can derive $y=(3,0.5,0.25,0.25,0)/4$ and $T(y)\not\subseteq M(y)$. Furthermore,
| 2,432
| 1,153
| 1,477
| 2,340
| 3,886
| 0.769523
|
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|
/n')
To:
password = word.strip("\n")
But you might as well just:
password = word.strip()
See the strip documenation:
Return a copy of the string with leading and trailing characters
removed. If chars is omitted or None, whitespace characters are
removed. If given and not None, chars must be a string; the characters
in the string will be stripped from the both ends of the string this
method is called on.
And make sure that red is in your words list Red was in mine so I had to add:
zfile.extractall(pwd=password.lower())
After those changes it all seems to work great:
import zipfile
zfile = zipfile.ZipFile("file.zip")
words = open("/usr/share/dict/words")
for word in words.readlines():
try:
password = word.strip("\n")
zfile.extractall(pwd=password.lower())
print "Password found: "+ password
exit(0)
except Exception, e:
pass
Q:
printing multiple sections of text between two markers in python
I converted this page (it's squad lists for different sports teams) from PDF to text using this code:
import PyPDF3
import sys
import tabula
import pandas as pd
#One method
pdfFileObj = open(sys.argv[1],'rb')
pdfReader = PyPDF3.PdfFileReader(pdfFileObj)
num_pages = pdfReader.numPages
count = 0
text = ""
while count < num_pages:
pageObj = pdfReader.getPage(count)
count +=1
text += pageObj.extractText()
print(text)
The output looks like this:
2019 SEASON
PREMIER DIVISION SQUAD NUMBERS
CLUB: BOHEMIANS
1
James Talbot
GK
2
Derek Pender
DF
3
Darragh Leahy
DF
.... some more names....
2019 SEASON
PREMIER DIVISION SQUAD NUMBERS
CLUB: CORK CITY
1
Mark McNulty
GK
2
Colm Horgan
DF
3
Alan Bennett
DF
....some more names....
2019 SEASON
PREMIER DIVISION SQUAD NUMBERS
CLUB: DERRY CITY
1
Peter Cherrie
GK
2
Conor McDermott
DF
3
Ciaran Coll
DF
I wanted to transform this output to a tab delimited file with three columns: team name, player name, and number. So for the example I gave, the output would look like:
Bohemians
| 2,433
| 2,209
| 284
| 1,892
| null | null |
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|
th $1/5$ solar abundances (“the low–Z models”); neither abundance set includes depletion onto dust grains. (We address the effects of dust in § \[sec:caveats\].) Because Cloudy does not predict the intensity of [ $1.7$ ]{}, we scaled the intensity from [ $4471$]{} Å, which shares the same upper level. For Case B and T$_e=5,000$ K, the [ $1.7$ ]{} line is a factor of $7.4 \times 10^{-3}$ fainter than [ $4471$]{} Å.
Models Using Individual Stars {#sec:indystars}
-----------------------------
We first consider the ratio of \[\] $15.6$ to \[\] $12.8$ . For reference, it requires $22$ eV to make singly–ionized neon, and $41$ eV to make doubly–ionized neon. We took ionizing spectra from the O star models of Pauldrach, Hoffmann, & Lennon (2001), as prepared by @snc, and also the CoStar model spectra of @sk as hardwired in Cloudy. These two stellar libraries predict dramatically different line ratios. Dwarf, giant, and supergiant Pauldrach stars all produce a maximum \[\]/\[\] ratio of $10$ at $=50,000$ K. By contrast, the CoStar dwarf and giant atmospheres yield \[\]/\[\]$=40$ at $=50,000$ K. At $=35,000$ K, the predicted line ratios disagree by an order of magnitude. @fs-m82 used Pauldrach atmospheres and an earlier version of Cloudy to make their figure 8, which our Pauldrach models reproduce.
The other mid–infrared line ratios also show this discrepancy. CoStar models predict ten times higher \[\]/\[\] and \[\]/\[\] ratios than Pauldrach models for most of the $25,000<{\hbox{T$_{eff}$}}<50,000$ K range; for \[\]/\[\] and \[\]/\[\], CoStar gives $2$ and $3$ times higher ratios, respectively. The near–infrared line ratios [ $1.7$ ]{}/ and [ $2.06$ ]{}/ are not sensitive to the choice of stellar atlas.
It is sobering that current O star models predict such different mid–infrared line ratio strengths. On the bright side, this sensitivity suggests that mid–IR line ratios may provide astrophysical tests of O star spectral models in simple regions. @giveon performe
| 2,434
| 2,881
| 2,803
| 2,356
| null | null |
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|
=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})$. Let us first show the validity of equation . The map $\textbf{D} (\widehat{\mathcal O^!}) (\vec X;\varnothing)\to \widehat{ \mathcal O}(\vec X;\varnothing)$ expanded in degrees of $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)$ is written as $\cdots \stackrel{{\partial}}{{{\longrightarrow}}} \textbf{D}(\widehat{\mathcal O^!})
(\vec X;\varnothing)^{-1} \stackrel{{\partial}}{{{\longrightarrow}}}
\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^0
\stackrel{proj}{{\longrightarrow}}\widehat{\mathcal O}(\vec X;\varnothing)$. As $\mathcal O$ and thus $\mathcal O^!$ are quadratic, we have the following identification, using the language and results of lemma \[O\_hat\_quadratic\]: $$\begin{aligned}
\widehat{\mathcal O}(\vec X;\varnothing)&=&\mathcal F(
\widehat{E})/(R,G) (\vec X;\varnothing) \\
\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^0&=&
\bigoplus_{ \substack{
\text{binary trees }T \\
\text{of type }(\vec X;\varnothing)
}} (\widehat{E^\vee}(T))\otimes {\mathrm{Det}}(T) =\mathcal
F(\widehat{E})(\vec X;\varnothing)\\ \textbf{D}(\widehat{\mathcal
O^!}) (\vec X;\varnothing)^{-1}&=&\bigoplus_{ \substack{
\text{trees }T\text{ of type }(\vec X;\varnothing), \\
\text{binary vertices, except}\\
\text{one ternary vertex}
}} \left(\widehat{\mathcal O^!}(T)\right)\otimes {\mathrm{Det}}(T) \\ &=&
\left\{
\begin{array}{c}
\text{space of relations in $\mathcal F(\widehat{E})(\vec X;\varnothing)$}\\
\text{generated by $R$ and $G$}
\end{array}
\right\}\end{aligned}$$ The last equality follows, because the inner product relations for $\mathcal O^!$, which are the relation space $G$ for the cyclic quadratic operad $\mathcal O^!$ from lemma \[O\_hat\_quadratic\], are the orthogonal complement of the inner product relations for $\mathcal O$. Hence, the map $proj$ is surjective with kernel ${\partial}\big(\textbf{D}(\widehat{\mathcal
| 2,435
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sociated channel matrices may be correlated. Thus, taking the dependent reconfiguration states into consideration is an interesting future work, where we can have correlated and non-identically distributed $R_{\psi}$. While this paper has adopted the VCR as the analytical channel model, as mentioned earlier, there are some potential limitations on the utility of VCR in practical mmWave systems. Thus, investigating the reconfigurable antennas for mmWave systems with other channel models, e.g., the S-V model, is an important and interesting future work. Furthermore, future work on effective channel estimation techniques with low latency for mmWave systems with reconfigurable antennas is needed. Also, the associated complexity of channel estimation needs to be taken into account for future system designs. For example, if the channel matrices associated with different reconfiguration states are (strongly) correlated, a potential scheme for reconfiguration state selection and beam selection requiring a low-complexity channel estimation may start by setting the reconfiguration state to a random initial state. After finding the optimal beams in the initial state, one can reselect another state based on the optimal beams for the initial state.
Proof of Proposition \[Prop:1\] {#App:proofaveGa}
===============================
With the Gaussian approximation of the distribution of $R_{\psi}$, we can then approximate $R_{{\widehat{\psi}}}$ as the maximum of $\Psi$ i.i.d. Gaussian random variables, and $$\begin{aligned}
\label{eq:Rcpsibar}
\bar{R}_{{\widehat{\psi}}}&\approx\int_0^\infty 1-\left(F_{R_\psi}(x)\right)^\Psi \mathrm{d}x\notag\\
&=
\int_0^\infty 1-\frac{1}{2^\Psi}\left(1+\mathrm{erf}\left(\frac{x-{\bar{R}_{\psi}}}{\sqrt{2{\sigma^2_{R_\psi}}}}\right)\right)^\Psi\mathrm{d}x,\end{aligned}$$ where $$\label{eq:CDFRpsi}
F_{R_\psi}(x)=1+\mathrm{erf}\left(\frac{x-{\bar{R}_{\psi}}}{\sqrt{2{\sigma^2_{R_\psi}}}}\right)$$ denotes the approximated cdf of $R_\psi$.
Substituting into completes the proof.
Proof of Corollary
| 2,436
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ed}$$ where $a(t)$ is the scale factor and $d\Omega^2 = d\theta ^2 + \sin ^2\theta \, d\phi ^2 $ is the metric of the 2-sphere. This metric is conformally invariant to Minkowski space-time, and from the relations written in the first Appendix, we can choose the following reduced basis of all independent order 6 scalar invariants: $(\mathcal{L}_4, \mathcal{L}_6, \mathcal{L}_7,\curv{L}_1,\curv{L}_3,\curv{L}_5,\curv{L}_8)$. We note that there is one scalar less than for a general conformally invariant space-time coming from the particular metric of FLRW for which there is the additional relation: $$\begin{aligned}
\mathcal{L}_1=\frac{1}{3}\big( -\mathcal{L}_7+2 \mathcal{L}_4 \big).\end{aligned}$$
### Linear Combination. $H^6$ correction.
With this metric, we can right the most general order 6 linear combination of all the independent scalars : $$\begin{aligned}
\begin{split}
J=&\sum \Big( v_i \mathcal{L}_i +x_i \curv{L}_i \Big) =\frac{3}{a(t)^6} \Bigg( \sigma_1(v_i,x_i) \; \dot{a}(t)^6 + \sigma_2(v_i,x_i) \; a(t) \; \dot{a}(t)^4 \; \ddot{a}(t) + \sigma_3(v_i,x_i) \; a(t)^2 \; \dot{a}(t)^2 \; \ddot{a}(t)^2
\\& + \sigma_4(v_i,x_i) \; a(t)^3 \; \ddot{a}(t)^3 + \sigma_5(v_i,x_i) \; a(t)^2 \; \dot{a}(t)^3 \; a^{(3)}(t)+ \sigma_6(v_i,x_i) \; a(t)^3 \; \dot{a}(t) \; \ddot{a}(t) \; a^{(3)}(t)
\\& + \sigma_7(v_i,x_i) \; a(t)^4 \; a^{(3)}(t)^2+ \sigma_8(v_i,x_i) \; a(t)^3 \; \dot{a}(t)^2 \; a^{(4)}(t)+ \sigma_9(v_i,x_i) \; a(t)^4 \; \ddot{a}(t) \; a^{(4)}(t) \Bigg),
\end{split}\end{aligned}$$ where the expressions of the $\sigma_j$ in terms of $(v_i,x_i)$ are presented in the second Appendix. Setting all of them to zero allows to check that our list of scalars is a basis. We can then impose $v_1=v_2=v_3=v_5=v_8=x_2=x_4=x_6=x_7=0$ to take into account the algebraic relations we have found.
Moreover, in this section, we are interested in linear combinations of order 6 scalars that lead to second order equations of motion. Therefore, we can also consider equivalence relations (up to boundary terms) between the scalars,
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he action is free, it is also a bijection. It is easy to check that a bijective quasi-isometry between two uniformly discrete metric spaces is in fact a bi-Lipschitz equivalence. This finishes the proof.
We now recall the definition of the real hyperbolic $n$-space. There are many definitions and we refer the reader to [@DruKap Chapter 4] for a more thorough discussion. We define ${\mathbb{H}}^n$, for $n\geq 2$, as follows. First we define the following quadratic form; for $x,y\in{\mathbb{R}}^{n+1}$: $$\langle x,y\rangle:=\sum_{i=1}^n x_iy_i-x_{n+1}y_{n+1},$$ and we set $${\mathbb{H}}^n:=\{x\in{\mathbb{R}}^{n+1}\colon \langle x,x\rangle=-1,x_{n+1}>0\}.$$ A metric $d$ on ${\mathbb{H}}^n$ can be defined using the formula, for $x,y\in{\mathbb{H}}^n$, $$\cosh d(x,y)=-\langle x,y\rangle.$$ We now state the main result of this subsection.
\[cor:hyperbolicnet\] Let $n\geq 2$ and let $N\subseteq {\mathbb{H}}^n$ be a net. Then ${\mathcal{F}}(N)$ has a Schauder basis.
It suffices to find a group $G$ acting freely and cocompactly on ${\mathbb{H}}^n$. Indeed, again by the Milnor-Schwarz lemma ([@DruKap Theorem 8.37]), such $G$ is then finitely generated and quasi-isometric to ${\mathbb{H}}^n$. It follows that $G$ is hyperbolic ([@DruKap Observation 11.125]). Therefore, as we demonstrated in Subsection \[subsec:examples\], $G$ is shortlex combable. Applying Theorem \[thm:shortlex\], we get that ${\mathcal{F}}(G)$ has a Schauder basis. Finally, an application of Proposition \[prop:actinggroup\] finishes the argument.
In order to find a group acting freely and cocompactly on ${\mathbb{H}}^n$, one can use several standard results from Riemannian geometry. Let $M$ be an arbitrary $n$-dimensional compact Riemannian manifold without boundary of constant sectional curvature $-1$ equipped with some Riemannian metric. By [@BrHa Theorem 3.32], its universal cover (refer to any standard textbook on algebraic topology, e.g. [@Hatch]) is isometric to ${\mathbb{H}}^n$. Another standard argument from algebraic topology (see e.g. [@Ha
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13(522): 829-844.
Zhang, Y., Duchi, J. C. and Wainwright, M. J. (2013). Communication-efficient algorithms for statistical optimization. Journal of Machine Learning Research, 14, 3321-3363.
[^1]: School of Mathematical Sciences, Soochow University, 215006, Suzhou, China, stamax360@outlook.com
[^2]: School of Mathematics, South China University of Technology, Guangzhou, 510640, P.R. China. mascwang@scut.edu.cn
[^3]: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, 117546, Singapore. stazw@nus.edu.sg
---
abstract: 'Cluster structure detection is a fundamental task for the analysis of graphs, in order to understand and to visualize their functional characteristics. Among the different cluster structure detection methods, spectral clustering is currently one of the most widely used due to its speed and simplicity. Yet, there are few theoretical guarantee to recover the underlying partitions of the graph for general models. This paper therefore presents a variant of spectral clustering, called $\ell_1$-spectral clustering, performed on a new random model closely related to stochastic block model. Its goal is to promote a sparse eigenbasis solution of a $\ell_1$ minimization problem revealing the natural structure of the graph. The effectiveness and the robustness to small noise perturbations of our technique is confirmed through a collection of simulated and real data examples.'
author:
- |
[Champion Camille$^{1}$, Blazère Mélanie$^1$, Burcelin Rémy$^2$, Loubes Jean-Michel$^1$, Risser Laurent$^3$]{}\
[$^1$ Toulouse Mathematics Institute (UMR 5219)]{}\
[University of Toulouse F-31062 Toulouse, France]{}\
[$^2$ Metabolic and Cardiovascular Diseases Institute (UMR 1048)]{}\
[University of Toulouse F-31432 Toulouse, France]{}\
[$^3$ Toulouse Mathematics Institute (UMR 5219)]{}\
[CNRS F-31062 Toulouse, France]{}
bibliography:
- 'example\_paper.bib'
title: Robust spectral clustering using LASSO regularization
---
[*Keywords:*]{
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T - w_T) (v_T - w_T)^T} ds dt}\\
\leq& \int_0^1 \int_0^s \E{\lrn{\nabla^2 f(x_T - v_T + s(v_T-w_T))}_2\lrn{v_T - w_T}_2^2} ds \\
\leq& \frac{1}{\epsilon} \E{\lrn{v_T - w_T}_2^2}\\
\leq& \frac{32}{\epsilon} \lrp{T^2 L^2 + TL_\xi^2} T\beta^2
\end{aligned}$$ wehere the second inequality is because $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\] and by Young’s inequality. The third inequality is by Lemma \[l:vt-wt\].
Summing the above, $$\begin{aligned}
&\E{f(x_T - w_T) - f(x_T - v_T)} \\
\leq& 8T^{3/2} L\beta + \frac{128}{\epsilon} T \beta^2 \lrp{\sqrt{T} L_\xi + TL} + \frac{32}{\epsilon} \lrp{T^2 L^2 + TL_\xi^2} T\beta^2\\
\leq& T^{3/2} \epsilon
\end{aligned}$$ where the last inequality is by our assumption on $T$, specifically, $$\begin{aligned}
&T \leq \frac{\epsilon^2}{128\beta^2}
\Rightarrow T^{3/2} L\beta \leq TL\epsilon\\
&T \leq \frac{\epsilon^2}{128\beta^2} \Rightarrow \frac{128}{\epsilon} T^2L \beta^2 \leq TL \epsilon\\
&T \leq \frac{\epsilon}{32\sqrt{L} \beta} \Rightarrow \frac{32}{\epsilon}(T^3 L^2 \beta^2) \leq TL\epsilon\\
& T \leq \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2} \Rightarrow \frac{128}{\epsilon} T^{3/2}\beta^2 L_\xi \leq T\LN^2 \epsilon\\
& T \leq \frac{\epsilon^2}{128\beta^2} \Rightarrow T \leq \frac{\epsilon^2}{128\cm^2} \Rightarrow \frac{32}{\epsilon} T^2L_\xi^2\beta^2 \leq T\LN^2\epsilon
\end{aligned}$$ where the last line uses the fact that $\beta \geq \cm^2$.
\[c:main\_nongaussian:1\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$.\
Let $T= \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$ and let $\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2
| 2,440
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-1}X^{(x)}_{\sigma_{B_1}} \in C_{0, r/|x|} \}}\geq\mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \in C_{0, r/\eta} \}}\eqqcolon \hat{J},$$ where $\hat{J}$ is a Bernoulli random variable with parameter $\hat{q}$. Stochastic dominance, $N\leq \hat{\Gamma}$ almost surely, follows by the same line of reasoning as in the proof of Theorem \[main\].
For the second result, we completely relax the geometrical requirements on $D$ at the expense of efficiency. With an abuse of our earlier notation, we introduce $$N(\varepsilon) = \min\Bigl\{n\geq 0\colon \rho_n\not\in D \text{ or }\inf_{z\in \partial D}|\rho_n-z|<\varepsilon \Bigr\}.$$ Intuitively, $N(\varepsilon)$ is the step that exits the inner $\varepsilon$-thickened boundary of $D$.
Suppose that $D$ is open and bounded (but not necessarily connected). Then for all $x\in D$, there exists a constant $q_\varepsilon = q_\varepsilon(\alpha,D)>0$ (independent of $x$) and a random variable $\Gamma^\varepsilon$ such that $N\leq \Gamma^\varepsilon$ almost surely, where $$\mathbb{P}_x(\Gamma^\varepsilon = k ) = (1-q_\varepsilon)^{k-1}q_\varepsilon, \qquad k\in\mathbb{N}.$$ Moreover, $q_\varepsilon = \mathcal{O}(\varepsilon^\alpha)$ as $\varepsilon\downarrow 0$. In particular $$\mathbb{E}_x[N(\varepsilon)] = \mathcal{O}(\varepsilon^{-\alpha}),\qquad \text{as $\varepsilon \downarrow 0$.}
\label{ea}$$
Define $$\delta\coloneqq \inf\Bp{r>0\colon D\subset B(x,r) \text{ for all } x\in D },$$ so that any sphere of radius $\delta$ centred at $x\in D$ contains $D$. Once again, we recall that, without loss of generality, we may choose our coordinate system such that $x = |x|\,{\rm\bf i}\in D$ is such that $\partial V(x)$ is a tangent hyperplane to $B_1$ and such that $0\in \partial B_1 \cap \partial V(x)\cap\partial D$. Then, taking account of scaling, and that, for all $x\in D$ such that $\inf_{z\in\partial D}|x-z|\geq \varepsilon$, with the particular choice of coordinates described above, $\delta/|x|\leq \delta/\varepsilon$, we have $$\mathbf{1}_{\{N(\varepsilon) = 1\
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({\rm rk}\, {\cal E}^{\alpha}_n)
\left( \frac{ \theta^{{\cal E},\alpha}_n }{\pi} \right)^2
\: - \: \frac{1}{8} \sum_n ( {\rm rk}\, T^{\alpha}_n )\left(
\frac{ \theta^{T,\alpha}_n }{\pi} \right)^2.\end{aligned}$$ In all cases the right-moving vacuum energy vanishes, since the right-moving bosons and fermions make equal and opposite contributions.
