On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ${\tilde{\xi }}_1,{\tilde{\xi }}_2,\dots$, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution ${\tilde{\mathfrak{e}}}_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{{\tilde{\xi }}_i}$ of the n past observations, while the latter is compared to the m-fold product ${\tilde{\mathfrak{e}}}_n^m$. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts ${\delta }_{{\tilde{\mathfrak{e}}}_n}$ and ${\tilde{\mathfrak{e}}}_n^m$, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ${\tilde{\xi }}_n$'s, and not under hypothetical product laws ${\mathfrak{p}}_0^{\infty }$, as ${\mathfrak{p}}_0$ varies among the admissible determinations of a random probability measure.     

Regularity in Lp Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:

(*)$Pu+{?}_{t}u=f(x,t),\text{ for }x\in \mathrm{\Omega }?{\mathbb{R}}^n,t\in I?\mathbb{R}.$

1) For strongly elliptic pseudodifferential operators (ψdo's) P on ${\mathbb{R}}^n$ of order $d\in {\mathbb{R}}_{+}$, a symbol calculus on ${\mathbb{R}}^{n+1}$ is introduced that allows showing optimal regularity results, globally over ${\mathbb{R}}^{n+1}$ and locally over $\mathrm{\Omega }\times I$:

$f\in H_{p,\mathrm{loc}}^{(s,s/d)}(\mathrm{\Omega }\times I)?u\in H_{p,\mathrm{loc}}^{(s+d,s/d+1)}(\mathrm{\Omega }\times I),$

for $s\in \mathbb{R}$, $1<p<\infty$. The $H_p^{(s,s/d)}$ are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal ${(?\mathrm{\Delta })}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition $\mathrm{supp}u?\stackrel{￣}{\mathrm{\Omega }}$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I;H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }}))$ with ${?}_{t}u\in L_p(\mathrm{\Omega }\times I)$. Here $H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }})={\dot{H}}_p^{2a}(\stackrel{￣}{\mathrm{\Omega }})$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a?\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ if $a=1/p$, but contains nontrivial elements from $d^a{\stackrel{￣}{H}}_p^a(\mathrm{\Omega })$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,?\mathrm{\Omega })$). The interior regularity of u is lifted when f is more smooth.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field

In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in $H^2({\mathbb{R}}^3)$ are obtained by introducing a new linearized system with respect to $(n^{\gamma }?{\stackrel{?}{n}}^{\gamma },n?\stackrel{?}{n},P?\stackrel{?}{P},u,H)$ for constants $\stackrel{?}{n}\ge 0$ and $\stackrel{?}{P}>0$, and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of $(n?\stackrel{?}{n},n^{\gamma }?{\stackrel{?}{n}}^{\gamma })$ in $H^2({\mathbb{R}}^3)$ norm.     

Formality theorem for g-manifolds

With any $\mathfrak{g}$-manifold M are associated two dglas $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))$ and $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$, whose cohomologies $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ and $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\mathrm{\Phi }:\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))\to \mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$ whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold M is an isomorphism of Gerstenhaber algebras from $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ to $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$.