THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3) Let B be a Banach space, B* be its dual, and denote the weak * topology of B* by σ(B*,B). Then the functional dimension of (B*,σ(B*,B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.     

The Generalized Moment Problem Associated with Correlation Measures

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case in which the algebra is a special commutative algebra of functions on the space of multiple finite configurations. If the above-mentioned functional is generated by a measure on the space of finite ordinary configurations, then this measure is the correlation measure for a measure on the space of infinite configurations. The positiveness of the functional gives conditions for the measure to be a correlation measure.     

逆序(L)集合范畴的格值函数空间及其性质

引入逆序(L)集合范畴概念,并研究该范畴中两种函数空间结构表示,即格值函数空间与伪格值函数空间,进一步指出在逆序(L)集合范畴中格值函数空间函子与格值积函子互为伴随及伪格值函数空间函子与格值交函子也互为伴随,从而逆序(L)集合范畴为Cartesian闭范畴.     

On the Curvature of the Indicatrix Surface in Three-Dimensional Minkowski Spaces

Periodica Mathematica Hungarica - As it is well-known a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. If the square of the Minkowski functional...     

广义函数空间上Drygas型函数方程的稳定性

在广义函数空间上重新定义了Drygas型函数方程的稳定性,然后利用高斯变换将广义函数空间上的函数方程稳定性转换为R~(n+1)空间上的光滑函数方程的稳定性.在求得正则化的函数方程稳定性后,利用广义函数与正则化函数间的关系给出在广义函数空间上的Drygas型函数方程的Hyers-Ulam-Rassias型稳定性.     

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.     

Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.     

On the stability of a generalized additive functional equation

We consider the Hyers–Ulam stability of a functional equation for continuous functions on a space on which a topological group acts, analogously to the additive functional equation on a group. We show, among other things, that our generalized additive equation, for continuous functions on a homogeneous space of a strongly amenable topological group, is stable provided that the canonical projection from that group to its homogeneous space is a fiber bundle.     

A Fixed Point Approach to the Fuzzy Stability of a Mixed Typ e Functional Equation

Through the paper, a general solution of a mixed type functional equation in fuzzy Banach space is obtained and by using the fixed point method a generalized Hyers-Ulam-Rassias stability of the mixed type functional equation in fuzzy Banach space is proved.     

两点齐性的Finsler流形

该文证明任何一个两点齐性的Finsler流形一定是黎曼流形．证明过程中作者将泛函分析中经典的Mazur定理推广到不一定是绝对齐次的Minkowski空间上．     

两点齐性的 Finsler 流形

该文证明任何一个两点齐性的 Finsler 流形一定是黎曼流形. 证明过程中作者将泛函分析中经典的Mazur 定理推广到不一定是绝对齐次的 Minkowski 空间上.     

A recursive algorithm by the moments method to evaluate a class of numerical integrals over an infinite interval

A special recursive algorithm is built by a three-term recursive formula with coefficients evaluated by the moments method.A new functionalc(·) is studied over any function space that contains the polynomial space and it is shown that such a functional is positive definite, enabling us to use the advantages of such a property on the zeros of orthogonal polynomials for such a functional. A comparison is presented of the numerical advantages of such a method with respect to the Laguerre polynomials.     

马尔可夫过程H-值可加泛函的向前向后鞅分解

本文研究了马尔可夫H-值可加泛函的向前向后鞅分解.利用Lyons-Meyer-Zheng鞅分解得到了泛函数极限定理所必需的极大不等式和紧性结果,在最小条件限度内得到了马尔可夫过程经验测度的泛函中心极限定理,将该定理从实值情形推广到了希尔伯特值情形.     

Stabilization and Practical Asymptotic Stability of Abstract Differential Equations

This article studies the problem of stabilization of the infinite-dimension time-varying control systems in Hilbert spaces. We consider the problem of practical asymptotic stability with respect to a continuous functional for a class of abstract nonlinear infinite-dimensional processes with multivalued solutions on a metric space when the origin is not an equilibrium point. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the practical stability of continuous semigroups in a Banach space.     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.     

Spatial functional normal mixed effect approach for curve classification

This paper proposes a spatial functional formulation of the normal mixed effect model for the statistical classification of spatially dependent Gaussian curves, in both parametric and state space model frameworks. Fixed effect parameters are represented in terms of a functional multiple regression model whose regression operators can change in space. Local spatial homogeneity of these operators is measured in terms of their Hilbert–Schmidt distances, leading to the classification of fixed effect curves in different groups. Assuming that the Gaussian random effect curves obey a spatial autoregressive dynamics of order one [SARH(1) dynamics], a second functional classification criterion is proposed in order to detect local spatially homogeneous patterns in the mean quadratic functional variation of Gaussian random effect curve increments. Finally, the two criteria are combined to detect local spatially homogeneous patterns in the regression operators and in the functional mean quadratic variation, under a state space approach. A real data example in the financial context is analyzed as an illustration.     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.     

Periodic solutions of neutral fractional differential equations

We characterize the existence of periodic solutions for some abstract neutral functional fractional differential equations with finite delay when the underlying space is a UMD space.     

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.     

A New Vector-Valued Padé-Type Approximation in the Inner Space

A new vector-valued Padé-type approximation is defined by introducing a generalized vector-valued linear functional from a scalar polynomial space to a vector space. Some proximants are provided with the generating function form and the determinant form.