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Register a new userAbout the nuclearity of ${mathcal S}_{(M_{p})}$ and ${mathcal S}_{omega}$
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We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain that for a weight function $omega$ satisfying the mild condition: $2omega(t)leq omega(Ht)+H$ for some $H>1$ and for all $tgeq0$, the space ${mathcal S}_omega$ in the sense of Bjorck is also nuclear.
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We reemphasize that the ratio $R_{smu} equiv overline{mathcal{B}}(B_stomubarmu)/Delta M_s$ is a measure of the tension of the Standard Model (SM) with latest measurements of $overline{mathcal{B}}(B_stomubarmu)$ that does not suffer from the persistent puzzle on the $|V_{cb}|$ determinations from inclusive versus exclusive $bto cellbar u$ decays and which affects the value of the CKM element $|V_{ts}|$ that is crucial for the SM predictions of both $overline{mathcal{B}}(B_stomubarmu)$ and $Delta M_s$, but cancels out in the ratio $R_{smu}$. In our analysis we include higher order electroweak and QED corrections und adapt the latest hadronic input to find a tension of about $2sigma$ for $R_{smu}$ measurements with the SM independently of $|V_{ts}|$. We also discuss the ratio $R_{dmu}$ which could turn out, in particular in correlation with $R_{smu}$, to be useful for the search for New Physics, when the data on both ratios improves. Also $R_{dmu}$ is independent of $|V_{cb}|$ or more precisely $|V_{td}|$.
The popular $mathcal{AB}$/push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of $mathcal{AB}$/Push-Pull method ($mathcal{S}$-$mathcal{AB}$) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This paper is devoted to the asymptotic properties of $mathcal{S}$-$mathcal{AB}$ with diminishing stepsize. Specifically, under the condition that each local objective is smooth and the global objective is strongly-convex, we first present the boundedness of the iterates of $mathcal{S}$-$mathcal{AB}$ and then show that the iterates converge to the global minimizer with the rate $mathcal{O}left(1/sqrt{k}right)$. Furthermore, the asymptotic normality of Polyak-Ruppert averaged $mathcal{S}$-$mathcal{AB}$ is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.
The gravitational $mathcal{S}$-matrix defined with an infrared (IR) cutoff factorizes into hard and soft factors. The soft factor is universal and contains all the IR and collinear divergences. Here we show, in a momentum space basis, that the intricate expression for the soft factor is fully reproduced by two boundary currents, which live on the celestial sphere. The first of these is the supertranslation current, which generates spacetime supertranslations. The second is its symplectic partner, the Goldstone current for spontaneously broken supertranslations. The current algebra has an off-diagonal level structure involving the gravitational cusp anomalous dimension and the logarithm of the IR cutoff. It is further shown that the gravitational memory effect is contained as an IR safe observable within the soft $mathcal{S}$-matrix.
We propose a generalization of S-folds to 4d $mathcal{N}=2$ theories. This construction is motivated by the classification of rank one 4d $mathcal{N}=2$ super-conformal field theories (SCFTs), which we reproduce from D3-branes probing a configuration of $mathcal{N}=2$ S-folds combined with 7-branes. The main advantage of this point of view is that realizes both Coulomb and Higgs branch flows and allows for a straight forward generalization to higher rank theories.
We study the Coulomb branch of class $mathcal{S}_k$ $mathcal{N} = 1$ SCFTs by constructing and analyzing their spectral curves.
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