Multivariate inequalities based on Sobolev representations

Here we derive very general multivariate tight integral inequalities of Chebyshev–Grüss, Ostrowski types and of comparison of integral means. These are based on well-known Sobolev integral representation of a function. Our inequalities engage ordinary and weak partial derivatives of the involved functions. We also give their applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions.     

Homogenization of a conductive suspension in a Stokes–Boussinesq flow

Radiant spherical suspensions have an ε-periodic distribution in a tridimensional incompressible viscous fluid governed by the Stokes–Boussinesq system. We perform the homogenization procedure when the radius of the solid spheres is of order ε³ (the critical size of perforations for the Navier-Stokes system) and when the ratio of the fluid/solid conductivities is of order ε⁶, the order of the total volume of suspensions. Adapting the methods used in the study of small inclusions, we prove that the macroscopic behavior is described by a Brinkman–Boussinesq type law and two coupled heat equations, where certain capacities of the suspensions and of the radiant sources appear.     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

A Markov chain Monte Carlo Algorithm for Approximating Exact Conditional Probabilities

Conditional inference eliminates nuisance parameters by conditioning on their sufficient statistics. For contingency tables conditional inference entails enumerating all tables with the same sufficient statistics as the observed data. For moderately sized tables and/or complex models, the computing time to enumerate these tables is often prohibitive. Monte Carlo approximations offer a viable alternative provided it is possible to obtain samples from the correct conditional distribution. This article presents an MCMC extension of the importance sampling algorithm, using a rounded normal candidate to update randomly chosen cells while leaving the remainder of the table fixed. This local approximation can greatly increase the efficiency of the rounded normal candidate. By choosing the number of cells to be updated at random, a balance is struck between dependency in the Markov chain and accuracy of the candidate.     

Characterization of mean transformations

The mean transformations M(A,?B) are linear mappings and they are analogues of the matrix means of A,?B?≥?0. They are defined by operator monotone functions. In this article several properties are described and a part of them characterize the concept.     

Parabolic Ito Equations with Mixed in Time Conditions

Abstract

We study linear stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the initial time is replaced by a condition that mixes the values of the solution at different times, including the terminal time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained.     

A Volume Inequality for Quantum Fisher Information and the Uncertainty Principle

Let A ₁,…,A _N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

Asymptotic results for weighted means of linear combinations of independent Poisson random variables

ABSTRACT

In this paper we prove the large deviation principle for a class of weighted means of linear combinations of independent Poisson distributed random variables, which converge weakly to a normal distribution. The interest in these linear combinations is motivated by the diffusion approximation in Lansky [On approximations of Stein's neuronal model, J. Theoret. Biol. 107 (1984), pp. 631–647] of the Stein's neuronal model (see Stein [A theoretical analysis of neuronal variability, Biophys. J. 5 (1965), pp. 173–194]). We also prove an analogue result for sequences of multivariate random variables based on the diffusion approximation in Tamborrino, Sacerdote, and Jacobsen [Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling, Phys. D 288 (2014), pp. 45–52]. The weighted means studied in this paper generalize the logarithmic means. We also investigate moderate deviations.