Vacuum energies in (NS,R) sectors (meaning, left-moving fermions of the first $E_8$ in an NS sector) can be computed similarly. For completeness, we list them below: in an untwisted sector, $$\begin{aligned}
E_{{\rm (NS,R)}, {\rm Id}} & = & 8\left(- \frac{1}{24} \right) \: + \:
8\left( - \frac{1}{24} \right) \: + \: 4\left( - \frac{1}{12} \right), \\
& = & -1,\end{aligned}$$ and in a twisted sector, $$\begin{aligned}
E_{{\rm (NS,R)}, \alpha} & = &
8\left(- \frac{1}{24} \right) \: + \:
\sum_n ({\rm rk}\, {\cal E}^{\alpha}_n)\left(
- \frac{1}{24} \: + \: \frac{1}{8}\left(
\frac{ \tilde{\theta}^{{\cal E},\alpha}_n }{\pi} \right)^2 \right)
\: + \: (8-r)\left( - \frac{1}{24} \right)
\\
& &
\: + \: \sum_n ( {\rm rk}\, T^{\alpha}_n )\left(
+ \frac{1}{24} \: - \: \frac{1}{8} \left(
\frac{ \theta^{T,\alpha}_n }{\pi} \right)^2 \right)
\: + \: (4-n) \left( - \frac{1}{12} \right),
\\
& = &
-1 \: + \: \frac{n}{8}
\: + \: \frac{1}{8} \sum_n ({\rm rk}\, {\cal E}^{\alpha}_n)\left(
\frac{ \tilde{\theta}^{{\cal E},\alpha}_n }{\pi} \right)^2
\: - \: \frac{1}{8} \sum_n ( {\rm rk}\, T^{\alpha}_n )\left(
\frac{ \theta^{T,\alpha}_n }{\pi} \right)^2,\end{aligned}$$ where $\tilde{\theta}$ denotes $\theta$’s as modified to include a sign in the boundary conditions. Vacuum energies in (NS,R) sectors (meaning, left-moving fermions of the first $E_8$ in an NS sector) can be computed similarly.
Fock vacua
----------
The fractional charges of the Fock vacua can and should be understood in terms of coupling to nontrivial bundles. Recall (see [*e.g.*]{} [@kw]) that a complex left-moving fermion $\lambda$ with boundary conditions $$\lambda(\sigma + 2 \pi) \: = \: e^{-i \theta} \lambda(\sigma)$$
| 2,442
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uation shows that indeed $G_1$ is twice the stochastic propagator $G_s^{ab}$ defined in eq. \[ne21\].
Remarks
-------
In this Section we have shown that the 1PI EA derived from a quantum field theory, cut off at terms quadratic in the difference field $\phi_-$, is identical in form to the effective action derived from a suitable Langevin equation. This allows us, for example, to compute the full Hadamard function from the stochastic approach.
It would be desirable to extend this equivalence to higher correlations, and to this effect we may wish to find stochastic equivalents to the higher effective actions to be introduced in next Section. However, the stochastic approach as presented above relies heavily on the representation \[d80\] for the 1PI EA. The structure of the 1PI EA is not replicated in the higher effective actions (except under restrictive conditions on the propagators, see [@CalHu95a]) and so this method fails in general.
In the following we shall present an alternative derivation of the stochastic approach which does not rely in any particular feature of the 1PI EA, and therefore it is readily generalized to higher effective actions. As a first step, we shall briefly introduce the 2PI effective action.
The 2PI EA and early stochastic formulations
============================================
The 2PI EA
----------
The 1PI generating functional introduced in last Section admits a path integral representation in terms of fields defined on the closed time-path
e\^[iW\_[1PI]{}]{}=D\^Ae\^[i]{} \[ne60\] where $S\left[\Phi^A\right]=S\left[\Phi^1\right]-S\left[\Phi^2\right]^*$ is the classical closed time-path action. In the 2PI theory we add nonlocal sources $K_{AB}$ coupled to $\left(1/2\right)\Phi_H^A\Phi_H^B$. Thus the CTP 2PI generating functional is
e\^[iW]{}=D\^Ae\^[i]{} \[ne61\]
The first and second derivatives of the 2PI generating potential $W$ read
W\^[,A]{}=\^A
W\^[,(AB)]{}=12
W\^[,AB]{}=iG\^[AB]{}
W\^[,A(BC)]{}=i2
W\^[,(AB)(CD)]{}=i4Observe that in these equations we are writing
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one of the regions $U$ or $V$ is included in the ball $B(X,|X-\A(X)|)$. So, one of the two points $Y$ and $\A(Y)$ belongs to the ball $B(X,|X-\A(X)|)$. This contradicts the fact that $\A(X)$ is the ancestor of $X$.
![\[fig:croisement2\] [*The hatched area corresponds to the one of the two sets $U$ and $V$ which is included in the ball $B(X,|X-\A(X)|)$. Here, it contains $\A(Y)$. Besides, let us remark the origin $O$ cannot belong to the ball $B(X,|X-\A(X)|)$.*]{}](croisement2.eps){width="8cm" height="5.5cm"}
$\Box$
**Acknowledgments:** This work has been financed by the GdR 3477 *Géométrie Stochastique*. The authors thank the members of the “Groupe de travail Géométrie Stochastique” of Université Lille 1 and J.-B. Gouéré for enriching discussions.
[10]{}
S. Athreya, R. Roy, and A. Sarkar. Random directed trees and forest - drainage networks with dependence. , 13:2160–2189, 2008. Paper no.71.
F. Baccelli and C. Bordenave. The radial spanning tree of a [P]{}oisson point process. , 17(1):305–359, 2007.
N. Bonichon and J.-F. Marckert. Asymptotic of geometrical navigation on a random set of points of the plane. , 43(4):889–942, 2011.
D. Coupier. Multiple geodesics with the same direction. , 16(46):517–527, 2011.
D. Coupier and P. Heinrich. Coexistence probability in the last passage percolation model is 6-8 log 2. , page in press.
D. Coupier and P. Heinrich. Stochastic domination for the last passage percolation tree. , 17(1):37–48, 2011.
D. Coupier and V.C. Tran. The 2d-directed spanning forest is almost surely a tree. , 2012. to appear.
P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. , 33(4):1235–1254, 2005.
S. Gangopadhyay, R. Roy, and A. Sarkar. Random oriented trees: a model of drainage networks. , 14(3):1242–1266, 2004.
C. D. Howard and C. M. Newman. Euclidean models of first-passage percolation. , 108(2):153–170, 1997.
C. D. Howard and C. M. Newman. Geodesics and spanning trees for [E]{}uclidean first-passage
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$. The advantage of this grading is that it is simply given by the left multiplication of ${\mathbf{h}}_{c+i}$. Thus, as is an isomorphism of left $U_{c+i}$-modules and hence of left ${\mathbb{C}}[{\mathbf{h}}_{c+i}]$-modules, it is automatically a graded isomorphism under the canonical grading.
Since ${\mathfrak{h}}^*$ has ${\mathbf{E}}$-degree $1$, the canonical grading on $\Delta_d(\mu)$, for any $d\in {\mathbb{C}}$, is a shift of the grading on $\widetilde{\Delta}_d(\mu)$. The shift is easy to compute. By definition, the generator $1\otimes \mu$ of $\widetilde{\Delta}_d(\mu)$ has $\deg_d(1\otimes \mu)=0$ whereas, by Proposition \[subsec-3.10\], the generator $1\otimes\mu$ of $\Delta_d(\mu)$ has $$\deg_{\mathrm{can}}(1\otimes \mu)=
D(d,\mu) =(n-1)/2+ d(n(\mu)-n({\mu^t})).$$ We may therefore regard $\Delta_d(\mu)$ as being in $\widetilde{{\mathcal{O}}}_d$, in which case $$\label{shft}
\Delta_d(\mu)= \widetilde{\Delta}_d(\mu)[D(d,\mu)].$$
Let $b\in B_{ij}$ with $\operatorname{{\mathbf{h}}\text{-deg}}(b)=r$ and suppose that $v\in e\Delta_{c+j}(\mu)$ has $\deg_{\mathrm{can}}(v)=s$. Then $${\mathbf{h}}_{c+i}\cdot b\otimes v = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b)\otimes v +b\,{\mathbf{h}}_{c+j}\otimes v
\ =\ ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b)\otimes v +b\otimes {\mathbf{h}}_{c+j}v
=(r+s)b\otimes v.$$ Thus $\deg_{\mathrm{can}}(b\otimes v) =\operatorname{{\mathbf{h}}\text{-deg}}(b)+
\deg_{\mathrm{can}}(v)$. Finally, implies that $$\begin{aligned}
\deg_{c+i}(b\otimes v) &=& \deg_{\mathrm{can}}(b\otimes v) - D(c+i, \mu) \ = \
\operatorname{{\mathbf{h}}\text{-deg}}(b) + \deg_{\mathrm{can}}(v) - D(c+i,\mu) \\ &=&
\operatorname{{\mathbf{h}}\text{-deg}}(b) + \deg_{c+j}(v) + D(c+j,\mu) - D(c+i,\mu) \\
&=& \deg (b\otimes v) + (j-i)(n(\mu) - n(\mu^t)),\end{aligned}$$ as required.
{#poincare-sa-sect}
Given a ${\mathbb{Z}}$-graded complex vector space $M =
\bigoplus_{r\in{\mathbb{Z}}}M_r$ such that $\dim_{{\mathbb{C}}} M_r$ is finite for all $r$ then, as in , we define the Poincaré series\[formal-Poincare-def
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|
(b) + 3*f(b).
-4*b**2 - b
Let c(j) = j + 1. Let s(v) = -2*v**2 + v + 2. Let u(q) = 2*q - 1. Let y be u(1). Give y*c(o) - s(o).
2*o**2 - 1
Let f(h) = 22*h. Let q(i) be the first derivative of -7*i**2/2 + 8. Give 5*f(y) + 16*q(y).
-2*y
Let t(k) = -22*k**2 + 50*k**2 + 6*k - 22*k**2 + 6*k**3. Let q(j) = -j**3 - j**2 - j. Give -34*q(n) - 6*t(n).
-2*n**3 - 2*n**2 - 2*n
Let g(j) = -9*j**3 + 9*j**2 + 13. Let u(n) = -4*n**3 + 4*n**2 + 6. What is 6*g(z) - 13*u(z)?
-2*z**3 + 2*z**2
Let m = -3 - -10. Let t(k) = 7*k**2 + m*k**2 - 2*k**2. Let z(j) = j**2. What is -2*t(u) + 27*z(u)?
3*u**2
Suppose 11*b = 9*b - 4. Let a(n) = -3*n**2 - 3*n. Let y(z) = -z**2 - z. Give b*y(t) + a(t).
-t**2 - t
Let w(m) = 2*m**3 - 2*m**2 + m. Let o(k) = -4*k**3 + 4*k**2 - 3*k. What is 2*o(g) + 5*w(g)?
2*g**3 - 2*g**2 - g
Let b(l) = 6*l**2 + 5*l - 5. Let k(j) = -7*j**2 - 6*j + 6. Suppose 8*h + 30 = 3*h. Determine h*b(m) - 5*k(m).
-m**2
Suppose -4*q = 3*f - 12, 0*q = -2*q. Let o(l) = -3*l**3 - 8*l**2 + 0*l**3 - 2 - 3. Let c(p) = -p**3 - 3*p**2 - 2. What is f*o(h) - 11*c(h)?
-h**3 + h**2 + 2
Let a(v) = -3*v**2 - 4*v - 2. Let u(z) = -z + 5. Let n be (4 + 0)/((-6)/12). Let f = -4 - n. Let c be u(f). Let q(m) = -m**2 - m. Give c*a(t) - 5*q(t).
2*t**2 + t - 2
Suppose 0 = -2*w, 0 = 3*h + 5*w - 14 + 5. Let k(n) = -9*n + 6. Let x(z) = -3*z + 2. Let d(u) = h*k(u) - 8*x(u). Let g(o) = o - 1. Calculate d(s) + 2*g(s).
-s
Let l(c) = -6*c**2 + 2*c**3 - 5*c**3 + 150*c + 5 - 142*c. Let t(o) = -6*o**3 - 11*o**2 + 15*o + 9. Give -11*l(n) + 6*t(n).
-3*n**3 + 2*n - 1
Let i(c) = 4*c - 1. Let s(m) = -7*m + 2. Give 5*i(h) + 3*s(h).
-h + 1
Let x(h) = -2*h**2 + 4. Let l(g) = -10*g**2 + 10 - g + 3 + 4*g**2. Determine 2*l(q) - 7*x(q).
2*q**2 - 2*q - 2
Let w(i) = 25*i**3 - 11*i**2 - 11*i + 11. Let c(p) = -5*p**3 + 2*p**2 + 2*p - 2. Give -22*c(n) - 4*w(n).
10*n**3
Let r(a) = -2*a**2 + 6. Let t(s) = 2*s**2 - 7. Calculate -6*r(g) - 5*t(g).
2*g**2 - 1
Let z(x) = 1. Let a(g) = g**2 - g - 3. Let v = 133 - 135. Give v*a(u) - 6*z(u).
-2*u**2 + 2*u
Let j(o) = -o**2 - 3*o - 3. Let p be j(
| 2,446
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|
} {f}\left(\rho_n + X^{(n+1)}_s\right)\,\mathrm{d}s\right],
\qquad x\in D,
\label{withrho}$$ where $X^{(n)}$ are independent copies of $(X, \mathbb{P}_0)$. Applying Fubini’s theorem, then conditioning each expectation on $\mathcal{F}_{n}\coloneqq \sigma(\rho_k\colon k\leq n)$ followed by Fubini’s theorem again, we have $$\begin{aligned}
\mathbb{E}_x\left[\int_0^{\sigma_D} {f}(X_s)\,\mathrm {d}s\right] & =\sum_{n\geq 0}\mathbb{E}_x\left[ \mathbf{1}_{\{\rho_n\in D\}} \left.
\mathbb{E}_{y}\left[\int_{0}^{\sigma_{B(y,r)}} {f}( X_s)\,\mathrm{d}s\right]
\right|_{y = \rho_n, r = r_n}\right]\\
& =\sum_{n\geq 0}\mathbb{E}_x\left[ \mathbf{1}_{\{\rho_n\in D\}}
V_{r_n}(\rho_n, {f}(\cdot))
\right]\\
& =\mathbb{E}_x\left[\sum_{n= 0}^{N-1} V_{r_n}(\rho_n, {f}(\cdot))
\right].
\end{aligned}$$ The proof is completed once we show that $V_r(x,g) = r^{\alpha}V_1(0, {f}(x + r\cdot)),$ for $r>0$, $x\in\mathbb{R}^d$ and bounded measurable ${f}$. To this end, we appeal to spatial homogeneity and the, now, familiar computations using the scaling property of stable processes: $$\begin{aligned}
V_r(x,{f}(\cdot)) & = \mathbb{E}_x\left[\int_0^{\sigma_{B(x,r)}} {f}(X_t) \,{\rm d}t\right]\nonumber \\
& = \mathbb{E}_0\left[\int_0^{\sigma_{B(0,r)}} {f}(x+X_t) \,{\rm d}t\right]\nonumber \\
& = \mathbb{E}_0\left[\int_0^{\sigma_{B(0,1)}} r^{\alpha}\,{f}(x+ r\,X_s) \,{\rm d}s\right]\nonumber \\
& =\int_{|y|<1} r^{\alpha} {f}(x + r\,y)\, V_1({0,\rm d}y)\nonumber \\
& =r^\alpha \,V_1(0, {f}(x + r\,\cdot)).
\label{timescale}
\end{aligned}$$ The proof is now complete.
Lemma \[integral\] now informs a Monte Carlo procedure based on simulating the quantity $$\
| 2,447
| 4,114
| 1,947
| 1,818
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e process probabilistic [@pra61_034301].
The main part of our paper is based on the formalism summarized in Section \[sec:formalism\]. Our description is completely independent of the dimensions of the Hilbert-spaces involved, and we do not even need to fix a basis. In Section \[sec:teleport\] we give a general condition for conditional teleportation in terms of the applicable entangled states and joint measurements. In Section \[sec:matching\] we show that partially entangled states are capable of conditional teleportation with fidelity one, but only if the outcome of the measurement performed by Alice and the state shared by the parties “match” each other. Section \[sec:concl\] summarizes our results.
States, channels and antilinear maps {#sec:formalism}
====================================
Consider a bipartite system with subsystems A and B. The subsystems are described by the Hilbert-spaces ${\cal H}_A$ and ${\cal H}_B$, thus the pure states of the system are in ${\cal H}_A\otimes {\cal H}_B$. Let $\dim {\cal
H}_A=\dim {\cal H}_B=N$. Let $\{|i\rangle_A\}$ and $\{|i\rangle_B\}\ (i=0,\ldots ,N-1)$ denote the computational bases on ${\cal
H}_A$ and ${\cal H}_B$, and let $$\label{eq:maxent}
|\Psi^+\rangle_{AB}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}|i\rangle_A|i\rangle_B$$ be a maximally entangled state of the system. All other maximally entangled states of the system can be obtained from $|\Psi^+\rangle$ by local unitary transformations. The set of density matrices of system A will be denoted by ${\cal S}_A$. A quantum channel $\$ _A $ is a completely positive, trace-preserving, and hermiticity preserving ${\cal S}_A \rightarrow {\cal S}_A$ map.
Keeping $|\Psi^{(+)}\rangle$ in mind, we may introduce the relative state representation [@pra54_2614] of states and operators on ${\cal H}_A$. Any pure state $|\Psi\rangle_A\in {\cal H}_A$ can be described by an (unnormalized) *index state* $|\Psi^*\rangle_B\in {\cal H}_B$ so that $$\label{eq:relrep_state}
|\Psi\rangle_A=\, _B\langle \Psi^*|\Psi^+\rangle_{AB}
| 2,448
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| 0.792662
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|
f there is a failover.
The thing is that none of the dependencies are distributed in the same way, and can therefore be expected to be running everywhere.
You should only need to expand on that config file to get it to work. Here I've done it and reindented it to show its structure better:
[{kernel,
[{distributed, [
{myapp_api, 1000, ['n1@myhost', {'n2@myhost', 'n3@myhost'}]},
{myapp, 1000, ['n1@myhost', {'n2@myhost', 'n3@myhost'}]},
]},
{sync_nodes_optional, ['n2@myhost', 'n3@myhost']},
{sync_nodes_timeout, 5000}
]}].
I haven't tested it directly, but I'm pretty sure this will work.
If what you wanted was for myapp_api to be everywhere but for myapp to run in one place, you could use global registration, give a name to myapp's public-facing process, get myapp_api to call these. myapp_api would then be able to route traffic to wherever myapp is when supported with the following config:
[{kernel,
[{distributed, [
{myapp, 1000, ['n1@myhost', {'n2@myhost', 'n3@myhost'}]},
]},
{sync_nodes_optional, ['n2@myhost', 'n3@myhost']},
{sync_nodes_timeout, 5000}
]}].
(See how myapp is the only app getting a distribution profile? Other apps will get to run on all nodes)
Q:
I made a mistake in my source control mapping. How do I correct it?
