Hausdorff moment problem and fractional moments: A simplified procedure

The purpose of this paper is the recovering of a probability density function with support [0, 1] from the knowledge of its sequence of moments, i.e. the classical Hausdorff moment problem. To avoid the well-known ill-conditioning, firstly the moment curve is calculated from the assigned sequence of moments; next the unknown density is approximated by Maximum Entropy (MaxEnt) technique selecting some proper points on the moment curve. Exploiting convergence in entropy, a simplified quick procedure is suggested to recover the approximate density. An application to Laplace Transform inversion is illustrated.     

A sharp uniqueness result for a class of variational problems solved by a distance function

We consider the minimization problem for an integral functional J, possibly nonconvex and noncoercive in , where is a bounded smooth set. We prove sufficient conditions in order to guarantee that a suitable Minkowski distance is a minimizer of J. The main result is a necessary and sufficient condition in order to have the uniqueness of the minimizer. We show some application to the uniqueness of the solution of a system of PDEs of Monge–Kantorovich type arising in problems of mass transfer theory.     

Maximum principle for functional equations in the space of discontinuous functions of three variables

The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed.     

Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term

We are concerned with singular elliptic equations of the form −Δu=p(x)(g(u)+f(u)+a|∇u|) in RN (N?3), where p is a positive weight and 0<a<1. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

Fuzzy数理论的几个重要定理

给出Fuzzy 数集上(下)确界的新定义,证明了确界定理、单调收敛定理、闭区间套定理及柯西收敛原理,从而得出了任何有界闭区间[a,b]都是完备的结果     

The median procedure in cluster analysis and social choice theory

Classical approachs for fitting and aggregation problems, specially in cluster analysis, social choice theory and paired comparisons methods, consist in the minimization of a remoteness function between relational data and a relational model. The notion of median, with its algebraic, metric, geometrical and statistical aspects, allow a unified treatment of many of base problems. Properties of median procedures are organized according to four directions: stabilities and axiomatic characterizations; Arrow-like properties; combinatorial properties; effective computational possibilities. Finally, interesting mathematical problems, related to the notion of median are surveyed.     

Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff DDE solver of order 4 to 14

This article presents a solver for delay differential equations (DDEs) called HBO414DDE based on a hybrid variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff ODE solver of order 4 to 14. The current version of our method solves DDEs with state dependent, non-vanishing, small, vanishing and asymptotically vanishing delays, except neutral type and initial value DDEs. Delayed values are computed using Hermite interpolation, small delays are dealt with by extrapolation, and discontinuities are located by a bisection method. HBO414DDE was tested on several problems and results were compared with those of known solvers like SYSDEL and the recent Matlab DDE solver ddesd and statistics show that it gives, most of the time, a smaller relative error than the other solvers for the same number of function evaluations.