Malliavin and Dirichlet structures for independent random variables

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of $U$-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.     

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ?Σ. Explicitly, let λ₁(?Σ) = inf u∈W^1,2 (Σ,g),R ?Σ udsg=0,u6≡0 R Σ(|?gu|2 + u2 )dvg R ?Σ u2 dsg and H = n u ∈ W^1,2 (Σ, g) : Z Σ(|?gu|2 + u2 )dvg ? α Z ?Σ u2dsg ≤ 1 and Z ?Σ u dsg = 0o ,where W^1,2 (Σ, g) denotes the usual Sobolev space and ?g stands for the gradient operator.By the method of blow-up analysis, we obtain sup u∈H Z ?Σ e πu2 dsg ( < +∞, 0 ≤ α < λ1(?Σ),= +∞, α ≥ λ1(?Σ).Moreover, the author proves the above supremum is attained by a function uα ∈ H∩C∞(Σ)for any 0 ≤ α < λ1(?Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang,Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for TrudingerMoser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015,3161–3193]     

A fast parallel clustering algorithm for molecular simulation trajectories

We implemented a GPU‐powered parallel k‐centers algorithm to perform clustering on the conformations of molecular dynamics (MD) simulations. The algorithm is up to two orders of magnitude faster than the CPU implementation. We tested our algorithm on four protein MD simulation datasets ranging from the small Alanine Dipeptide to a 370‐residue Maltose Binding Protein (MBP). It is capable of grouping 250,000 conformations of the MBP into 4000 clusters within 40 seconds. To achieve this, we effectively parallelized the code on the GPU and utilize the triangle inequality of metric spaces. Furthermore, the algorithm's running time is linear with respect to the number of cluster centers. In addition, we found the triangle inequality to be less effective in higher dimensions and provide a mathematical rationale. Finally, using Alanine Dipeptide as an example, we show a strong correlation between cluster populations resulting from the k‐centers algorithm and the underlying density. © 2012 Wiley Periodicals, Inc.     

Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

Ground state solution for differential equations with left and right fractional derivatives

In this work, we study the existence of positive solutions for a class of fractional differential equation given by (1) where . Using the mountain pass theorem and comparison argument, we prove that (1) at least has one nontrivial solution. Copyright © 2015 John Wiley & Sons, Ltd.     

The contribution of K.-H. Elster to generalized conjugation theory and nonconvex duality

This article surveys the main contributions of K.-H. Elster to the theory of generalized conjugate functions and its applications to duality in nonconvex optimization.