I created a team project, say ProjectA
When I mapped it to my local folder, I found that the root server folder: ProjectA folder is not mapped. However the ProjectA folder within that IS mapped to the project. This is a problem as the BuildProcessTemplate is not included in source control and I cannot build my application.
How do I remove the mappings and start again?
A:
On the source control explorer window in visual studio there is a drop down near the top for workspaces. When you originally mapped that folder, it created a workspace for you. If you want to change the mappings, it can be done by clicking the drop down and selecting the Workspaces... option. You can select your workspace and then click Edit.. to change mappings. You can also de
| 2,449
| 1,075
| 1,863
| 2,022
| 3,253
| 0.773749
|
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|
otential was already considered in [@Aldazabal:2006up] and arises from the Chern-Simons term $$\int D_8 \wedge (Q \cdot H_3 +P_1^2 \cdot F_3 )_2 \quad . \label{D8tadpole}$$ By imposing absence of sources for this potential, this leads to the quadratic constraint $$Q^{cd}_{[a}H_{b]cd}+P^{cd}_{[a}F_{b]cd}=0 \quad . \label{d8}$$ By analysing this constraint, one finds that it implies exactly eq. . In IIB/O3 the constraints from $D_{9,1}$ and $D_{10}$ identically vanish, while the constraint arising from the $D_{10,2}$ potential in IIB is $$(P^{1,4}\cdot F_3)^{ab}=P^{a,abcd}F_{acd}+P^{b,abcd}F_{bcd}=0 \quad ,$$ which leads exactly to the condition . In the IIA/O6 model, the only non-trival constraint comes from $D_{9,1}$, and again it can be shown that it is perfectly compatible with all the IIB constraints.
What this analysis shows is that when the $P$ fluxes are included, one can consistently impose all the NS-NS constraints, but this also imposes for consistency that the quadratic constraints arising from the $D_8$ and $D_{10,2}$ potentials in IIB have to vanish. On the other hand, in the previous section we have shown that the $P$ fluxes also induce charges for the $\alpha=-3$ branes that can be different from zero. We now want to analyse the tadpole conditions for these branes.
$P$ fluxes and tadpoles
-----------------------
Using the T-duality rules for the $P$ fluxes and the $E$ potentials that we have found in this paper, one can determine, starting from eq. , all the tadpole conditions for the $\alpha=-3$ branes listed in Table \[Ebranestable\] in the presence of $P$ fluxes. In the IIB/O3 theory, there are three $7_3$-branes, each orthogonal to one of the three tori $T_{(i)}^2$, corresponding to the components $E_{4\, x^jy^j x^ky^k}$ of the potential $E_8$. Denoting the number of each of these branes as $N_{( 7_3 )_i}$, from eq. one gets [@Aldazabal:2006up] $$N_{ (7_3 )_i}+\tfrac{1}{2}[-h_0\bar{f}_i+\bar{h}_0f_i+\bar{a}_jg_{ji}-a_j\bar{g}_{ji}]=0 \quad , \label{P7}$$ where it is understood that the in
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| 2,180
| 2,099
| 0.782737
|
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|
\ .$$ The dominant contributions should be the gluon and electroweak penguins mediated by $H$ and $Q$. In contrast to the KM Model, the vector coupling of the vector quark means that the $Z$ boson penguin will be suppressed by $O(m_K^2/m_Z^2)$ due to vector current conservation. The gluon penguin contributes only to $A_0$, but the isospin-breaking electromagnetic penguin (EMP) gives rise to both $A_0$ and $A_2$. Due to its suppression by $O(\alpha/\alpha_s)$, the latter affects $\epsilon'$ solely through its contribution to $\Omega$. Including the effects of evolution from the vector quark mass down to the charm mass scale, we estimate the EMP contribution to be $\Omega_{\rm EMP}
\stackrel{<}{\sim} O(1)$. There is an additional contribution to $\Omega$ from $\eta, \eta'$ isospin-breaking, with $\Omega_{\eta-\eta'}
= 0.25$. We shall ignore the electromagnetic penguin contribution here (inclusion of the electromagnetic penguin will be studied elsewhere[@UsAgain]), and set $\Omega=\Omega_{\eta-\eta'}$ to simply the analysis. The inclusion of $\Omega_{\rm EMP}$ will not change our conclusion qualitatively.
The gluon penguin diagram, which involves the virtual vector quark $Q$ and the charge Higgs boson, produces an effective Hamiltonian at the electroweak scale: $${\cal H}^{\Delta S=1}= (G_F/\sqrt{2}) \tilde{C}
( \bar s T^a \gamma_\mu(1+\gamma_5)d )
\times
\sum_q (\bar q T^a \gamma^\mu q ) \ ,$$ $$\tilde{C}=\alpha_s
\sum_i{\xi_{di}\xi^*_{si}\over 6\pi}
{m_W^2\over M_Q^2}F({m_{H_i}\over M_Q^2})
\ .$$ $$F(x)=
{x^2(2x-3)\log x\over (1-x)^4}
+ {16x^2-29x+7 \over 6(1-x)^3} ; \, F(1)={3\over4}.$$ Written in terms of the operators in Ref.[@renormgroup] (but of flipped chirality), $${\cal H}^{\Delta S=1}=
(G_F / \sqrt{2}) \sum_{i=3}^{6} \tilde{C}_i \tilde Q_i \ ,$$ with $\tilde{C}_{4,6} = \tilde{C}/4$, $\tilde{C}_{3,5} = -\tilde{C}/(4N_c) $, $Q_{3 (5)} = (\bar s_i d_i)_{V+A}
\sum_q (\bar q_j q_j)_{V+A (V-A)}$ and $Q_{4 (6)} = (\bar s_i d_j)_
| 2,451
| 2,021
| 3,050
| 2,206
| 3,945
| 0.769081
|
github_plus_top10pct_by_avg
|
this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd.
We let $u=2u'-2$ and $v=2v'-3$. Then $u+v=n$, and we have $$u-v=2(u'-v')+1\equiv-1\ppmod{2^{l(v')+1}}$$ and $l(v)\ls l(v')+1$. So $S^{(u,v)}$ is irreducible. Furthermore, $v\gs7$, $v\equiv3\ppmod4$ and $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2,$$ as required.
: In this case, let $$n'=\frac{n+13}2,\quad a'=
\begin{cases}
\mfrac{a+6}2&(a\equiv2\ppmod4)\\[5pt]
\mfrac{a+8}2&(a\equiv0\ppmod4).
\end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$. So by induction there is a pair $u',v'$ such that $$v'\equiv1\pmod4,\qquad v'\gs7,\qquad u-v\equiv-1\pmod{2^{l(v')}},\qquad\mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Because $u'-v'$ is odd and $u'-a'$ is even, this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd.
We let $u=2u'-6$ and $v=2v'-7$. Then $u+v=n$, and we have $$u-v=2(u'-v')+1\equiv-1\pmod{2^{l(v')+1}},$$ and $l(v)\ls l(v')+1$. So $S^{(u,v)}$ is irreducible. Furthermore, we have $v\equiv3\ppmod4$, $v\gs7$ and $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2.\tag*{\raisebox{-10pt}{\qedhere}}$$
Now we can prove our main result.
Suppose we have a pair $a,b$ of positive even integers with $a\gs4$. If $a\gs6$, $b\gs4$ and $a+b\equiv0\ppmod8$, then the result follows from Proposition \[ab0\]. If $a\gs8$, $b\gs4$ and $a+b\equiv2\ppmod8$, then the result follows from Proposition \[ab1\]. If $a=6$ and $a+b\equiv2\ppmod8$, then by Theorem \[main\] the Specht module $S^{(a+b,3)}$ is an irreducible summand of $S^\la$.
The second case of the corollary is precisely the condition for $S^{(a+b,3)}$ to be an irreducible summand of $S^\la$, while the third case is the condition for $S^{(a+b-4,5,2)}$ to be a summand. So in any of the given cases, $S^\la$ certainly has an irreducible Specht module as a summand. To complete the proof, it suffices to show that if $a=4$, $b=2$ or $a+b\equiv4$ or $6\ppmod8$, then the only possible Specht modules which c
| 2,452
| 1,833
| 2,217
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| 2,393
| 0.780236
|
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|
tation theory [@gst:basic; @gst:tree; @gst:Dynkin-A; @gst:acyclic] which will be continued in [@gst:acyclic-Serre]. The perspective from enriched derivator theory offers additional characterizations of stability, and these together with a more systematic study of the stabilization will appear in [@gs:enriched]. It is worth noting that in [@ps:linearity], what we here call “$\Phi$-stable monoidal derivators” are shown to admit a linearity formula for the traces and Euler characteristics of $\Phi$-colimits, so the abstract study of stability has computational as well as conceptual importance.
The content of the paper is as follows. In §\[sec:char\] we characterize pointed and stable derivators by the commutativity of certain (co)limits or Kan extensions. In \[sec:galois\] we define the Galois correspondence of relative stability. In \[sec:enriched-derivators\] we define enriched derivators and weighted colimits, and in \[sec:stab-via-wcolim\] we use them to give the second class of characterizations of stability. Finally, in §\[sec:fun\] we study further the characterizations in terms of iterated adjoints to constant morphism morphisms.
**Prerequisites.** We assume *basic* acquaintance with the language of derivators, which were introduced independently by Grothendieck [@grothendieck:derivators], Heller [@heller:htpythies], and Franke [@franke:adams]. Derivators were developed further by various mathematicians including Maltsiniotis [@maltsiniotis:seminar; @maltsiniotis:k-theory; @maltsiniotis:htpy-exact] and Cisinski [@cisinski:direct; @cisinski:loc-min; @cisinski:derived-kan] (see [@grothendieck:derivators] for many additional references). Here we stick to the notation and conventions from [@gps:mayer]. For a more detailed account of the basics we refer to [@groth:intro-to-der-1].
Stability and commuting (co)limits {#sec:char}
==================================
In this section we obtain characterizations of pointed and stable derivators in terms of the commutativity of certain left and right Kan ext
| 2,453
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imate, parameter inference is also available to assist in assessing the quality of fit. The deterministic map to the sufficient statistics from the set of firing events and responses permits the transformation from the final particle set $\{X_{1:T}^{(i)}\}_{i=1}^{N}$ to an $N$-component Gaussian-gamma mixture approximating the posterior distribution for the observation parameters. A similar transformation for evaluating the posterior distribution for the excitability parameters is derived from the approximation to the prior; see Section \[sec:DetailFireProc\] for details.
### Equivalent particle specification and degeneracy
The Bayesian conjugate structure for the observation process suggests storing and updating the sufficient statistics when assimilating the latest observations. Given the prior statistics ${\bar{\mathcal{A}}}_0$ and ${\mathcal{A}}_0$, there is a deterministic map from $({\mathbf{x}}_{1:t-1},~
y_{1:t-1})$ to $({\bar{\mathcal{A}}}_{t-1},~{\mathcal{A}}_{t-1})$. Hence estimates relating to the observation process at time $t$ are equivalently expressed with respect to the samples $\{{\bar{\mathcal{A}}}_{t-1}^{(i)},~
{\mathcal{A}}_{t-1}^{(i)},~ {\mathbf{x}}_t^{(i)}\}_{i=1}^N$; the storage required for this set does not increase with with number of observations assimilated. Since the method relies on these sufficient statistics, Algorithm \[tab:Alg\] may be considered as a case of particle learning [@Car10]. For notational clarity, however, the particle set is described in terms of the historical firing events, ${\mathbf{x}}_{1:t-1}$, unless otherwise specified.
Assimilating the observation, $y_{\tau}$, at the supramaximal stimulus, $s_{\tau}$, before any of the other non-baseline observations ensures an update for *all* MUs, $j$, from the initial vague priors for each $\mu_j$. After this, each $m_j\approx
y_{\tau}/u$, ensuring more sensible predictions in and hence , when a new MU fires. This helps to mitigate against the inevitable particle degeneracy that occurs with particle learning. Further
| 2,454
| 611
| 2,078
| 2,428
| 1,762
| 0.785898
|
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|
rbation $\delta\rho_1 =
\delta\psi_1^*+\delta\psi_1$ then follows as $$\delta\hat\rho_1 = \frac{e^{-\frac{k^2a^2}{2}} (4 k^2(1+\gamma^2)-8i\gamma{\boldsymbol{k}}\cdot {\boldsymbol{V}}_p)}{4{\boldsymbol{k}}\cdot{\boldsymbol{V}}_p({\boldsymbol{V}}_p \cdot {\boldsymbol{k}}+i\gamma k^2 +2i\gamma)- k^2(4+k^2)(1+\gamma^2)}.
\label{Rho1Delta}$$ Using the convolution theorem, we can express the self-induced force (\[eq:fp\_perturb\]) (in the co-moving frame, i.e. with $\mathbf r_p=0$) in terms of $\delta\hat\rho_1$ as $${\boldsymbol{F}}^{(1)}= - \frac{g_p^2}{(2\pi)^2}\int d^2{\boldsymbol{k}} i{\boldsymbol{k}}\delta\hat{\rho}_1({\boldsymbol{k}}) e^{-\frac{k^2a^2}{2}} .$$ This force can be decomposed into the normal and tangential components relative to the particle velocity ${\boldsymbol{V}}_p$: $ {\boldsymbol{F}}^{(1)} = F_{\|} {\boldsymbol{e}}_{\|}+F_{\perp} {\boldsymbol{e}}_{\perp}$. Due to symmetry, the normal component vanishes upon polar integration, and we are left with the tangential, or drag, force
$$\begin{aligned}
F_{\|}&=& -\frac{g_p^2}{(2\pi)^2} \int_0^\infty dk\int_0^{2\pi} d\theta e^{-k^2a^2}\frac{i k^2\cos(\theta) \left[4 k^2(1+\gamma^2) - 8i\gamma kV_p\cos(\theta)\right]}{4kV_p\cos(\theta)(kV_p\cos(\theta)+ i\gamma k^2 + 2i\gamma) - k^2(4 + k^2)(1+\gamma^2)}.
\label{eq:force_int}\end{aligned}$$
$V_p$ is the modulus of ${\boldsymbol{V}}_p$. At zero temperature, i.e. when $\gamma=0$, the drag force reduces to the one that has also been calculated for a point particle in Refs. [@astrakharchik2004motion] and in [@pinsker2017gaussian] for a finite-$a$ particle: $$F_\parallel = - \frac{g_p^2}{\pi^2}\int_0^\infty dk\int_0^{2\pi} d\theta\frac{i k^2 \cos{\theta}e^{-k^2a^2}}{4V_p^2 \cos^2{\theta}- (4+k^2)} \ ,$$ which is zero for particle speed smaller than the critical value given by the long-wavelength sound speed, $V_p < c=1$. Above the critical speed, the integral has poles and acquires a non-zero value given by $$F_\parallel = -\frac{g_p^2 k_{max}^2}{4V_p} e^{-\frac{a^2 k_{max}^2}{2}} \left[I_0\left(
| 2,455
| 3,320
| 2,636
| 2,270
| null | null |
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|
a_{\hat{S}} + t/\sqrt{2n}]$$ Both confidence intervals satisfy (\[eq::honest\]).
We now compare $\hat\beta_{{\widehat{S}}}$ and $\hat{C}_{{\widehat{S}}}$ for both the splitting and non-splitting procedures. The reader should keep in mind that, in general, $\hat{S}$ might be different between the two procedures, and hence $\beta_{{\widehat{S}}}$ may be different. The two procedures might be estimating different parameters. We discuss that issue shortly.
[**Estimation.**]{} First we consider estimation accuracy.
\[lemma::est-accuracy\] For the splitting estimator: $$\sup_{P\in {\cal P}_{n}}\mathbb{E}|\hat\beta_{{\widehat{S}}}-\beta_{{\widehat{S}}}| \preceq n^{-1/2}.$$ For non-splitting we have $$\label{eq::lower1}
\inf_{\hat\beta}\sup_{P\in {\cal P}_{n}}
\mathbb{E}|\hat\beta_{{\widehat{S}}}-\beta_{{\widehat{S}}}| \succeq
\sqrt{\frac{\log D}{n}}.$$ The above is stated for the particular selection rule ${\widehat{S}}= \operatorname*{argmax}_s
\hat{\beta}_s$, but the splitting-based result holds for general selection rules $w\in\mathcal{W}_n$, so that for splitting $$\sup_{w\in {\cal W}_n}\sup_{P\in {\cal
P}_{n}}\mathbb{E}|\hat\beta_{{\widehat{S}}}-\beta_{{\widehat{S}}}| \preceq n^{-1/2}$$ and for non-splitting $$\label{eq::lower2}
\inf_{\hat\beta}\sup_{w\in {\cal W}_{2n}}\sup_{P\in {\cal P}_{n}}
\mathbb{E}|\hat\beta_{{\widehat{S}}}-\beta_{{\widehat{S}}}| \succeq
\sqrt{\frac{\log D}{n}}.$$
Thus, the splitting estimator converges at a $n^{-1/2}$ rate. Non-splitting estimators have a slow rate, even with the added assumption of Normality. (Of course, the splitting estimator and non-splitting estimator may in fact be estimating different randomly chosen parameters. We address this issue when we discuss prediction accuracy.)
[**Inference.**]{} Now we turn to inference. For splitting, we use the usual Normal interval $\hat{C}_{{\widehat{S}}} = [\hat\beta_{{\widehat{S}}}-z_\alpha s/\sqrt{n},\
\hat\beta_{{\widehat{S}}}+z_\alpha s/\sqrt{n}]$ where $s^2$ is the sample variance from ${\cal D}_{2,n}$. We then have
| 2,456
| 4,097
| 2,386
| 1,977
| 2,882
| 0.776347
|
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|
led introduction to the topics missing from Ref. but necessary to understand supersymmetric field theories is of greater use and interest to most readers of this Proceedings volume. With these notes, our aim is thus to equip any interested reader with a few handy concepts and tools to be added to the backpack to be carried on his/her explorer’s journey towards the quantum geometer’s universe of XXI$^{\rm st}$ century physics, in search of the new principle beyond the symmetry principle of XX$^{\rm th}$ century physics.[@GovCOPRO2]
Also by lack of space and time, even of the anticommuting type if the world happens to be supersymmetric indeed, we shall thus stop short of discussing explicitly any supersymmetric field theory in 4-dimensional Minkowski spacetime, even the simplest example of the $\mathcal N=1$ Wess-Zumino model[@WZ] that may be constructed using the hand-made tools of an amateur artist-composer in the art of supersymmetries. From where we shall leave the subject in these notes, further study could branch off into a variety of directions of wide ranging applications, beginning with general supersymmetric quantum mechanics and the general superspace and superfield techniques for $\mathcal N=1$ and $\mathcal N=2$ supersymmetric field theories with Yang–Mills internal gauge symmetries and the associated Higgs mechanism of gauge symmetry breaking, to further encompass the search for new physics at the LHC through the construction of supersymmetric extensions of the Standard Model, or also reaching towards the duality properties of supersymmetric Yang–Mills and M-theory, mirror geometry, topological string and quantum field theories,[@GovTQFT] etc., to name just a few examples.[@Strings; @Witten1; @MATH]
Let us thus point out a few standard textbooks and lectures for large and diversified accounts of these classes of theories and more complete references to the original literature. Some important such material is listed in Refs. and . In particular, the lectures delivered at the Workshop were
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standard Gaussian random variable. If $\varphi$ is concave, then we have
Corollary \[t0\] follows directly from Theorem \[t6\] by and the fact that $1+\dots+\frac{1}{n}{\leqslant}\log n+1.$
For the general case where the mean of $X_1$ is uncertain (that is, $\underline{\mu}\ne \overline{\mu}$) and $\varphi$ may not be convex or concave, we formulate a new CLT for where $\mu_i$ equals $\overline{\mu}$ or $\underline{\mu}$ depending on previous $\{X_j: j<i\}$ and the solution to the heat equation, and $\sigma_i$ depends furthermore on the set of the possible first two moments of $X_1$. As above, let $\{X_i\}_{i=1}^\infty$ be an i.i.d. sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $\E[ |X_1|^3]<\infty$. Define and for each possible mean $\mu$ of $X_1$, define We impose the following assumption:
[**Assumption A**]{}. *Regarded as functions of $\mu$, $\overline{\sigma}_\mu^2$ and $\underline{\sigma}_\mu^2$ are continuous at, or can be continuously extended to, $\mu=\overline{\mu}$ and $\mu=\underline{\mu}$.*
Denote There is no conflict of notation between [121]{} and [122]{} by Assumption A. We assume further that
[**Assumption B**]{}. *All the four quantities in [122]{} are positive.*
Let be the set of all possible pairs of mean and variance of $X_1$. Define and On the basis of Assumptions A and B, we have $\sigma_0^2>0$. We have the following theorem.
\[t1\] Under the above setting, we have the following CLT: for each $\varphi\in lip(\mathbb{R})$, In [15]{}, $Z$ is a standard Gaussian random variable, with $\mu_i=\mu_i((X_j, \mu_j, \sigma_j): j<i)$ are defined as $\sigma_i=\sigma_i((X_j, \mu_j, \sigma_j): j<i, \mu_i)$ are defined as where $V_{i-1}:= V(t_{i-1}, W_{i-1})$ and $V(\cdot, \cdot)$ is the solution to the heat equation
The proof of Theorem \[t1\] is deferred to Section 5.2.
