Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

A Local Property of Hausdorff Centered Measure of Self-Similar Sets

We analyze the local behavior of the Hausdorff centered measure for selfsimilar sets. If E is a self-similar set satisfying the open set condition, then Cs(E∩B(x,r)) ≤(2r)s for all x ∈ E and r 0, where Csdenotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.     

广义n参数Wiener过程和广义OUPn的导出过程的注记

记为非负单增函数且一阶偏导数均存．本文证明：（ｉ）广义ｎ参数Ｗｉｅｎｅｒ过程（ＧＢｎ）和广义ｎ参数Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ过程（广义ＯＵＰｎ）在Ｄ中导出的过程是马氏过程；（ｉｉ）如果ＧＢｎ（或广义ＯＵＰｎ）在Ｄ中导出的过程成为某个ＧＢｋ（或某个广义ＯＵＰｋ），则ｆｊ（ｔ１…，ｔｋ）未必恒为正常数．（ｊ＝ｋ＋１，…，ｎ）．     

On point energies, separation radius and mesh norm for -extremal configurations on compact sets in

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class of d-dimensional compact sets embedded in R^d′, 1dd^′. We assume that these points interact through a Riesz potential V=|·|^-s, where s>0 and |·| is the Euclidean distance in . With and denoting, respectively, the separation radius and mesh norm of s-extremal configurations, which are defined to yield minimal discrete Riesz s-energy, we show, in particular, the following.(A) For the d-dimensional unit sphere and s<d-1, and, moreover, if sd-2. The latter result is sharp in the case s=d-2. In addition, point energies for s-extremal configurations are asymptotically equal. This observation relates to numerical experiments on observed scar defects in certain biological systems.(B) For and s>d, and the mesh ratio is uniformly bounded for a wide subclass of . We also conclude that point energies for s-extremal configurations have the same order, as N→∞.     

Density Modulo 1 of Sublacunary Sequences

We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence s _n , n = 1, 2, 3, ..., generated by the ordered numbers of the form 2ⁱ3^j, i, j = 1, 2, 3, ..., we prove that the set of real numbers α such that inf _n∈ℕ n‖s _n α‖ > 0 is a set of Hausdorff dimension 1. The divergence of the series implies that the Lebesgue measure of those numbers is zero.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin.     

The volume measure for flat connections as limit of the Yang–Mills measure

We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.     

有缺失数据的条件独立正态母体中参数的最优同变估计

在缺失数据机制是可忽略的假设下,导出了有单调缺失数据的条件独立正态模型中协方差阵和精度阵的Cholesky分解的最大似然估计和无偏估计.通过引入一类特殊的变换群并在更广义的损失下,获得了其最优同变估计.这表明最大似然估计和无偏估计是非容许的.最后,通过数值模拟验证了相关结果的有效性.     

Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains

Given a bounded strongly pseudoconvex domain D in with smooth boundary, we characterize ‐Bergman Carleson measures for , , and . As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric.     

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.