A note about minimum relative entropy solutions of finite moment problems

The finite Hausdorff moment problem admits minimum relative entropy solutions if and only if the data satisfy some simple geometrical conditions. A procedure to compute such solutions is rigorously analyzed here.     

Hausdorff moment problem and fractional moments

Hausdorff moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, in this work a stable algorithm to obtain centered moments from integer moments is found. The algorithm transforms a direct method into an iterative Jacobi method which converges in a finite number of steps, as the iteration Jacobi matrix has null spectral radius. The centered moments are needed to calculate fractional moments from integer moments. As an application few fractional moments are used to solve finite Hausdorff moment problem via maximum entropy technique. Fractional moments represent a remedy to ill-conditioning coming from an high number of integer moments involved in recovering procedure.     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.     

Local moment problem

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

Existence and stability of a discrete probability distribution by maximum entropy approach

The discrete maximum entropy (ME) probability distribution which can take on a finite number of values and whose first moments are assigned, is considered. The necessary and sufficient conditions for the existence of a maximum entropy solution are identical to the general ones for the finite moment problem. The entropy decreasing by adding one more moment is studied. Unstability of the distribution recovering is proved when an increasing number of moments is used.     

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.     

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.     

EXISTENCE AND UNIQUENESS OF ENTROPY SOLUTION TO PRESSURELESS EULER SYSTEM WITH A FLOCKING DISSIPATION

We study the existence and uniqueness problem for the nonhomogeneous pressureless Euler system with the initial density being a Radon measure. Our uniqueness result is obtained in the same space as the existence theorem. Besides, by counterexample we prove that Huang-Wang's energy condition is also necessary for our nonhomogeneous system.     

Long-Time Dynamics for a Nonlinear Viscoelastic Kirchhoff Plate Equation

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.     

On Generalized Vector Equilibrium Problems

In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.     

Well-posedness and Global Attractors for the Burgers-Ginzburg-Landau Equations

In this paper we consider the Burger-Ginzburg-Landau equations, and prove the existence of the global attractor in with finite Hausdorff and fractal dimensions.     

有界多连通区域具线性阻尼Navier-Stokes方程的有限维行为

考虑二维有界多连通区域上具线性阻尼Navier-Stokes方程,在适当的边界条件下证明了解的存在唯一性及整体吸引子A的存在性,并给出了A的Hausdorff维数与Fractal维数.     

Fractal Measures for Parabolic IFS

Let h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d?2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3.     

Power moments of negative order for the principal spectral function of a string

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem. Odessa Academy of Food Technologies, Odessa. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1209–1222, September, 1996.     

A primal-dual algorithm for the fermat-weber problem involving mixed gauges

We give a new algorithm for solving the Fermat-Weber location problem involving mixed gauges. This algorithm, which is derived from the partial inverse method developed by J.E. Spingarn, simultaneously generates two sequences globally converging to a primal and a dual solution respectively. In addition, the updating formulae are very simple; a stopping rule can be defined though the method is not dual feasible and the entire set of optimal locations can be obtained from the dual solution by making use of optimality conditions. When polyhedral gauges are used, we show that the algorithm terminates in a finite number of steps, provided that the set of optimal locations has nonepty interior and a counterexample to finite termination is given in a case where this property is violated. Finally, numerical results are reported and we discuss possible extensions of these results.     

Existence of solutions of Two-Phase free boundary problems for fully nonlinear elliptic equations of second order

In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hⁿ⁻¹ Hausdorff measure and a normal in a measuretheoretic sense Hⁿ⁻¹ almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C^1,α surface near Hⁿ⁻¹-a.e. point.     

THE FINITE DIMENSION BEHAVIOR OF THE SCHRDINGER EQUATION WITH NONLOCAL INTEGRAL NONLINEARITY

1.IntroductionInthispaperweconsiderthefollowinginitialproblem,lUt box OIUlpU 7U["III'dX IkU~gi(1.l)j--colul'dx iku~g,(1.l)nit=0~u',(1.2)whereP,7andkarerealnumber,k>0,theexponentpsatisfiesPZ0,ifP50,05P<4,ifP>0.(l.3)ForthenonlinearSchrodingerequation.lat aam P(lul')u ikm=g,(1.4)itslongtimebehaviorhasbeenobservedbynumericalmethodandphysicalargUmefits,(1)chaoticattractorexists;(2)afinitedimensional"space"confinestheattractors(see[l,2]),ajndithasalsobeenprovedbymathematicalmethodthatthereex…     

离散FitzHugh-Nagumo方程的整体吸引子和维数

该文对ＦｉｔｚＨｕｇｈ Ｎａｇｕｍｏ方程初边值问题用有限差分格式离散空间变量,证明了离散模型整体吸引子的存在性,并给出了与犿无关的Ｈａｕｓｄｏｒｆｆ维数和Ｆｒａｃｔａｌ维数上界估计。