From the definition of $\mu_i$ in [13]{}, the first term of $f_{i-1, b}(\mu, \sigma^2)$ in [17]{} is ${\leqslant}0$
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a})} + 2\delta^2L^2 \E{\lrn{w_{k\delta}}_2^2} + 2\delta \beta^2\\
\leq& \E{a(w_{k\delta})} -m\delta \E{a(w_{k\delta})} + 2\delta^2L^2 a(w_{k\delta}) + 2\delta^2 L^2 R^2 + 2\delta \beta^2\\
\leq& (1-m\delta/2)a(w_{k\delta}) + {m\delta} R^2 + 2\delta \beta^2
\end{aligned}$$
Where the first inequality uses the upper bound on $\lrn{\nabla^2 a(y)}_2$ above, the second inequality uses the fact that $w_{(k+1)\delta} = \lrp{y_{k\delta} - \delta \nabla U(y_{k\delta}) = \xi(w_{k\delta}, \eta_k)}$, and $\E{ \xi(w_{k\delta}, \eta_k) | w_{k\delta}} = 0$, the third inequality uses claim 2. at the start of this proof, the fourth inequality uses item 2 of Assumption \[ass:xi\_properties\]. The fifth inequality uses claim 3. above, the sixth inequality uses our assumption that $\delta \leq \frac{m}{16L^2}$.
Taking expectation wrt $w_{k\delta}$, $$\begin{aligned}
& \E{a(w_{(k+1)\delta})} \leq \E{a(w_{k})}-m\delta \lrp{\E{a(w_{k\delta})} - 2R^2 + 2\beta^2/m}\\
\Rightarrow \qquad &
\E{a(w_{(k+1)\delta})} - (2R^2/2 + 2\beta^2/m) \leq (1-m\delta) \lrp{\E{a(w_{k\delta})} - (2R^2 + 2\beta^2/m}
\end{aligned}$$
Thus, if $\E{\|w_0\|_2^2} \leq 2R^2 + 2\beta^2/m$, then $\E{a(w_0)} - \lrp{2R^2 + 2\beta^2/m} \leq 0 $, then $\E{a(w_{k\delta})} - \lrp{2R^2 + 2\beta^2/m}\leq 0$ for all $k$, which implies that $$\begin{aligned}
\E{\lrn{w_{k\delta}}_2^2} \leq 2\E{a(w_{k\delta})} + 4 R^2 \leq 8\lrp{R^2 + \beta^2/m}
\end{aligned}$$ for all $k$.
[Divergence Bounds]{}
\[l:divergence\_xt\] Let $x_t$ be as defined in (or equivalently or ), initialized at $x_0$. Then for any $T\leq \frac{1}{16L}$, $$\begin{aligned}
\E{\lrn{x_T - x_0}_2^2} \leq 8 \lrp{T\beta^2 + T^2 L^2 \E{\|x_0\|_2^2}}
\end{aligned}$$ If we additionally assume that $\E{\lrn{x_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{x_T - x_0}_2^2} \leq 16 T\beta^2
\end{aligned}$$
By Ito’s Lemma
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equences \[Sec:consequences\]
=================================
In the previous section, we compared the magnitude of the SUSY threshold corrections to the bottom quark mass. Particularly, we have shown that the various approximations made to obtain the common form in \[Eq:common-app\] all seem to be valid approximations with the exception of neglecting the $B_1^{\widetilde{g}}$ terms in the gluino-sbottom contribution and possibly the contributions from the Higgses. In this section, we will highlight some of the consequences of including these terms in the corrections to the bottom quark mass.
Fits to the bottom quark mass {#fits-to-the-bottom-quark-mass .unnumbered}
-----------------------------
A good choice of scale to integrate out the massive SUSY particles is the $M_Z$ scale. At the $M_Z$ threshold one then has to match the value of $m_b$ before and after integrating out the massive states. This leads to the relation m\_b(M\_Z)\^[SM]{} = m\_b(M\_Z)\^[MSSM]{}(1 + m\_b/m\_b) \[eq:thresholds\] . $m_b(M_Z)^{\rm below}$ can be determined by taking the value of $m_b(m_b) = 4.19$ GeV and running it to the $M_Z$ scale. This is evaluated using the package to be $m_b(M_Z)^{\rm below} = 2.69$ GeV. The hope then is that the right choice of bottom Yukawa coupling and the appropriate set of SUSY boundary conditions at some UV scale will give rise to the necessary $m_b(M_Z)^{\rm above}$ and $\Delta m_b/m_b$ to satisfy \[eq:thresholds\].
When fitting the bottom quark mass, it is common to use the full, exact one-loop correction. This is done in most numerical spectrum calculators, such as [@Allanach:2014nba] and [@Porod:2003um]. Physical interpretations are often based however on the approximate formula given in \[Eq:common-app\]. As was shown in the previous section, additional terms, namely the $B_1^{\widetilde{g}}$ terms from the gluino-sbottom contributions and the contributions from the Higgses, should also be included for a full description. These “missing" terms contribute $\sim$12% to the correction.
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t {{W}}$ and its subrings from . Since $e$, $e_-$ and $\delta$ are homogeneous under this action, each $Q_{c+\ell}^{c+\ell+1}$ and hence each $B_{ij}$ and $N(k)$ is also graded under this action. As in , this induces a graded structure, again called $\operatorname{{\mathbf{E}}\text{-deg}}$, on $\operatorname{{\textsf}{ogr}}B_{ij}$ and $\operatorname{{\textsf}{ogr}}N(k)$. Since the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ are ${\mathbf{E}}$-homogeneous, the ${\mathbf{E}}$-grading on $N(k)$ descends to gradings on $\overline{N(k)}$ and $\underline{N(k)}$. Similarly, each $A^{u}\delta^{u}$ and $J^u\delta^u$ has an ${\mathbf{E}}$-grading induced from that on ${\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$ and hence so does $A=\bigoplus_{u\geq 0}A^u\delta^u.$
However, the ${\mathbf{E}}$-grading on $B_{k0}$ and hence on $N(k)$ is [*not*]{} equal to the adjoint ${\mathbf{h}}$-grading. The problem is that, in , the adjoint ${\mathbf{h}}$ action does not “see” the element $\delta$. Thus if we wish to relate the Poincaré series of $N(k)$ to that of $J^{k}\delta^{k}$ we need the following slight modification of Proposition \[poincare-SA\].
Let $k\geq 0$, set $N=n(n-1)/2$ and write $K=kN$.
1. If $b\in B_{ij}$ for $i\geq j\geq 0$ is homogeneous under the ${\mathbf{h}}$-grading then it is homogeneous in the ${\mathbf{E}}$-grading and $\operatorname{{\mathbf{E}}\text{-deg}}b = (i-j)N + \operatorname{{\mathbf{h}}\text{-deg}}b.$
2. Under the ${\mathbf{E}}$-grading, $\overline{N(k)}$ has Poincaré series $$p(\overline{N(k)}, v) = v^{K}
\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-k(n(\mu)
- n(\mu^t))}[n]_v!}{\prod_{i=2}^n
(1-v^{-i})}.$$ while $\underline{N(k)}$ has Poincare series $p(\underline{N(k)},v) = v^{K}\displaystyle
\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) -
n(\mu^t))}}{\prod_{i=2}^n
(1-v^{i})}.$
\(1) If $b_1\in B_{ik}$ and $b_2\in B_{kj}$ then ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}(b_1b_2) = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_1)b_2 + b_1({\mathbf
| 2,461
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95), 133–164.
S. Kojima and Y. Yamashita, [Shapes of stars]{}, Proc. Amer. Math. Soc., 117 (1993), 845–851.
S. Kojima, H. Nishi and Y. Yamashita, [Configuration spaces of points on the circle and hyperbolic Dehn fillings]{}, to appear in Topology.
W. Neumann and D. Zagier, [*Volumes of hyperbolic 3-manifolds*]{}, Topology, 24 (1985), 307–332.
W. Thurston, [*Geometry and Topology of 3-manifolds*]{}, Lecture Notes, Princeton Univ., 1977/78,
---
abstract: |
We introduce a new class of non-standard variable-length codes, called *adaptive codes*. This class of codes associates a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. An efficient algorithm for constructing adaptive codes of order one is presented. Then, we introduce a natural generalization of adaptive codes, called *GA codes*.
[**Keywords:**]{} adaptive mechanisms, compression rate, data compression, entropy, prefix codes, variable-length codes.
title: '****'
---
Introduction
============
The theory of variable-length codes [@bp1] originated in concrete problems of information transmission. Especially by its language theoretic branch, the field has produced a great number of results, most of them with multiple applications in engineering and computer science. Intuitively, a *variable-length code* is a set of strings such that any concatenation of these strings can be uniquely decoded. We introduce a new class of non-standard variable-length codes, called *adaptive codes*, which associate a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. The paper is organized into six sections. After this introductory section, the definition of adaptive codes and several theoretical remarks are given in Section 2, as well as some characterization results for adaptive codes. The main results of this paper are presented in Section 3, where we focus on designing an algorithm for constructing adaptive codes of order one. In Sect
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]{} on that side. Such polynomials are of the form $$G=x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b)
\quad.$$
For the first assertion, simply note that under the stated hypotheses only one monomial in $F$ is dominant in $F\circ\alpha(t)$; hence, the limit is supported on the union of the coordinate axes. A simple dimension count shows that such limits may span at most a 6-dimensional locus in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, and it follows that such germs do not contribute a component to the PNC.
For the second assertion, note that the dominant terms in $F\circ\alpha(t)$ are precisely those on the side of the Newton polygon with slope equal to $-b/c$. It is immediate that the resulting polynomial can be factored as stated.
If the point $p=(1:0:0)$ is a singular or an inflection point of the support of ${{\mathscr C}}$, and $b/c\ne 1/2$, we find the type IV germs of §\[germlist\]; also cf. [@MR2001h:14068], §2, Fact 4(ii). The number $S$ of ‘cuspidal’ factors in $G$ is the number of segments cut out by the integer lattice on the selected side of the Newton polygon. If $b/c=1/2$, then a dimension count shows that the corresponding limit will contribute a component to the PNC (of type IV) unless it is supported on a conic union (possibly) the kernel line.
If $p$ is a [*nonsingular, non-inflectional*]{} point of the support of ${{\mathscr C}}$, then the Newton polygon consists of a single side with slope $-1/2$; these are the type II germs of §\[germlist\]. Also cf. [@MR2001h:14068], Fact 2(ii).
Components of type V {#typeVcomps}
====================
Having dealt with the 1-PS case in the previous section, we may now assume that $$\tag{$\dagger$}
\alpha(t)=\begin{pmatrix}
1 & 0 & 0\\
q(t) & t^b & 0\\
r(t) & s(t)t^b & t^c
\end{pmatrix}$$ with the conditions listed in Lemma \[faber\], and further [*such that $q,r$, and $s$ do not all vanish identically.*]{}
Our task is to show that contributing germs of this kind must in fact be of the form specified in §\[germlist\] a
| 2,463
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the single vector meson production, which is determined by the photon flux and the $\gamma h \rightarrow V h$ cross section.
In what follows we will consider the color dipole formalism to describe the diffractive vector meson photoproduction, which successfully describe the HERA data and recent LHC data [@amir; @bruno1; @bruno2]. In this approach the description of the single vector meson production can be factorized as follows: i) a photon is emitted by one of the incident hadrons, ii) the photon fluctuates into a quark-antiquark pair (the dipole), iii) this color dipole interact with the other hadron by the exchange of a color single state, denoted Pomeron ($I\!\!P$) and, iv) the pair converts into the vector meson final state. The $\gamma h \rightarrow V h$ cross section is given by $$\begin{aligned}
\sigma (\gamma h \rightarrow V h) = \int_{-\infty}^0 \frac{d\sigma}{d{t}}\, d{t}
= \frac{1}{16\pi} \int_{-\infty}^0 |{\cal{A}}_T^{\gamma h \rightarrow V h }(x,\Delta)|^2 \, d{t}\,\,,
\label{sctotal_intt}\end{aligned}$$ with the scattering amplitude is given by $$\begin{aligned}
{\cal A}_{T}^{\gamma h \rightarrow V h}({x},\Delta) = i
\int dz \, d^2{\mbox{\boldmath $r$}}\, d^2{\mbox{\boldmath $b$}}_h e^{-i[{\mbox{\boldmath $b$}}_h-(1-z){\mbox{\boldmath $r$}}].{\mbox{\boldmath $\Delta$}}}
\,\, (\Psi^{V*}\Psi)_{T} \,\,2 {\cal{N}}_h({x},{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_h) \,\,,
\label{sigmatot2}\end{aligned}$$ where $(\Psi^{V*}\Psi)_{T}$ denotes the overlap of the transverse photon and vector meson wave functions. The variable $z$ $(1-z)$ is the longitudinal momentum fractions of the quark (antiquark) and $\Delta$ denotes the transverse momentum lost by the outgoing pion (${t} = - \Delta^2$). The variable ${\mbox{\boldmath $b$}}_h$ is the transverse distance from the center of the target $h$ to the center of mass of the $q \bar{q}$ dipole and the factor in the exponential arises when one takes into account non-forward corrections to the wave functions [@non]. As in our previous studies [@bruno
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7.6 1.24 0.14 2.79 0.82 2.46
Combined (1393) 172.1 1.6 11.82 0.30 0.68 0.11 0.70
: The $v_H^{*}$ (km/s), $r_H^{*}$ (kps), and resultant $\sigma_8$, $\Delta\sigma_8$, and t-test comparison with the WMAP value of $\sigma_8$.[]{data-label="summary"}
We have estimated $\Omega_{\hbox{\scriptsize asymp}}$ by averaging $\rho_{\hbox{\scriptsize asymp}}(r)$ over a sphere of radius $r_{II}$, and found $\Omega_{\hbox{\scriptsize{asymp}}} =
0.197_{\pm0.017}$. In calculating this average, we assumed that there is only a single galaxy within the sphere, however. While this is a gross under counting of the number of galaxies in the universe, $\rho_{\hbox{\scriptsize{asymp}}}$ is an asymptotic solution, and $\rho_{II}^1 \to0$ rapidly with $r$. Additional galaxies may change the form of $\rho_{\hbox{\scriptsize asymp}}$, but these changes are expected to be equally short ranged; we expect that our calculation is an adequate estimate of $\Omega_{\hbox{\scriptsize asymp}}$. Such is not the case for $\Omega_{\hbox{\scriptsize Dyn}}$, however. Direct calculation of $\Omega_{\hbox{\scriptsize Dyn}}$ would require knowing both the detailed structure of galaxies, and the distribution of galaxies in the universe. Instead, we note that $\Omega_m = \Omega_{\hbox{\scriptsize{asymp}}}+
\Omega_{\hbox{\scriptsize{Dyn}}}$, and using $\Omega_m=0.238_{-0.026}^{+0.025}$ from WMAP, find $\Omega_{\hbox{\scriptsize{Dyn}}}=0.041^{+0.030}_{-0.031}$.
Concluding Remarks
==================
Given how sensitive $\sigma_8$ is to $v_H^{*}$, $r_H^{*}$, and $\alpha_\Lambda$, that our predicted values of $\sigma_8$ is within experimental error of the WMAP value is surprising. Even in the absence of a direct experimental search for $\alpha_\Lambda$, this agreement provides a compelling argument for the validity of our extension of the GEOM. It a
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defect states start to overlap significantly, and the particle initially tied to one defect oscillates to the other. If the amplitude then rises again and reaches the “collapse” value $$F_{\rm collapse} = j_{0,1} \frac{\hbar\omega}{ed}$$ at $t = T_{\rm pulse}$, and is kept constant thereafter, the particle has been transferred to the final state, and will stay there. The time $T_{\rm pulse}$ is chosen such that $$\label{eq:intphi}
\frac{1}{\hbar} \int_0^{T_{\rm pulse}} \! {{\rm d}}\tau \,
\Delta\varepsilon^{F(\tau)} = \varphi\; ;$$ to obtain a transfer from one defect to the other, again $\varphi=\pi$ has to be chosen.
For a matter-of-principle demonstration of this scenario, one may employ the envelope function $$\begin{aligned}
\label{eq:env}
F(t) & = & F_{\rm collapse}
\\ &\times&
\frac{\exp\left(-\frac {t^2}{2T_{\rm ramp}^2}\right) + a +
\exp\left(-\frac{\left(t-T_{\rm pulse}\right)^2}{2T_{\rm ramp}^2}\right)}
{a+1 + \exp\left(-{\frac {T_{\rm pulse}^{2}}{2T_{\rm ramp}^2}}\right)}
\nonumber\end{aligned}$$ for $0 \le t \le T_{\rm pulse}$, where $T_{\rm ramp}$ quantifies the characteristic time interval during which the amplitude is ramped down and up again, with $T \ll T_{\rm ramp} \ll T_{\rm pulse}$ being understood. The parameter $a$ has been introduced in order to allow for a nonvanishing amplitude at intermediate times, at $t \approx T_{\rm pulse}/2$.
The above considerations rely heavily on the adiabatic principle; if the parameter variation proceeds too fast, complete population transfer is not achieved. This is illustrated in Fig. \[fig:versuch\] for a system with defect parameters $\gamma = \pm3$ and $\nu/W = 0.1$, subjected to forcing with frequency $\hbar\omega/W = 7.5$ and the envelope function (\[eq:env\]), setting $a = 1.5$, $T_{\rm ramp} = 50\,T$, and $T_{\rm pulse} = 595\,T$. Under these conditions, the transfer remains incomplete; slight oscillations visible in the occupation probabilities of the defect sites indicate non-adiabatic dynamics. However, Fig. \[fig:t
| 2,466
| 4,795
| 1,932
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ekar variables the full Hamiltonian for general relativity is a sum of constraints $$H_{\text{G}}^{\text{tot}}= \int d^3 {\bf x} \, (N^i G_i + N^a C_a + N h_{\text{sc}}),$$ where $$\begin{aligned}
C_a &= E^b_i F^i_{ab} - (1-\gamma^2)K^i_a G_i ,\nonumber \\
G_i &= D_a E^a_i\end{aligned}$$ and the scalar constraint has a form $$\begin{aligned}
\label{ham}
&H_{\text{G}}:=\int d^3{\bf x} \, N(x) h_{\rm sc}= \nonumber \\
&\frac{1}{16 \pi G} \int d^3{\bf x} \, N(x)\left( \frac{E^a_i
E^b_j}{\sqrt{|\det E|}} {\varepsilon^{ij}}_k F_{ab}^k -
2(1+\gamma^2) \frac{E^a_i E^b_j}{\sqrt{|\det E|}} K^i_a
K^j_b \right) \end{aligned}$$ with $F=dA + \frac{1}{2}[A,A]$. The full Hamiltonian of theory is a sum of the gravitational and matter part. With convenience as a matter part we choose the scalar field with the Hamiltonian $$H_{\phi}=\int d^3{\bf x} \, N(x)\left( \frac{1}{2}\frac{\pi^2_{\phi}}{\sqrt{|\det E|}} + \frac{1}{2} \frac{E^a_i
E^b_i \partial_a \phi \partial_b \phi }{\sqrt{|\det E|}} + \sqrt{|\det E|} V(\phi) \right).$$ We assume here that field $\phi$ is homogeneous and start his evolution from the minimum of potential $ V(\phi)$. The second assumption states that contribution from potential term is initially negligible. So the density of Hamiltonian $H_{\phi}$ is simplified to the form $\mathcal{H}_{\phi}=(1/2)\pi^2_{\phi}/\sqrt{|\det E|}$. The term $1/\sqrt{|\det E|}$ for the classical FRW universe corresponds to $1/a^3$ where $a$ is the scale factor. On the quantum level term $1/\sqrt{|\det E|}$ is quantised and have discrete spectrum. In the regime $a \gg a_i$ we can however use the approximation $1/\sqrt{|\det E|}=D/a^3$ where $$D(q)=q^{3/2} \left\{ \frac{3}{2l} \left( \frac{1}{l+2}\left[(q+1)^{l+2}-|q-1|^{l+2} \right]-
\frac{q}{1+l}\left[(q+1)^{l+1}-\mbox{sgn}(q-1)|q-1|^{l+1} \right] \right) \right\}^{3/(2-2l)}
\label{correction}$$ and $q=(a/a_*)^2$ with $a_*=\sqrt{\gamma j / 3}l_{\text{Pl}} $. Function (\[correction\]) depends on the ambiguity parameter $l$. As it was shown by Bojowald [@Bojowald:2
| 2,467
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| 2,103
| null | null |
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|
}\,}+ \langle \mathcal{L}_{L_+}{\bf u}^{(k)}, {\bf v}^{(k^\prime-1) } \rangle,\end{aligned}$$
where in the last line we used the fact that $\overline{L_+}=-L_-$. Note that this relationship does not hold between $H_{\pm}$, so this type of proof will not work in Poincaré coordinates.
We would like to discard the first term on the RHS of Eq. , which would show that ${\mathcal{L}}_{L_+}$ and ${\mathcal{L}}_{L_{-}}$ are adjoints of each other. We can do this by converting the Lie derivative into a covariant derivative and then a total divergence. Since $L_{\pm}$ are KVFs, they are automatically divergence-free, so we can pull them inside the covariant derivative: $$\int_{\Sigma_u} {\mathrm{dVol}\,}\mathcal{L}_{L_-}\left(\overline{u_i^{(k)}} v^i_{(k^\prime)}\right)
=
\int_{\Sigma_u} {\mathrm{dVol}\,}L^j_- D_j \left( \overline{u_i^{(k)}} v^i_{(k^\prime)}\right)
=
\int_{\Sigma_u} {\mathrm{dVol}\,}D_j \left( L^j_-\overline{u_i^{(k)}} v^i_{(k^\prime)}\right)
\,.$$
This step is identical if we are considering scalars/vectors/tensors, since the argument of the Lie derivative has all indices contracted. Using Stokes’ theorem, the integral of the total derivative becomes a boundary integral, evaluated at $\psi=0,\pi$. This boundary contribution vanishes for $h<-1$ in the highest-weight module. To see this, one must count the powers of $\sin\psi$ which depends on $h$ (see App. \[app:global-basis\]), and take into account the volume element’s contribution, $\sqrt{-\gamma}\propto(\sin\psi)^{-2}$.
We repeat the procedure of extracting lowering operators from the ket as in Eq. , and arrive at $$\langle {\bf u}^{(k)}, {\bf v}^{(k^\prime)} \rangle = \langle
\left(\mathcal{L}_{L_+}\right)^{k^\prime}{\bf u}^{(k)}, {\bf v}^{(0)} \rangle
\,.$$ Recall that the vector basis terminates at the highest weight. Therefore when $k^\prime > k$, $\left(\mathcal{L}_{L_+}\right)^{k^\prime} {\bf u}^{(k)}$ will vanish. Similarly when $k^\prime < k$, we can extract all lowering operators from the bra and raise the weight of the states in
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|
, and hence 1-balanced, and it satisfies the 1-size property as $|\mathcal{E}_t|= n$. Suppose that $\{\beta_1, \beta_2\}\subset H_t(\alpha)$ for some $\alpha\in \mathbb{Z}_n$, [with $\beta_1\neq \beta_2$]{}. Then there exists $j_1,j_2\in \{0,\ldots, s-1\}$ such that $\beta_1 = \alpha + j_1 k_t\, (\operatorname{mod}\, n)$ and $\beta_2 = \alpha + j _2 k_t\, (\operatorname{mod}\, n)$. Thus, $\beta_2 - \beta_1 = (j_2-j_1) k_t\, (\operatorname{mod}\, n)$. [Note that $j_1, j_2$ must be distinct as $\beta_1,\beta_2$ are distinct.]{} Next suppose that [$k_{t_1}\neq k_{t_2}$ for some $t_1,t_2\in\{1,\ldots, n\}$, and take any $j_1,j_2\in \{1,\ldots, s-1\}$. [By definition of $k_t$]{} and working in $\mathbb{Z}$, we see that $$1{\leqslant}| j_2 k_{t_2} - j_1 k_{t_1} | {\leqslant}(s-1)\big( \lceil\sqrt{n}\rceil + \lceil n^{1/4} \rceil\big) < n,$$ ]{} and it follows that $$\label{no-mod}
j_1k_{t_1}\neq j_2k_{t_2}\, (\operatorname{mod}\, n).$$ [Finally, suppose that some distinct $\beta_1,\beta_2$ satisfy $\{\beta_1, \beta_2\}\subset H_{t_1}(\alpha)\cap H_{t_2}(\alpha)$ where $k_{t_1}\neq k_{t_2}$. Then $\beta_2 - \beta_1 = j k_{t_1}\, (\operatorname{mod}\, n)$ for some $j_1\in\{1,\ldots, s-1\}$, and $\beta_2 - \beta_1 = j_2 k_{t_2}\, (\operatorname{mod}\, n)$ for some $j_2\in \{ 1,\ldots, s-1\}$, but this contradicts (\[no-mod\]). ]{} Therefore, by definition of $k_t$, for every [$\{\beta_1, \beta_2\}\subset\mathbb{Z}_n$]{}, we have $${\ensuremath{\operatorname{\mathtt{vis}}(\beta_1, \beta_2)}}=|\{t\in \{1,2,\ldots, n\}\mid \{\beta_1, \beta_2 \}\subset H_t(\alpha) ~\text{for some}~ \alpha\in \mathbb{Z}_n\}|{\leqslant}{\mathcal{O}}(n^{3/4}).$$
- Consider an $R$-dimensional torus $\Gamma(n, R)$, which is a graph whose vertex set is the Cartesian product of $\mathbb{Z}_\ell^R={\mathbb{Z}_\ell\times\ldots\times\mathbb{Z}_\ell}$, where $\ell=n^{1/R}\in\mathbb{Z}$, and two vertices $(x_1,\ldots,x_R)$ and $(y_1,\ldots,y_R)$ are connected if for some $j\in \{1,2\ldots,R\}$ $x_j=y_j\pm 1$ mod $n$ and for all $i\neq j$ we hav
| 2,469
| 2,956
| 2,072
| 2,349
| null | null |
github_plus_top10pct_by_avg
|
as the defining relations of new local operators $s_x$ and $s_y$, taking into account of the fact that $$\bar{h}_x=\bar{h}_x(x),\hs{2ex}\bar{h}_y=0.$$ The canonical commutation relations are still valid, as well as the conservation laws. Considering the conservation laws of $d_\m$ and $h_\m$ $$\partial_y d_x+\partial_x d_y=0,\hs{2ex}\partial_y h_x+\partial_x h_y=0$$ we have $$d\bar{h}_x=-\partial_y s_x-\partial_x s_y.$$ Now $s_x$ is an operator of weight zero under the dilation. The two-point function is $${\left\langle}s_x s_x {\right\rangle}=\mbox{constant},$$ which implies that $$\partial_y s_x=0$$ is valid as an operator equation. We arrive at $$\bar{h}_x(x)=-\partial_x s_y.$$ Note that $s_y$ is an operator of weight $d$ under the dilation along $y$ direction such that $$\partial_y{\left\langle}s_y s_y {\right\rangle}=f(x)\partial_y y^{-2d}\neq 0.$$ But $s_y$ is invariant under the Galilean boost as well, which means that the above relation should be vanishing. The only way to be self-consistent is to set $s_y=0$ and therefore $\bar{h}_x=0$. This implies that a 2D theory with $c=0$ and the symmetries $$y\rightarrow y+v x,\hs{2ex}y\rightarrow \lambda y,\hs{2ex}y\rightarrow y+\delta y,$$ is inconsistent and does not exist.
### Other cases: $c\neq 0$
Next we turn to the $c\neq 0$ cases, in which we can normalize the dilation so that $c=1$. However, we keep $c$ unfixed in the following discussion in this section. One should note that the final results cannot be symmetric in $c$ and $d$, since the boost symmetry tells the difference between $x$ and $y$ directions.
We start from the dilation currents $d_\m$ $$d_x=c xh_x+d y\bar{h}_x+s_x,\hs{2ex}d_y=c xh_y+d y\bar{h}_y+s_y.$$ The conservation law of $d_\m$ leads to the relation $$c h_y+d\bar{h}_x=-\partial_y s_x-\partial_x s_y.$$ Moreover we have $$\bar{h}_y=0.$$ We can shift the current $h_\m$ as follows h\_y&&h’\_y=h\_y + (\_y s\_x+\_x s\_y),\
h\_x && h’\_x=h\_x - (\_x s\_x+\_y s\_y). This will not change the commutation relations and the conservation laws
| 2,470
| 3,586
| 2,819
| 2,231
| null | null |
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|
} }
],
"DO" : [
{ "commerce_order_update_status" : { "commerce_order" : [ "commerce_order" ], "order_status" : "completed" } }
]
}
}
In short as soon as the balance is 0 I want the order to be complete.
A:
In the end I created a new rule:
{ "rules_complete_order" : {
"LABEL" : "Complete Order",
"PLUGIN" : "reaction rule",
"TAGS" : [ "Commerce Checkout" ],
"REQUIRES" : [ "commerce_payment", "commerce_order", "entity" ],
"ON" : [ "commerce_payment_transaction_insert" ],
"IF" : [
{ "commerce_payment_order_balance_comparison" : {
"commerce_order" : [ "commerce-payment-transaction:order" ],
"value" : "0"
}
}
],
"DO" : [
{ "commerce_order_update_status" : {
"commerce_order" : [ "commerce-payment-transaction:order" ],
"order_status" : "completed"
}
}
]
}
}
Q:
Computing Line Integral using Stokes
I'm trying to compute the line integral
$$
\int_{C(0,4)^+} (x^2+y^3)dx + (y^4+x^3)dy,
$$
where $C(0,4)$ is the circle, positively oriented, with center 0 and radius 4.
So far, I've tried using Stokes theorem: $\int_{\partial M} \omega = \int_M d\omega$. So $C(0,4)$ is the boundary of the closed ball $B(0,4)$. We compute:
$$
d((x^2+y^3)dx + (y^4+x^3)dy)) = d(x^2+y^3) \wedge dx + d(y^4+x^3) \wedge dx = (3x^2-3y^2) dx \wedge dy
$$
So we need to compute the integral
$$
\int_{B(0,4)^+} (3x^2-3y^2) dx \wedge dy.
$$
Here is where I'm stuck. I tried to use a coordinate patch $\alpha : (0,2\pi) \times (0,4) \to B(0,4) : (\theta,r) \mapsto (r \cos \theta, r \sin \theta)$, and compute $\alpha^\ast(\omega)$, but I'm not sure whether I'm on the right track since the expressions become rather complicated and nothing cancels out since we have $x^2-y^2$, and for polar coordinates $x^2+y^2$ would be convenient. Any tips?
A:
I think this is an application of Green's Theorem, which is a specific case of Stokes' Theorem. Namely,
$$
\int_{\partial D} P \mathrm{d}x + Q \mathrm{d}y = \iint_{D} \fra
| 2,471
| 2,630
| 69
| 2,442
| 924
| 0.797569
|
github_plus_top10pct_by_avg
|
) (y_T - v_T)^T} ds dt}\\
=& \E{f(x_T - y_T)+ \underbrace{\lin{\nabla f(x_0 - y_0), y_T - v_T}}_{\circled{1}} + \underbrace{\lin{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0), y_T - v_T}}_{\circled{2}} }\\
&\quad + \E{\underbrace{\int_0^1\int_0^s \lin{\nabla^2 f(x_T - y_T + s(y_T-v_T)), (y_T - v_T) (y_T - v_T)^T} ds dt}_{\circled{3}}}
\end{aligned}$$
We will bound each of the terms above separately. $$\begin{aligned}
& \E{\circled{1}}\\
=& \E{\lin{\nabla f(x_0 - y_0), y_T - v_T}}\\
=& \E{\lin{\nabla f(x_0 - y_0), n \delta \nabla U(y_0) - n \delta \nabla U(v_0) + \int_0^T -\nabla U(w_0) dt + \int_0^T \cm dV_t + \int_0^T N(w_0) dW_t + \sum_{i=0}^{n-1} \sqrt{\delta} \xi(v_0,\eta_i)}}\\
=& \E{\lin{\nabla f(x_0 - y_0), n \delta \nabla U(y_0) - n \delta \nabla U(v_0) }}\\
=& 0
\end{aligned}$$ where the third equality is because $\int_0^T dB_t$, $\int_0^T dW_t$ and $\sum_{k=1}^T \xi(v_0,\eta_i)$ have zero mean conditioned on the information at time $0$, and the fourth equality is because $y_0 = v_0$ by definition in and . $$\begin{aligned}
& \E{\circled{2}}\\
=& \E{\lin{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0), y_T - v_T}}\\
\leq& \sqrt{\E{\lrn{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0)}_2^2} }\sqrt{\E{\lrn{y_T - v_T}_2^2}}\\
\leq& \frac{2}{\epsilon}\sqrt{2\E{\lrn{x_T - x_0}_2^2 + \lrn{y_T - y_0}_2^2}} \sqrt{\E{\lrn{y_T - v_T}_2^2}}\\
\leq& \frac{2}{\epsilon}\sqrt{\lrp{32T\beta^2 + 4T\beta^2}} \cdot \lrp{6 \sqrt{d\delta}\beta {\log n}}\\
\leq& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta} {\log n}}
\end{aligned}$$
Where the second inequality is by $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\] and Young’s inequality. The third inequality is by Lemma \[l:divergence\_xt\] and Lemma \[l:divergence\_yt\] and .
Finally, we can bound $$\begin{aligned}
& \E{\circled{3
| 2,472
| 2,910
| 1,691
| 2,230
| null | null |
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|
f Specht modules in our family.
In fact, we speculate that every Specht module has a filtration in which the factors are isomorphic to indecomposable Specht modules; this would imply in particular that every indecomposable summand has a Specht filtration. This speculation is certainly true in the case of Specht modules labelled by hook partitions; this follows from [@gm2 §2].
$2$-quotient separated partitions
---------------------------------
In [@jm Definition 2.1], James and Mathas make the following definition: a partition $\la$ is *$2$-quotient separated* if it can be written in the form $$(c+2x_c,c-1+2x_{c-1},\dots,d+2x_d,d^{2y_d},(d-1)^{2y_{d-1}+1},\dots,1^{2y_1+1}),$$ where $c+1\gs d\gs0$, $x_c\gs\cdots\gs x_d\gs0$ and $y_1,\dots,y_d\gs0$. (Note that the definition includes the case $c=0$, where we have $\la=(2x_0)$ if $d=0$, or $(1^{2y_1})$ if $d=1$.)
Informally, the $2$-quotient separated condition means that the Young diagram of $\la$ can be decomposed as in the following diagram, where horizontal ‘dominoes’ can appear in the first $c-d+1$ rows, and vertical ‘dominoes’ can appear in the first $d$ columns. $$\begin{tikzpicture}[scale=0.4]
\draw(0,0)--(0,7)--++(9,0);
\foreach \x in {0,1,2,3,4}\draw(\x,\x+2)--++(1,0)--++(0,1);
\foreach \x in {0,1}\draw(\x,\x)--++(1,0)--++(0,2);
\foreach \x in {0,1,2}\draw(\x+3,\x+4)--++(2,0)--++(0,1);
\foreach \x in {0,1}\draw(\x+6,\x+5)--++(2,0)--++(0,1);
\end{tikzpicture}$$
The definition of a $2$-quotient separated partition was made as part of the study of decomposition numbers: for the Iwahori–Hecke algebra $\calh_{\bbc,-1}({\mathfrak{S}_}n)$, whose representation theory is very similar to that of ${\mathfrak{S}_}n$ in characteristic $2$, the composition factors of a Specht module labelled by a $2$-quotient separated partition are known explicitly. The reason we recall the definition here is that every known example of a decomposable Specht module is labelled by a $2$-quotient separated partition. (Note that the partition $(a,3,1^b)$ considered in this
| 2,473
| 1,966
| 1,110
| 2,390
| 568
| 0.805448
|
github_plus_top10pct_by_avg
|
below Eq. \[e.2to2operator\]. The *upper dotted orange line* is for $y^{\phi_1}_\chi = y^{\phi_2}_\chi/20$, in which case the vertical axis is understood to be $(y_\chi^{\phi_2}/\Lambda)^2$. **Right:** Direct detection bounds on the $2\rightarrow2$ regime of $n_\phi = 1$ dmDM, same labeling as the left plot. The vertical axis is proportional to $\sigma_{\chi N \to \bar \chi N}$, and is understood to be $(y_\chi^1 y_\chi^2/\Lambda)^2$ for the $n_\phi = 2$ model outlined in [Section \[s.indirectconstraints\]]{}. []{data-label="f.mappingbounds"}](2to3_alltransformedboundsLambdam2__5 "fig:"){width="8cm"} ![**Left:** Direct detection bounds on the $2\rightarrow3$ regime of dmDM. The vertical axis is proportional to $\sigma_{\chi N \to \bar \chi N \phi}$. *Solid lines*: 90% CL bounds by XENON100 (red), LUX (black) and CDMSlite (purple), as well as the best-fit regions by CDMS II Si (blue, green). The large-dashed black line indicates the irreducible neutrino background [@Billard:2013qya]. *Small-dashed magenta line*: Upper bound for $y_\chi = y_\chi^\mathrm{relic}(m_\chi)$ and neutron star cooling bound $\Lambda < 10 \tev$. *Lower dotted orange line*: upper bound for $2\to3$ dominated direct detection and neutron star bound with all equal Yukawa couplings. This line can be arbitrarily moved, as discussed below Eq. \[e.2to2operator\]. The *upper dotted orange line* is for $y^{\phi_1}_\chi = y^{\phi_2}_\chi/20$, in which case the vertical axis is understood to be $(y_\chi^{\phi_2}/\Lambda)^2$. **Right:** Direct detection bounds on the $2\rightarrow2$ regime of $n_\phi = 1$ dmDM, same labeling as the left plot. The vertical axis is proportional to $\sigma_{\chi N \to \bar \chi N}$, and is understood to be $(y_\chi^1 y_\chi^2/\Lambda)^2$ for the $n_\phi = 2$ model outlined in [Section \[s.indirectconstraints\]]{}. []{data-label="f.mappingbounds"}](2to2dmDM_alltransformedboundsLambdam2_no2to3dominancebound "fig:"){width="8cm"}
[Fig. \[f.compare2to2to2to3\]]{} defines an experiment-dependent parameter map that we can
| 2,474
| 1,009
| 1,619
| 2,406
| 380
| 0.811601
|
github_plus_top10pct_by_avg
|
xtrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-.
\end{cases}$$ where $(x,\omega,E)\in G\times S\times I$ when taking the limits.
Define $$z_0=\begin{cases}
x_0-t(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in G\times S\times I \\
x_0-\tau_+(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_+ \\
x_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-.
\end{cases}$$ In each of the limits concerned, we have thus assumed that $(z_0,\omega_0,E_0)\in\Gamma_-$, which implies, using the fact that $\ol{G}$ is $C^1$-manifold with boundary and $\omega_0\cdot\nu(z_0)<0$, that there is an open finite cone $C\subset{\mathbb{R}}^3$ and $\lambda_0>0$ such that $z_0+\lambda_0 C\subset{\mathbb{R}}^3\backslash G$ and $-\omega_0\in C$. The claim then follows from Lemma \[le:esccont\]. Note in particular that when considering the limit $\lim t(x,\omega)=\tau_+(x_0,\omega_0)=\tau_-(y_0,\omega_0)$ one takes in the lemma $y_0=x_0-\tau_+(x_0,\omega_0)\omega_0$ and (hence) $y_+=x_0$.
\[prop-ex\] Define $$(\Gamma_+)_c:=\{(y,\omega,E)\in \Gamma_+\ |\ (y-\tau_+(y,\omega)\omega,\omega,E)\in \Gamma_-\}.$$ Then $\Gamma_+\backslash (\Gamma_+)_c$ has zero-measure in $\Gamma$ and there is a continuous extension $\ol{t}$ of $t:D\to {\mathbb{R}}$ onto ${\mathcal{D}}_-:=D\cup \Gamma_-\cup (\Gamma_+)_c$ given by $$\ol{t}(x,\omega)=\begin{cases}
t(x,\omega), & \textrm{if}\ (x,\omega,E)\in D, \\
\tau_+(x,\omega), & \textrm{if}\ (x,\omega,E)\in (\Gamma_+)_c, \\
0, & \textrm{if}\ (x,\omega,E)\in \Gamma_-.
\end{cases}$$
The result is a straightforward consequence of the above Lemmas.
Note that in the case where $G$ is convex and its boundary is $C^1$-regular the above extension $\ol{t}$ is continuous on $\ol G\times S$. For example, for the ball $G=B(0,r)$ we have $\tau_+(x,\omega)=2|{\left\langle}x,\omega{\right\rangle}|$ and $t(x,\omega)={\left\langle}x,\omega{\right\rangle}+\sqrt{{\left\langle}x,\omega{\right\rangle}^2+r^2-{\left\Vert x\right\Vert}^2}$ ([@tervo14 Example 3.1]). We find that for $(y,\omega)\in \Gamma'=\partial G\
| 2,475
| 1,209
| 1,558
| 2,249
| null | null |
github_plus_top10pct_by_avg
|
inside another vacuous ball, thereby requiring the algorithm to continue. In that case, the comparison with the case of exiting a single sphere breaks down.
![Histogram of the proportion of runs of the walk-on-spheres algorithm with $\alpha=1$ for which $N>n$ for the unit-ball domain (left) and the Swiss cheese domain (right). The red curve shows the tail of Geom($p(1,2)$), this is $(1-p(1,2))^n$, as in Remark 2 of Theorem \[main\].[]{data-label="fig:hist"}](test1_hist2 "fig:") ![Histogram of the proportion of runs of the walk-on-spheres algorithm with $\alpha=1$ for which $N>n$ for the unit-ball domain (left) and the Swiss cheese domain (right). The red curve shows the tail of Geom($p(1,2)$), this is $(1-p(1,2))^n$, as in Remark 2 of Theorem \[main\].[]{data-label="fig:hist"}](test6_hist2 "fig:")
![Mean number of steps for the walk-on-spheres algorithm started at $x=(\sqrt{0.29},-\sqrt{0.7})$ inside the circle domain (left) and inside the Swiss cheese domain (right). The dashed curve on both plots is $1/p(\alpha,2)$ as in Corollary \[indicators\].[]{data-label="fig:steps"}](steps_compare_ball "fig:") ![Mean number of steps for the walk-on-spheres algorithm started at $x=(\sqrt{0.29},-\sqrt{0.7})$ inside the circle domain (left) and inside the Swiss cheese domain (right). The dashed curve on both plots is $1/p(\alpha,2)$ as in Corollary \[indicators\].[]{data-label="fig:steps"}](steps_compare_cheese "fig:")
Appendix: Proof of Theorem \[hasacorr\] {#appendix-proof-of-theorem-hasacorr .unnumbered}
=======================================
Our proof of Theorem \[hasacorr\] uses heavily the joint conclusion of Theorems 2.10 and 3.2 in [@bucur], namely that the Theorem \[hasacorr\] is true in the case that $D$ is a ball. Our proof is otherwise constructive proving existence and uniqueness separately.
[*Existence:*]{} On account of the fact that $D$ is bounded, we can define a ball of sufficiently large radius $R>0$, say $B^* = B(X_0, R)$, centred at $X_0$, such that $D$ is a subset of $B^*$ and hence $\sigma_D
| 2,476
| 228
| 2,185
| 1,693
| 430
| 0.809955
|
github_plus_top10pct_by_avg
|
rformed with a 3.0-T magnet; 28%, with a 1.5-T magnet; and 5%, with a \< 1.0-T magnet. Demographics, overall outcomes, SSc tendon tear types, and prevalence of concomitant SS and/or IS tendon injury of the cohort are shown in [Table 1](#table1-2325967120913036){ref-type="table"}. Patients with isolated SSc tendon repair were significantly younger compared with patients who had multitendon repairs (57.2 vs 64.3 years; *P* \< .0001). Significant improvements were seen in VAS pain scores (*P* \< .001) ([Table 1](#table1-2325967120913036){ref-type="table"}). We found that 90 patients (64.3%) had concurrent SS tendon repair; of these, 14 patients (10.0% of the total cohort) underwent IS tendon repair.
######
Overall Cohort Demographics, Outcome Scores, and Rotator Cuff Tear Characteristics (N = 140)*^a^*
Parameter Finding
--------------------------------------------- ------------------
Age, y, mean ± SD 61.8 ± 9.9
Follow-up, mo, mean ± SD 51.7 ± 19.5
Female, n (%) 59 (42.1)
PROMIS-UE score (95% CI) 50.7 (49.4-52.0)
Preoperative VAS pain score (95% CI) 4.8 (4.4-5.2)
Postoperative VAS pain score (95% CI) 0.8 (0.6-1.1)
Complete SSc tendon tear, n (%) 34 (24.3)
Tendon tear pattern, n (%)
Partial SSc tear with SS and/or IS repair 65 (46.4)
Partial isolated SSc tear 41 (29.3)
Complete SSc tear with SS and/or IS repair 25 (17.9)
Complete isolated SSc tear 9 (6.4)
*^a^*IS, infraspinatus; PROMIS-UE, Patient-Reported Outcomes Measurement Information System for Upper Extremity; SS, supraspinatus; SSc, subscapularis; VAS, visual analog scale.
The prevalence of GC grade 2 change or higher was significantly greater in patients with multitendon repair relative to isolated SSc tendon repair for the SSc (*P* = .008), SS (
| 2,477
| 156
| 2,461
| 2,782
| null | null |
github_plus_top10pct_by_avg
|
y $v\in C^1(\ol G\times S\times I)$, \[csda25\] &-,v\_[L\^2(GSI)]{}+\_x,v\_[L\^2(GSI)]{}+CS\_0,v\_[L\^2(GSI)]{}\
&+,v\_[L\^2(GSI)]{}-K\_C,v\_[L\^2(GSI)]{}\
=& ,S\_0[E]{}\_[L\^2(GSI)]{}- \_[GS]{} S\_0v|\_[E=0]{}\^[E=E\_[m]{}]{} dx d\
&-,\_x v\_[L\^2(GSI)]{} +\_[GSI]{}()v dddE\
&+,CS\_0v\_[L\^2(GSI)]{} +,\^\* v\_[L\^2(GSI)]{}-,K\_C\^\*v\_[L\^2(GSI)]{}\
=&[**f**]{},v\_[L\^2(GSI)]{} where \^\*=and (K\_C\^\*v)(x,,E)= \_[SI]{}(x,,’,E,E’) e\^[C(E’-E)]{}v(x,’,E’)d’ dE’. Assuming that the inflow boundary condition $\phi_{|\Gamma_-}={\bf g}$ and the initial condition $\phi(\cdot,\cdot,E_{\rm m})=0$ are valid, the equation (\[csda25\]) is equivalent to \[csda26\] & ,S\_0[E]{}\_[L\^2(GSI)]{}+ (,,0),S\_0(,,0) v(,,0)\_[L\^2(GS)]{} -,\_x v\_[L\^2(GSI)]{}\
&+\_[GSI]{}()\_+v dddE +,CS\_0v\_[L\^2(GSI)]{}\
&+,\^\* v\_[L\^2(GSI)]{}-,K\_C\^\*v\_[L\^2(GSI)]{}\
=&[**f**]{},v\_[L\^2(GSI)]{}+\_[GSI]{}()\_-[**g**]{} v dddE. Clearly \_[GSI]{}()\_-[**g**]{} v dddE= [**g**]{},\_-(v)\_[T\^2(\_-)]{} and \_[GSI]{}()\_+v dddE= \_+(),\_+(v)\_[T\^2(\_+)]{}.
One thus deduces that the relevant bilinear from $B$ and linear form $F$ are \[csda27\] B(,v)=& ,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{}\
&+C,S\_0v\_[L\^2(GSI)]{}+,(\^\*-K\_C\^\*) v\_[L\^2(GSI)]{}\
&+\_+(),\_+(v)\_[T\^2(\_+)]{} +(,,0),S\_0(,0) v(,,0)\_[L\^2(GS)]{}, and $$F(v)={\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)}.$$ The variational equation corresponding to the problem (\[csda3A\]), (\[finalbc\]), (\[finalic\]) (in the classical sense) is $$B(\phi,v)=F(v)\quad \forall v\in C^1(\ol G\times S\times I).$$
We show that the bilinear form $B:C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)\to{\mathbb{R}}$ obeys the following [*boundedness and coercivity*]{} conditions:
\[csdath1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$) and (\[csda9\]), (\[csda9aa\]), (\[csda9a\]) are valid. Then there exists a constant $
| 2,478
| 569
| 2,549
| 2,564
| null | null |
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|
st(site1);
getPageURLList(site2);
}
Calls the same method that gets called when there is only one link
private void getPageURLList(string site)
{
webBrowser.DocumentCompleted += createList;
webBrowser.Navigate(site);
}
I'm pretty sure the issue is "Navigate" is getting called a second time before createList even starts the first time.
The reason I am using WebBrowser is because these pages use Javascript to sort the links in the table so HTTPRequests and the HTMLAgilityPack don't seem to be able to grab those links.
So I guess my question is: How can I keep my WebBrowser from navigating to a new page until after I finish what I'm doing on the current page?
A:
You have to make a bool variable to know when the first proccess has completed. And then start the other. Application.DoEvents() will help you.
Note that all this events run in the main thread.
Q:
(Munkres) Clarification with respect to what a saturated subset means (quotient maps)
In Munkres, Section 22 (The Quotient Topology) he says the following:
Another way of describing the quotient map is as follows: We say that a subset $C$ of $X$ is saturated (with respect to the surjective map $p:X \rightarrow Y)$ if $C$ contains every set $p^{-1}(\{y\})$ that it intersects.
He further says: Thus $C$ is saturated if it equals the complete inverse image of a subset of $Y$.
Question: Shouldn't it be the complete inverse image of $Y$? Since $p^{-1}({y})$ implies it takes all the points of $Y$?
A:
The definition of saturated is correct. It should not be the complete inverse image of $Y$, because the complete inverse image of $Y$ would simply be $X$ itself, and why go to all that trouble with a new definition that produces nothing other than $X$ itself?
Here's an example. For $p : \mathbb R^2 \to \mathbb R$ defined by $p(x,y)=x$, a saturated subset is any subset of $\mathbb R^2$ which can be expressed as a union of vertical lines. So, for example, the union of the two vertical lines $x=0$ and $x=1$ is satura
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are equivalent but the second one is a bit more convenient to handle in practice. The quantitative equivalence of the two definitions as well as the existence of the limit could be seen using the global conformal transformations explained in . We review it in appendix A.
Energy correlators are defined as follows In all correlation functions are Wightman or non-ordered functions. Non-ordered correlation functions are typical to the in-in type of computations that we discuss here . A universal state we would like to consider is obtained by acting with the stress tensor carrying momentum $q$ on the vacuum where $\eps^{\mu}$ is the polarization tensor. Since the stress tensor is conserved, symmetric and traceless we choose $\eps^2 = 0$, $\eps .q = 0$. The momentum $q$ is always assumed to be time-like with positive energy $q^0 >0$.
Throughout the paper we freely switch between the polarization tensor being $\eps^{\mu} \eps^{\nu}$ with $\eps^2 = 0$, $\eps.q = 0$ and $\eps^{\mu \nu}$ with $\eps^{\mu}_{~ \mu} = 0$, $q_{\mu}\eps^{\mu \nu} = 0$. We found it more convenient to use the first choice when writing the most general structures allowed by symmetries while the second is more convenient when thinking about the positivity of energy correlators in the reference frame where $q=(q^0, \vec 0)$ as will be explained below.
Let us discuss symmetries of energy correlators with the state . They are
[a)]{} Lorentz invariance
This is manifest in the way we defined energy correlators above .
[b)]{} Projective covariance In the formula above we relaxed the condition $n=(1,\vec n)$ and considered $n$ to be a light-like vector $n^2 = 0$. The property could be easily checked to follow from conformal properties of the stress tensor. Qualitatively, one can understand it as follows. The leading contribution from the spin part of $T(x).\bar n . \bar n$ behaves like $(x . \bar n)^2$. Combined with the limiting prefactor this forms $(x . \bar n)^4$ that picks up the corresponding term in the large $(x . \bar n)$ expansion of
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g \triangleright h_2$ that can be written as $${\label{eq:KS4'}}
g^{-1}(g\triangleleft {h_1}) \in {\rm Stab}_G (h_2)$$ for any $g\in G$, $h_1$, $h_2\in H$. Thus if $\alpha$ is an action as automorphisms then $(H, G, \alpha, \beta)$ is a matched pair if and only if [(\[eq:3\])]{} and [(\[eq:KS4’\])]{} hold. The condition [(\[eq:KS4’\])]{} gives important information regarding $\beta$: the elements $g^{-1}\beta (g, h)$ act trivially on $H$ for any $g\in
G$ and $h\in H$.
Now we can describe all matched pairs $(C_3, C_m, \alpha, \beta)$.
[\[pr:2.4.45\]]{} Let $m$ be a positive integer, $\alpha : C_m \times C_3
\rightarrow C_3$, $\beta : C_m \times C_3 \rightarrow C_m$ two maps and $t\in [m-1]$ such that $m|t^3 -1$. Then:
1. Let $\alpha$ be the trivial action and $\beta = \beta_t
: C_m \times C_3 \rightarrow C_m$ given by $${\label{eq:2.4.90}}
\beta (b^i, a) = b^{it}, \quad \beta (b^i, a^2) = b^{it^2}, \quad
\beta (b^i, 1) = b^i$$ for any $i= 0, \cdots, m-1$. Then $(C_3, C_m, \alpha, \beta_t)$ is a matched pair. There are no other matched pairs $(C_3, C_m,
\alpha, \beta)$ if $m$ is odd.
2. Assume that $m$ is even. Let $\beta$ be the trivial action and $\alpha : C_m \times C_3 \rightarrow C_3$ given by $\alpha (b^j, 1) = 1$ and $${\label{eq:2.4.740}}
\alpha (b^j, a) = \left \{\begin{array}{rcl}
a, \, & \mbox {\rm if $j$ is even }\\
a^2, \, & \mbox {\rm if $j$ is odd }
\end{array} \right.$$ $${\label{eq:2.4.750}}
\alpha (b^j, a^2) = \left \{\begin{array}{rcl}
a^2, \, & \mbox {\rm if $j$ is even }\\
a, \, & \mbox {\rm if $j$ is odd }
\end{array} \right.$$ for all $j = 1, \cdots, m-1$. Then $(C_3, C_m, \alpha, \beta)$ is a matched pair.
3. Assume that $m = 6 u$ for some positive integer $u$ and that $\alpha$ is described by [(\[eq:2.4.740\])]{} and [(\[eq:2.4.750\])]{}. Then there exist two matched pairs $(C_3, C_m,
\alpha, \beta)$, $(C_3, C_m, \alpha, \beta')$, where $\beta$ and $\beta'$ are given by $${\label{eq:24.11}}
\beta (b^{2k+1}, a) = b^{2u
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YM theory of the gauge symmetry of Sec.\[GaugeSymm\] can be interpreted as a gravity theory. This will be the main issue to focus on below.
Nevertheless, motivated by this potential identification, we denote the covariant derivative as D\_[a]{} = \_[a]{}\^\_, where we used the notation \_[a]{}\^ \_[a]{}\^ + A\_[a]{}\^. \[def-e-A\]
The kinetic term of a scalar field is \^[ab]{}D\_a D\_b = \^ \_ \_ , where the effective metric $\hat{g}_{\m\n}$ naturally arises. It is defined by \_ = \_\^a \_[ab]{} \_\^b, \[metric\] where $\hat{e}_{\m}{}^a$ is by definition the inverse of $\hat{e}_{a}{}^{\m}$.
Field Strength vs Torsion
-------------------------
The field strength of the non-Abelian gauge symmetry constructed above is F\_[ab]{}(x) = \_[a]{} A\_[b]{}(x) - \_[b]{} A\_[a]{}(x) + \[A\_[a]{}(x), A\_[b]{}(x)\]. In the representation (\[Teps\]), it is F\_[ab]{}(x) = F\_[ab]{}\^[()]{}(x) \_, where F\_[ab]{}\^[()]{}(x) = \_a\^\_\_b\^ - \_b\^\_\_a\^. \[field-strength\]
With the analogy between $\hat{e}_{\mu}{}^a$ and the vielbein $e_{\mu}{}^a$, we define \^\_ \_\^ - \_\^. \[torsion\] It is the torsion for the Weitzenböck connection \^\_ \_a\^\_\_\^a \[connection\] used in teleparallel gravity when $\hat{e}_{\mu}{}^a$ is identified with the vielbein $e_{\mu}{}^a$. The field strength and the “torsion” are essentially the same quantity: F\_[ab]{}\^[()]{}(x) = - \_a\^ \_b\^ \^\_, if we think of $\hat{e}_{\mu}{}^a$ and $\hat{e}_a{}^{\m}$ as the quantities used to switch between the two bases $\del_{\m}$ and $D_a$.
The “connection” (\[connection\]) satisfies the relation D\_ \_\^a \_ \_\^a - \_\^ \_\^a = 0, and has zero “curvature”: d - = 0.
Field-Dependent Killing Form
----------------------------
An interesting feature of the algebra (\[commutator\]) for space-time diffeomorphism is that the Killing form (invariant inner product) has to be field-dependent.
For two elements of the Lie algebra $T_f \equiv f^{\m}(x) \del_{\m}$ and $T_{f'} \equiv f'{}^{\n}(x) \del_{\n}$, it is clear that the Killing form should b
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thbb{G}}}_a$; this fact was mentioned in §\[germlist\]. The following picture represents schematically the curves described above.
 
[^1]: [**Acknowledgments.**]{} Work on this paper was made possible by support from Mathematisches Forschungsinstitut Oberwolfach, the Volkswagen Stiftung, the Max-Planck-Institut für Mathematik (Bonn), Princeton University, the Göran Gustafsson foundation, the Swedish Research Council, the Mittag-Leffler Institute, MSRI, NSA, NSF, and our home institutions. We thank an anonymous referee of our first article on the topic of linear orbits of plane curves, [@MR94e:14032], for bringing the paper of Aldo Ghizzetti to our attention.
---
abstract: 'In this article, I present the questions that I seek to answer in my PhD research. I posit to analyze natural language text with the help of semantic annotations and mine important events for navigating large text corpora. Semantic annotations such as named entities, geographic locations, and temporal expressions can help us mine events from the given corpora. These events thus provide us with useful means to discover the locked knowledge in them. I pose three problems that can help unlock this knowledge vault in semantically annotated text corpora: *i.* identifying important events; *ii.* semantic search; and *iii.* event analytics.'
author:
- |
Dhruv Gupta\
\
\
subtitle: 'Detecting Events in Semantically Annotated Corpora for Search & Analytics'
title: Event Search and Analytics
---
Introduction
============
Information retrieval systems have largely relied on word statistics in text corpora to satisfy information needs of users by retrieving documents with high relevance for a given keyword query. In my PhD research I hypothesize that information needs of users can be satisfied to a greater extent by using *events* as a means of navigating text corpora. Events in our context would be an act performed by certain actor(s) at a specific location during a specific tim
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measurement errors and the expected variation of helium abundance. Thus, these starburst regions appear to contain massive stars ($> 39,000$ K if main sequence stars.) By contrast, NGC 253, NGC 4102, and the nucleus of M82 have [ $1.7$ ]{}/$<0.15$, and thus are inferred to have softer ionizing continua ($\lesssim 37,000$ K if main sequence stars.) NGC 3504 and the two off–nuclear regions of M82 have line ratios intermediate to these extremes.
Figure \[fig:heh\_hek\] illustrates that [ $2.06$ ]{}/ does not trace [ $1.7$ ]{}/ as the models predict. The nucleus of M82 demonstrates that [ $2.06$ ]{} may be strong while [ $1.7$ ]{} is weak, contrary to the expected behavior (but expected if [ $2.06$ ]{} is pumped.) However, for most galaxies, [ $2.06$ ]{} is *too weak* for the measured [ $1.7$ ]{}. This is the first direct demonstration that [ $2.06$ ]{}/ is a poor diagnostic of in starburst galaxies. Radiative transfer considerations have predicted that the behavior of [ $2.06$ ]{} should not be a simple function of [@shields]. @lph320 confirm this complex behavior for planetary nebulae, though they attempt to constrain the dependence on T$_e$ and density by also considering optical lines [@doherty95]. However, the data do not contradict the expectation that a very low [ $2.06$ ]{}/ ratio (below $\sim0.2$) indicates that the continuum is fairly soft, because there would be few ionizing photons and also few resonantly scattered photons.
We further consider the reliability of the [ $2.06$ ]{}/ ratio in figure \[fig:ne\_206\], by comparing it to the mid–infrared line ratio \[\] 15.6/\[\] 12.8. Here, too, [ $2.06$ ]{}/ is too low for a given \[\]/\[\] (compared to model predictions) and there is no obvious correlation between the two ratios. An alternative interpretation of figure \[fig:ne\_206\] would be that [ $2.06$ ]{}/ is correct and \[\]/\[\] is systematically overproduced; we feel this is unlikely because, as we will demonstrate in § \[sec:midir-test
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een $V'$ and $V.$ As usual, by identifying $H$ with $H'$ we have $V\underset d\hookrightarrow H\cong H'\underset d\hookrightarrow V'$ see e.g., [@Bre11].
Let ${\mathfrak{a}}:[0,T]\times V \times V \to {\mathbb{C}}$ be a *non-autonomous sesquilinear form*, i.e., ${\mathfrak{a}}(t;\cdot,\cdot)$ is for each $t\in[0,T]$ a sesquilinear form, $$\label{measurability}
{\mathfrak{a}}(\cdot;u,v) \text{ is measurable for all } u,v\in V,$$ such that $$\label{eq:continuity-nonaut}
|{\mathfrak{a}}(t,u,v)| \le M \Vert u\Vert_V \Vert v\Vert_V\ \text{ and } {\operatorname{Re}}~{\mathfrak{a}}(t,u,u)\ge \alpha \|u\|^2_V \quad \ (t,s\in[0,T], u,v\in V),$$ for some constants $M, \alpha>0$ that are independent of $t, u,v.$ Under these assumptions there exists for each $t\in[0,T]$ an isomorphism ${\mathcal{A}}(t):V\to V^\prime$ such that $\langle {\mathcal{A}}(t) u, v \rangle = {\mathfrak{a}}(t,u,v)$ for all $u,v \in V.$ It is well known that $-{\mathcal{A}}(t),$ seen as unbounded operator with domain $V,$ generates an analytic $C_0$-semigroup on $V'$. The operator ${\mathcal{A}}(t)$ is usually called the operator associated with ${\mathfrak{a}}(t,\cdot,\cdot)$ on $V^\prime.$ Moreover, we associate an operator $A(t)$ with ${\mathfrak{a}}(t;\cdot,\cdot)$ on $H$ as follows $$\begin{aligned}
D(A(t))={}&\{u\in V {\, \vert \,}\exists f\in H \text{ such that } {\mathfrak{a}}(t;u,v)=(f{\, \vert \,}v)_H \text{ for all } v\in V \}\\
A(t) u = {}& f.\end{aligned}$$ It is not difficult to see that $A(t)$ is the part of ${\mathcal{A}}(t)$ in $H.$ In fact, we have $D(A(t))= \{ u\in V : {\mathcal{A}}(t) u \in H \}$ and $A(t) u = {\mathcal{A}}(t) u.$ Furthermore, $-A(t)$ with domain $D(A(t))$ generates a holomorphic $C_0$-semigroup on $H$ which is the restriction to $H$ of that generated by $-{\mathcal{A}}(t).$ For all this results see e.g. [@Tan79 Chapter 2] or [@Ar06 Lecture 7].
We now assume that there exist $0< \gamma<1$ and a continuous function $\omega:[0,T]\longrightarrow [0,+\infty)$ such that $$\label{eq 1:Dini-condition}
|{\mathfra
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athbb E}}\big(\widetilde{\chi}_r^{2}\big)} \sqrt{{{\mathbb P}}(\widetilde{\chi}_{r}\geq 1)} ~,$$ the desired limit follows from (\[step1\]) if we prove that $\limsup_{r\rightarrow+\infty}{{\mathbb E}}(\widetilde{\chi}_{r}^{2})$ is finite, which is the result of Lemma \[lemme:moment2\]. $\Box$
The section ends with the proofs of Lemmas \[lemme:moment2\] and \[lemme:approximationDSF\].
Let $A_r$ and $B_r$ be as in the proof of Theorem \[theo:sublin\]. Let $W_r$ be the intersection point of the two tangents to $S(O,r)$ that pass through the points $A_r$ and $B_r$. See the left part of Figure \[fig:arc\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:arc\] [*On the left: The sphere $S(O,r)$ with center $O$ and radius $r$ is represented in bold. The tangents (the dotted lines) to $S(O,r)$ respectively at $A_r$ and $B_r$, intersect the horizontal axis on $W_r$. On the right: The segment $[X,\mathcal{A}(X)]$ crosses $I_{r}$ (in bold) on $J(X)$. On this picture, $X$ is outside $I_{r}\oplus B(0,c)$.*]{}](arc.eps "fig:"){width="5.5cm" height="5cm"} ![\[fig:arc\] [*On the left: The sphere $S(O,r)$ with center $O$ and radius $r$ is represented in bold. The tangents (the dotted lines) to $S(O,r)$ respectively at $A_r$ and $B_r$, intersect
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eq \mathbb{P}(\Gamma >n ) = (1-p)^n, \qquad n\in\mathbb{N}.$$
- The randomness in the geometric random variables $\Gamma$ is heavily correlated to $N$. The fact that each of the $\Gamma$ are geometrically distributed has the advantage that $$\sup_{x\in D}\mathbb{E}_x[N] \leq \sup_{x\in D}\mathbb{E}_x[\Gamma] = \frac{1}{p}.$$ However, it is less clear what kind of distributional properties can be said of the random variable $
\sup_{x\in D}\Gamma,
$ which almost surely upper bounds $\sup_{x\in D}N$.
Finally, it is worth stating formally that the walk-on-spheres algorithm is unbiased and therefore, providing $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$, the [strong law of large numbers]{}applies and a straightforward Monte Carlo simulation of the solution to is possible. Moreover, providing $\mathbb{E}_x[{g}(X_{\tau_D})^2]<\infty$, the [central limit theorem]{}offers the rate of convergence.
\[rate1\] When $D$ is bounded and convex and ${g}$ is continuous and in $ L^1_\alpha(D^{\mathrm{c}})$, $$\label{WoSMC2}
\lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n {g}(\rho^{i}_{N^{i}}) = \mathbb{E}_x[{g}(\rho_{N})]=\mathbb{E}_x[{g}(X_{\tau_D})] = u(x),$$ almost surely where $(\rho^{i}_n, n\leq N^{i})$, $i\geq 1$ are [*iid*]{}copies of the walk-on-spheres with $\rho_0^i = x\in D$, $i\geq 1$ and $u(x)$ is the solution to (\[aDirichlet\]). Moreover, when $$\int_{D^\mathrm{c}}\frac{{g}(x)^2}{1+|x|^{\alpha+d}}\,{\rm d}x<\infty,
\label{f2}$$ then $\operatorname{Var}({g}(\rho_N))<\infty$ and, in the sense of weak convergence, $$\lim_{n\to\infty}n^{1/2}\left(\frac 1n \sum_{i=1}^n {g}(\rho^{i}_{N^{i}})- u(x)\right)= \operatorname{Normal}(0, \operatorname{Var}({g}(\rho_N))).$$
The first part is a straightforward consequence of the earlier mentioned [strong law of large numbers]{}and the fact that Theorem \[corr\] ensures that $\mathbb{E}_x[{g}(\rho_N)]=\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$. For the second part, we need to show that implies $\mathbb{E}_x[{g}(\rho_N)^2]=\mathbb{E}
| 2,487
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| 2,138
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$. Let $a_j$, $j=1,2,\ldots$ be an ascending sequence in $\Gamma$. Since $\Gamma_i=\bar{\Gamma}_i$ for all $i\in I$, it follows that $\lim a_j\in \Gamma_i$ for all $i$, and hence $\lim a_i\in \Gamma$, as desired.
On the other hand, suppose $J=\{1,\ldots,k\}$ and let $\Gamma=\Union_{i=1}^k\Gamma_i$. Then $\Gamma$ satisfies obviously condition (1) of a multicomplex.
By Lemma \[finitem\] the sets ${\mathcal M}(\Gamma_i)$ are finite, and $\Union_{i=1}^k\Gamma_i$ is the set of all $a\in\NN^n_\infty$ for which there exists $j\in J$ and $m\in {\mathcal M}(\Gamma_j)$ such that $a\leq m$. Thus it follows from Corollary \[m\] the $\Gamma$ is a multicomplex.
\[closure\] Let $A\subset \NN^n_\infty$ be an arbitrary subset of $\NN^n_\infty$. Then there exists a unique smallest multicomplex $\Gamma(A)$ containing $A$.
Let $\Gamma$ be a multicomplex, and let $I(\Gamma)$ be the $K$-subspace in $S=K[x_1,\ldots,x_n]$ spanned by all monomials $x^a$ such that $a\not\in \Gamma$. Note that if $a\in \NN^n$ and $b\in \NN^n\setminus\Gamma$, then $a+b\in \NN^n\setminus \Gamma$, that is, if $x^a\in I(\Gamma)$ then $x^ax^b\in I(\Gamma)$ for all $x^b\in S$. In other words, $I(\Gamma)$ is a monomial ideal. In particular, the monomials $x^a$ with $a\in \Gamma$ form a $K$-basis of $S/I(\Gamma)$.
For example for the above multicomplex $\Gamma=\{a\: a\leq (0,\infty )\; \text{or}\; a\leq (2,0)\}$ in $\NN^2_\infty$ we have $I(\Gamma)=(x_1^3,x_1x_2)$.
Conversely, given an arbitrary monomial ideal $I\subset S$, there is a unique multicomplex $\Gamma$ with $I=I(\Gamma)$. Indeed, let $A =\{a\in \NN^n\:
x^a\not\in I\}$; then $\Gamma=\Gamma(A)$.
The monomial ideal of a multicomplex behaves with respect to intersections and unions of multicomplexes as follows:
\[ideal\] Let $\Gamma_j$, $j\in J$ be a family of multicomplexes. Then
1. $I(\Sect_{j\in J}\Gamma_j)=\sum_{j\in J} I(\Gamma_j)$,
2. if $J$ is finite, then $I(\Union_{j\in J}\Gamma_j)=\Sect_{j\in J}I(\Gamma_j)$.
Next we describe the relationship between simplicial complexes and multicompl
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+ 36 = 0, a - t - 2*t - 10 = i. What is the un
- 0 - 8/(-2). Suppose 1251 = -2*b + 1307. Calculate b*z(i) + h*f(i).
-4*i
Let p(h) = 4*h - 3. Suppose b - 4 = -0*u - 2*u, 2*b - 5*u = 53. Let t(s) = 9*s - 7. Calculate b*p(g) - 6*t(g).
2*g
Let w(n) = -5*n**2 + 4*n + 4. Let d(v) = -6*v**2 + 3*v + 3. Suppose 32 + 36 = 17*y. Give y*d(h) - 3*w(h).
-9*h**2
Let z(u) = 2*u - 141 + 75 + 71 - 6*u. Let x = 8 - 11. Let y(o) = -o**3 - 8*o**2 + 21*o + 14. Let g be y(-10). Let s(q) = -5*q + 6. Determine g*z(c) + x*s(c).
-c + 2
Let i(q) = 20*q + 7. Let a(g) be the second derivative of 5*g**3/3 + 2*g**2 - 16*g. Let c(f) = -5*a(f) + 2*i(f). Let n(o) = 3*o + 2. Determine 2*c(u) + 7*n(u).
u + 2
Let a(n) = 3*n + 30. Let t(f) = -3*f - 31. Determine -6*a(x) - 7*t(x).
3*x + 37
Suppose 0 = -4*d + 4*i - 32, -5*d - i - 10 = -0*i. Let y(v) = v + 1. Let c(r) = -4*r - 6. What is d*y(a) - c(a)?
a + 3
Let o(x) = -x - 5. Let d(k) = 414*k + 450. What is -d(b) - 90*o(b)?
-324*b
Let m(d) = -2*d**2 + d + 1. Let b(j) = -3*j**2 - 2*j + 10. Give b(c) - m(c).
-c**2 - 3*c + 9
Suppose -2*q + 5 + 1 = 0. Suppose 4*z + 5*t = 65, -t + 13 = 2*z - 18. Let h(b) = 1. Let a(y) = -y - 5. Give q*a(j) + z*h(j).
-3*j
Let k(u) = -15. Let w(r) = r - 58. What is 9*k(v) - 2*w(v)?
-2*v - 19
Let u(p) = 9*p**3 + 10*p - 1. Let c(s) = 8*s**3 + s**2 + 9*s - 1. Let j(g) = 5*c(g) - 4*u(g). Let k(b) = 6*b**3 + 7*b**2 + 7*b - 2. Calculate 7*j(t) - 5*k(t).
-2*t**3 + 3
Let g(b) = b**3 + b**2 + 1. Let f(w) = 2*w**3 + 2*w**2 + w + 3. Let n(y) = -y**2 + 7*y + 496. Let x be n(26). Give x*f(q) - 6*g(q).
-2*q**3 - 2*q**2 + 2*q
Let z(f) = -1365*f**2 - 112*f - 112. Let h(d) = 124*d**2 + 10*d + 10. Determine -56*h(o) - 5*z(o).
-119*o**2
Let l(x) = 0 + 67*x - 56*x - 8 - 9*x**2 + 0. Let h be (8/(-5))/((-2)/10). Let b(c) = -6*c**2 + 7*c - 5. Calculate h*b(i) - 5*l(i).
-3*i**2 + i
Let h(b) = 2*b + 1. Suppose 7*d + 23 = 2. Let u(r) = -r. Let i(f) = f - 10. Let c be i(6). Calculate c*u(m) + d*h(m).
-2*m - 3
Let d(t) = -7*t**2 + 7*t + 2. Let b(u) = 9*u**2 - 8*u - 2. Calculate 5*b(j) + 6*d(j).
3*j**2 +
| 2,489
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wards new stimuli. It was implemented on a Fischer Technik mobile robot, which uses a Motorola 68HC11 microcontroller. The robot has a two wheel differential drive system and four light sensors facing in the cardinal directions.
{width=".3\textwidth"}
In the experiments described below, the robot received a number of different light stimuli, which varied in the frequency of the flashes. It classified these stimuli autonomously and decided whether or not to respond (turn towards the source) according to how novel they were. Each of the sensors on the robot, in this case four light sensors, had its own novelty filter, as shown in figure \[SysLayout\]. At each cycle, the current reading on each sensor was concatenated with the previous five to form a six element input vector, known as a delay line or lag vector. This vector was classified by the novelty filter and an output produced. In the case of the TKM, which keeps an internal history of previous inputs, only the most recent reading was needed as input.
The output of the filter was a function of how many times that neuron had fired before, due to the habituating synapse. Each of the four novelty filters fed their output to a comparator function which propagated the strongest signal, providing that it was above a pre-defined threshold, to the action mechanism. If none of the stimuli were strong enough, the cycle repeated. Owing to memory constraints, the clustering mechanism was limited to just twelve neurons arranged in a ring. All three of the networks described in section \[NNs\] were the same size.
A bypass function was associated with each sensor. If a neuron had not fired before (that is, its synapse had not been habituated) the comparator function favoured it, so that the system responded rapidly to new signals. If two new signals were detected simultaneously, the stronger one was used.
Experiments and Results \[Results\]
===================================
Three separate experiments were carried out. The first, t
| 2,490
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r-order expansion coefficients are more involved, due to the above item (c). This will also be explained in detail in Section \[s:bounds\].
Bounds on $\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for the ferromagnetic models {#s:reduction}
=================================================================================
From now on, we restrict ourselves to the ferromagnetic models. In this section, we explain how to prove Proposition \[prp:Pij-Rj-bd\] assuming a few other propositions (Propositions \[prp:GimpliesPix\]–\[prp:exp-bootstrap\] below). These propositions are results of diagrammatic bounds on the expansion coefficients in terms of two-point functions. We will show these diagrammatic bounds in Section \[s:bounds\].
The strategy to prove Proposition \[prp:Pij-Rj-bd\] is model-independent, and we follow the strategy in [@h05] for the nearest-neighbor model and that in [@hhs03] for the spread-out model. Since the latter is simpler, we first explain the strategy for the spread-out model. In the rest of this paper, we will frequently use the notation $$\begin{aligned}
{\vbx{|\!|\!|}}=|x|\vee1.\end{aligned}$$ We also emphasize that constants in the $O$-notation used below (e.g., $O(\theta_0)$ in [(\[eq:pi-bd\])]{}) are independent of $\Lambda\subset{{\mathbb Z}^d}$.
Strategy for the spread-out model
---------------------------------
Using the diagrammatic bounds below in Section \[s:bounds\], we will prove in detail in Section \[ss:proof-so\] that the following proposition holds for the spread-out model:
\[prp:GimpliesPix\] Let $J_{o,x}$ be the spread-out interaction. Suppose that $$\begin{aligned}
{\label{eq:IR-xbd}}
\tau\leq2,&&
G(x)\leq\delta_{o,x}+\theta_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$ and $q\in(\frac{d}2,d)$. Then, for sufficiently small $\theta_0$ (with $\theta_0L^{d-q}$ being bounded away from zero) and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned}
{\label{eq:pi-bd}}
\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq
\begin{cases}
O(\
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re is the orbit $Gp = \{ \phi_{g}(p) | g\in G\}$, all points which are related to $p$ by an ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$ transformation. $Gp$ is a 3-dimensional submanifold of $\mathcal{M}$, and the collection of all the orbit spaces forms a foliation. In this case, each leaf $\Sigma_{u}$ is a surface of constant $\theta$ (or $u$). Thus we can perform a $3+1$ decomposition of the spacetime, and look for basis functions of ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$ acting on a hypersurface $\Sigma_u$.
#### Highest weight states. {#sec:step-highest-weight}
Second, we simultaneously diagonalize $\{{\mathcal{L}}_{Q_{0}}, {\mathcal{L}}_{H_0}, \Omega\}$ in the space of scalar, vector, and symmetric tensor functions. We label the eigenstates by $m,h,k$ respectively, $$\begin{aligned}
\label{eq:codiagonalize-three-operator}
{\mathcal{L}}_{Q_0}\,{\xi}^{(m\,h\,k)} &= im\,{\xi}^{(m\,h\,k)} \,, \\
\Omega\,{\xi}^{(m\,h\,k)} &= h(h+1)\,{\xi}^{(m\,h\,k)} \,, {\nonumber}\\
{\mathcal{L}}_{H_0}\,{\xi}^{(m\,h\,k)} &= (-h+k)\,{\xi}^{(m\,h\,k)} \,. {\nonumber}\end{aligned}$$ Then using the raising operator ${\mathcal{L}}_{H_{+}}$, we also impose the highest-weight condition, $k=0$, $$\label{eq:general-highest-weight-solution}
\mathcal{L}_{H_+}\,{\xi}^{(m\,h\,0)} = 0 \,.$$ The solutions ${\xi}^{(m\,h\,0)}$ that satisfy both Eq. and are the highest-weight basis functions. At each point on $\Sigma_{u}$, the spaces of scalars, vectors, and symmetric tensors have dimensions 1, 3, and 6. Thus the space of solutions of this system of equations is a linear vector space of dimension 1, 3, and 6 for scalars, vectors, and symmetric tensors, for each choice of $(m, h)$. Correspondingly, for each $(m,
h)$, there will be 1, 3, and 6 free coefficients $c_{\beta}$ for the solution, with $\beta$ ranging over the appropriate dimensionality.
#### Descendants. {#sec:step-descendants}
Next, we obtain basis functions with arbitrary weight by applying the lowering operator $\mathcal{L}_{H_-}$ to the highest-weight state
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tivity of an NMR style experiment using Xe, the blue line is the sensitivity using $^3\text{He}$. The dashed lines show the limit from magnetization noise for each sample. These lines assume the parameters in Table \[Tab: experiments\]. The ADMX region shows the part of QCD axion parameter space which has been covered (darker blue) [@Asztalos:2009yp] or will be covered in the near future (lighter blue) [@ADMXwebpage; @snowdarktalk] by ADMX.](nucleonplot.pdf){width="6"}
Figure \[Fig:Nucleon\] shows constraints on $g_\text{aNN}$ and the potential sensitivity of our proposals. The width of the line shows axion model-dependence in the axion-nucleon coupling. The solid lines are preliminary sensitivity curves with the sensitivity limited by magnetometer noise. Both lines assume samples of volume $\left(10 \text{ cm}\right)^3$ with 100 percent nuclear polarization. Other sample parameters are described in Table \[Tab: experiments\]. The dashed lines show the limits from sample magnetization noise, so where they are higher than the corresponding solid line, they are the limit on sensitivity. The solid curves are cutoff at high frequencies by the requirement that the Larmor frequency be achievable with the assumed maximum magnetic field.
---- --------- ------------------------------------------------ ----------------- ------- -------- ----------------------------------------------
Element Density Magnetic Moment $T_2$ Max. B Magnetometer
($n$) ($\mu$) Sensitivity
1. Xe $1.3 \times 10^{22} \frac{1}{{\text{cm}}^{3}}$ $0.35 \, \mu_N$ 100 10 T $10^{-16} \frac{\text{T}}{\sqrt{\text{Hz}}}$
2. $^3$He $2.8 \times 10^{22} \frac{1}{{\text{cm}}^{3}}$ $2.12 \, \mu_N$ 100 20 T $10^{-17} \frac{\text{T}}{\sqrt{\text{Hz}}}$
---- --------- ------------------------------------------------ ----------------- ------- -
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rst, **generic information**: the latent patterns in the AOG were pre-fine-tuned using massive object images in a category, instead of being learned from a few part annotations. Thus, these patterns reflected generic part appearances and did not over-fit to a few part annotations.
Second, **less model drifts:** Instead of learning new CNN parameters, our method just used limited part annotations to mine the related patterns to represent the part concept. In addition, during active QA, Equation (\[eqn:predict\]) usually selected objects with common poses for QA, *i.e.* choosing objects sharing common latent patterns with many other objects. Thus, the learned AOG suffered less from the model-drift problem.
Third, **high QA efficiency:** Our QA process balanced both the commonness and the accuracy of a part template in Equation (\[eqn:predict\]). In early steps of QA, our approach was prone to asking about new part templates, because objects with un-modeled part appearance usually had low inference scores. In later QA steps, common part appearances had been modeled, and our method gradually changed to ask about objects belonging to existing part templates to refine the AOG. Our method did not waste much labor of labeling objects that had been well modeled or had strange appearance.
Summary and discussion
======================
In this paper, we have proposed a method to bridge and solve the following three crucial issues in computer vision simultaneously.
- Removing noisy representations in conv-layers of a CNN and using an AOG model to reveal the semantic hierarchy of objects hidden in the CNN.
- Enabling people to communicate with neural representations in intermediate conv-layers of a CNN directly for model learning, based on the semantic representation of the AOG.
- Weakly-supervised transferring of object-part representations from a pre-trained CNN to model object parts at the semantic level, which boosts the learning efficiency.
Our method incrementally mines object-part patterns from conv-laye
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times
K(\Z/2,1)$ is given by the composition $L^{62}_{\I_3} \to
K(\I_3,1) \stackrel{\pi_{b,\ast}}{\longrightarrow} K(\I_b,1)
\subset K(\Z/4,1) \times K(\Z/2,1)$, where $\kappa_1 \in
K(\I_b;\Z/2)$ determines the inclusion $K(\I_b,1) \subset
K(\Z/2,1) \subset K(\Z/4,1)$.
Let us formulate the results in the following lemma.
### Lemma 4 {#lemma-4 .unnumbered}
–1. Let $n \ge 2^{13}-2$ and $k$, $n-4k=62$ satisfy the conditions of Theorem 1 (in particular, an arbitrary element in the group $Imm^{sf}(n-k,k)$ admits a retraction of the order $62$. Then for arbitrarily small positive numbers $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_1 >> \varepsilon_2$ (the numbers $\varepsilon_1$, $\varepsilon_2$ are the calibers of the regular deformations in the construction of the $PL$–mapping $\hat h
:\hat X \to \R^n$ and of the immersion $g: N^{n-2k}
\looparrowright \R^n$ correspondingly) there exists the mapping $m_a =(\kappa_a \times \mu_a): \Sigma_{h}^{reg} \to K(\Z/4,1)
\times K(\Z/2,1)$ under the following condition. The restriction $m_a \vert_{\partial \Sigma_h^{reg}}$ (by $\partial
\Sigma_h^{reg}$ is denoted the part of the singular polyhedron consisting of points on the diagonal) has the target $K(\Z/2,1)
\times K(\Z/2,1) \subset K(\Z/4,1) \times K(\Z/2,1)$ and is determined by the cohomological classes $\kappa_{\hat X,1},
\kappa_{\hat X,2}$.
–2. The mappings $\kappa_a$, $\mu_a$ induces a mapping $(\mu_a
\times \kappa_a): L^{62} \to K(\Z/4,1) \times K(\Z/2,1)$ on the self-intersection manifold of the immersion $g$. $$$$
Let us prove that the mapping $(\mu_a \times \kappa_a)$ constructed in Lemma 4 determines a $\Z/2 \oplus
\Z/4$–structure for the $\D_4$–framed immersion $g$. We have to prove the equation (9).
Let us recall that the component $L^{62}_{int}$ of the self-intersection manifold of the immersion $g$ is a $\Z/2 \int
\D_4$–framed manifold with trivial Kervaire invariant: the corresponding element in the group $Imm^{\Z/2 \int \D_4}(62,
n-62)$ is in the image of the transfer homomorphism. Therefo
| 2,495
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)\end{array}$$
with the last two being disjoint unions, and $A$ is closed iff $A$ contains all its cluster points, $\textrm{Der}(A)\subseteq A$, iff $A$ contains its closure. Hence $$\begin{gathered}
A=\textrm{Cl}(A)\Longleftrightarrow\textrm{Cl}(A)=\{ x\in A\!:((\exists N\in\mathcal{N}_{x})(N\subseteq A))\vee((\forall N\in\mathcal{N}_{x})(N\bigcap(X-A)\neq\emptyset))\}\end{gathered}$$ Comparison of Eqs. (\[Eqn: Def: Boundary\]) and (\[Eqn: Def: Derived\]) also makes it clear that $\textrm{Bdy}(A)\subseteq\textrm{Der}(A)$. The special case of $A=\textrm{Iso}(A)$ with $\textrm{Der}(A)\subseteq X-A$ is important enough to deserve a special mention:
**Definition 2.4.** ***Donor set.*** *A proper, nonempty subset $A$ of $X$ such that* $\textrm{Iso}(A)=A$ *with* $\textrm{Der}(A)\subseteq X-A$ *will be called* *self-isolated* *or* *donor.* *Thus sequences eventually in a donor set converges only in its complement; this is the opposite of the characteristic of a closed set where all converging sequences eventually in the set must necessarily converge in it. A closed-donor set with a closed neighbour has no derived or boundary sets, and will be said to be* *isolated in $X$.*$\qquad\square$
**Example 2.5.** In an isolated set sequences converge, if they have to, simultaneously in the complement (because it is donor) and in it (because it is closed). Convergent sequences in such a set can only be constant sequences. Physically, if we consider adherents to be contributions made by the dynamics of the corresponding sequences, then an isolated set is secluded from its neighbour in the sense that it neither receives any contributions from its surroundings, nor does it give away any. In this light and terminology, a closed set is a *selfish* set (recall that a set $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ converges in $A$; conversely a set is open in $X$ iff the only nets that converge in $A$ are eventually in it), **whereas a set with a derived set that intersects itself and its compleme
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\], which we restate here in a stronger form.
\[bigmatroid\] Let $s \ge 0$ be an integer and $M$ be a matroid. Either
- $M$ has a $U_{s,2s}$-minor,
- $M$ has an $s U_{1,2}$-minor, or
- $M$ has a minor $N$ so that $|M| - |N| \le 4^{4^{2s^2}}$ and every element of $N$ is a loop or a coloop.
The result is trivial for $s \le 1$. Suppose that $s \ge 2$ and let $h = \tfrac{1}{2}4^{4^{2s^2}}$; note that $h \ge 4^{s(s4^s)^s}$. Let $B$ be a basis for $M$ and let $X = E(M) - B$. If $r_M(X) \ge h$ then $M$ clearly has a rank-$h$ minor with two disjoint bases, and the result follows from Theorem \[selfdual\]. Otherwise, let $X'$ be a basis for $M|X$; now $r^*_{M \con X'}(B) = r_M(X') \le h$, so there exists $B' \subseteq B$ so that $|B'| \le h$ and every element of $B-B'$ is a coloop of $N = M \con X' \del B'$. Since $|B'|,|X'| \le h$ and every element of $X-X'$ is a loop of $N$, we have the required minor.
Complete Matroids
=================
Let $a \ge 2$ be an integer. We say a matroid $M$ is *$a$-complete* if $M$ has a basis $B$ such that, for every $I \subseteq B$ with $2 \le |I| \le a$, there is some $e \in E(M)$ for which $I \cup \{e\}$ is a circuit of $M$. We call such a $B$ a *joint-set* of $M$. For example, a $2$-complete matroid with joint-set $B$ is the same as a spanning $B$-clique restriction. We will freely use the easily proved fact that, if $M$ is an $a$-complete matroid with joint-set $B$, and $B' \subseteq B$ is a basis of a contraction-minor $M'$ of $M$, then $M'$ is $a$-complete with joint-set $B'$.
Huge $2$-complete matroids do not contain large $3$-complete minors. However, in Lemma \[upgradecomplete\] we prove that, for each integer $a>3$, a huge $3$-complete matroid does contain a large $a$-complete minors. Then, in Lemma \[3completewincor\], we prove that a huge $3$-complete matroid has either a large balanced uniform matroid or a large projective geometry as a minor.
\[buildcomplete\] Let $m > a \ge 2$ be integers and let $h = \binom{m}{a+1}$. If $M$ is an $a$-complete matroid with
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y Theorem \[mixhyp\] (since ${\mathbb{E}}{\mathbf{v}}_i \in \Lambda_+$ for all $i$ by convexity). In particular the polynomial ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$ is real–rooted.
The second assertion is an immediate consequence of the first combined with Lemma \[rk1le\].
Bounds on zeros of mixed characteristic polynomials
===================================================
To prove Theorem \[hypprob\], we want to bound the zeros of the *mixed characteristic polynomial* $$\label{mip}
t \mapsto h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{1}}),$$ where $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, ${\mathbf{1}}\in {\mathbb{R}}^m$ is the all ones vector, and ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+({\mathbf{e}})$ satisfy ${\mathbf{v}}_1+\cdots+{\mathbf{v}}_m ={\mathbf{e}}$ and $\tr({\mathbf{v}}_i) \leq \epsilon$ for all $1\leq i \leq m$.
\[hypid\] Note that a real number $\rho$ is larger than the maximum zero of if and only if $\rho {\mathbf{e}}+{\mathbf{1}}$ is in the hyperbolicity cone $\Gamma_{++}$ of $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$. Hence the maximal zero of is equal to $$\inf \{ \rho >0 : \rho {\mathbf{e}}+{\mathbf{1}}\in \Gamma_{++}\}.$$
For the remainder of this section, let $h \in {\mathbb{R}}[x_1,\ldots, x_n]$ be hyperbolic with respect to ${\mathbf{e}}$, and let ${\mathbf{v}}_1,\ldots, {\mathbf{v}}_m \in \Lambda_{++}$. To enhance readability in the computations to come, let $\partial_j := D_{{\mathbf{v}}_j}$ and $$\xi_j[g] := \frac {g}{\partial_j g}.$$
Note that a continuously differentiable concave function $f : (0,\infty) \to {\mathbb{R}}$ satisfies $$f(t+\delta) \geq f(t)+ \delta f'(t+\delta), \quad \mbox{ for all } \delta \geq 0.$$ Hence by Theorem \[direct\] $$\label{concon}
\xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j) \geq \xi_i[h]({\mathbf{x}}) + \delta \partial_j \xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j)$$ for all ${\mathbf{x}}\in \Lambda_{+}$ and $\delta \geq 0$. The following elementary identity is
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-1 \right) E_i \cdot E_j.$$ Following the Hirzebruch-Jung method, one obtains two exceptional divisors $E_1$ and $E_2$ and the following numerical data: $$A = \begin{pmatrix} -2 & 1 \\ 1 & -3 \end{pmatrix},
\quad \nu_1 = \dfrac{4}{5}, \quad \nu_2 = \dfrac{3}{5},
\quad \b_1 = -\dfrac{11}{5}, \quad \b_2 = -\dfrac{2}{5},$$ where $A$ is the intersection matrix of the resolution and $H=\operatorname{div}_X(x)-4\operatorname{div}_X(z)$ was chosen for the computation of $\b_1$ and $\b_2$. This completely describes the map $R_{X,P}$, since $H$ generates the group $\operatorname{Weil}(X,P)/\operatorname{Cart}(X,P)$ –for instance $R_{X,P}(-3H) = -1/5$, cf. [@Blache95 p. 312].
Smooth versus singular surfaces
-------------------------------
The next two examples illustrate the relevance of the codimension 2 singular locus of a surface when it comes to computing birational invariants of coverings.
\[ex:4A2\] Let us consider the curve $\cC\subset S=\PP^2_{(1,2,1)}$ with equation $$F(x,y,z):=-4 x^{3} z^{3} - 27 z^{6} - 18 x y z^{3} + x^{2} y^{2} + 4 y^{3}=0.$$ This curve of degree $6$ has 4 points of type $\mathbb{A}_2$ and does not pass through the singular point $P$ of the plane. The only possible $k$ for which $H^1(Y,\cO_Y(L^{(k)}))$ may not vanish is $k=5$ where the ideal is the maximal one. We have $$\pi^{(5)}: H^0(S,\mathcal{O}_{S}(5-4)) \longrightarrow\CC^4$$ and each point is in a different line passing through $P$. Hence $\ker\pi^{(5)}=0$ and $$\dim\operatorname{coker}\pi^{(5)}=\dim H^1(Y,\cO_Y(L^{(k)}))=2.$$ As a remark, note that the homology of the complement of the curve $\cC$ in $\PP^2_{(1,2,1)}$ is $\ZZ_3$. One might think that order 6 coverings ramified along $\cC$ might not exist, however they do since the homology of the complement of $\cC\cup \operatorname{Sing}(\PP^2_{(1,2,1)})$ is $\ZZ_6$.
\[ex:cusp23\] Let $\mathcal{C}$ be the curve of degree $d=6$ in $\PP^2_{w}$, $w=(3,2,1)$ given by $G = x^2 + y^3$. Consider the cyclic branched covering $\rho: \tilde{X} \to \PP^2_w$ ramifying on $\mathcal{C}$
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1}}{a_{t-1}}\left({\mathbf{x}}_{t}^\top C_{t-1}{\mathbf{x}}_{t}+{\mathbf{x}}_{t}^\top{\mathbf{1}}\right),
~2 a_{t-1}\right] \quad & \mathrm{otherwise}.\\
\end{array}
\right.\label{eq:ObsMarginal}\end{aligned}$$ Here, $\mathsf{t}(y; m, v, n)$ denotes the Student’s t-density function on $n$ degrees of freedom with centrality parameter $m$ and scaling factor $\sqrt{v}$. The statistics ${\bar{\mathcal{A}}}_{t-1}$ and ${\mathcal{A}}_{t-1}$ are deterministic functions of $y_{1:t-1}$ and ${\mathbf{x}}_{1:t-1}$, and are sufficient in that $f(y_t|{\mathbf{x}}_{1:t},y_{1:t-1}) \equiv f(y_t|{\mathbf{x}}_{t},{\bar{\mathcal{A}}}_{t-1},{\mathcal{A}}_{t-1})$.
The posterior-predictive mass function for the next excitation vector, $\mathbb{P}\left({\mathbf{x}}_t|{\mathbf{x}}_{1:t-1},y_{1:t-1},s_{1:t}\right)$\
$= \mathbb{P}\left({\mathbf{x}}_t|{\mathbf{x}}_{1:t-1},s_{1:t}\right)$, is given by the following intractable marginalisation: $$\begin{aligned}
\mathbb{P}\left({\mathbf{x}}_t|{\mathbf{x}}_{1:t-1},s_{1:t}\right) = \int \mathbb{P}\left({\mathbf{x}}_t|\eta_{1:u},\lambda_{1:u},s_t\right) \pi\left(\eta_{1:u},\lambda_{1:u}| {\mathbf{x}}_{1:t-1},s_{1:t-1}\right)~d\eta_{1:u}~d\lambda_{1:u}, \label{eq:ECmarg}\end{aligned}$$ where $\pi\left(\eta_{1:u},\lambda_{1:u}|
{\mathbf{x}}_{1:t-1},s_{1:t-1}\right)$ is the posterior for the excitability parameters given the firing vectors to time $t-1$. Section \[sec:DetailFireProc\] presents a fast numerical quadrature scheme for evaluating to any desired accuracy. The marginalisations over the parameters in and together provide the predictive: $$\begin{aligned}
f\left(y_t|~ {\mathbf{x}}_{1:t-1},~ y_{1:t-1},~ s_{1:t}\right) & = \sum_{{\mathbf{x}}_t\in\mathcal{X}_t} f\left(y_t|~ {\mathbf{x}}_{1:t},~ y_{1:t-1},~ s_{1:t}\right) \mathbb{P}\left({\mathbf{x}}_{t} |~ {\mathbf{x}}_{1:t-1},~ s_{1:t}\right),\label{eq:Ypred}\end{aligned}$$ Combination of with the historical firing event mass function $\mathbb{P}\left({\mathbf{x}}_{1:t}|y_{1:t},s_{1:t}\right)$ would provide the quantity $
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