Trees as Brelot spaces

A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many results on the potential theory of trees. By restricting the functions just to the vertices, we obtain several new results on the potential theory of trees considered in the usual sense. We study trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks. Besides the usual case of harmonic functions on trees (the kernel of the Laplace operator), we also consider as “harmonic” the eigenfunctions of the Laplacian relative to a positive eigenvalue showing that these also yield a Brelot structure and creating new classes of functions for the study of potential theory on trees.     

On the existence of the biharmonic Green kernels and the adjoint biharmonic functions

We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.     

调和函数的Lipschitz型空间和Landau-Bloch型定理

主要研究调和函数和Poisson方程的解的性质.讨论了调和函数的Lipschitz型空间,建立了调和函数的Schwarz-Pick型引理,并利用所得结果证明了与调和Hardy空间有关的一个Landau-Bloch型定理.最后,还利用正规族理论讨论了与Poisson方程的解有关的Landau-Bloch型定理的存在性.     

New exact solutions to the nonautonomous Liouville equation

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.     

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.     

紧致齐性空间上的调和分析（Ⅰ）：富里埃级数的Poisson求和

首先介绍了紧致齐性空间上调和分析的若干基础性结果,并给出这些结果的较简洁的证明。接着,我们定义了紧致齐性空间上函数的卷积(熟知n维球面是一个紧致齐性空间),这一定义看来对研究紧致齐性空间上的调和分析向题是相当有用的。最后,用定义的卷积,研究了紧致齐性空间上Fourier级数的Poisson求和。     

Théorie du potentiel sur les produits infinis et les groupes localement compacts

Let F. be a connected amd locally connected locally compact group having a countable basis for its topology. Does E admit a translation invariant Brelot harmonic sheaf? For which E does the elliptic Bauer theory coincides with the Brelot theory for all invariant harmonic elliptic sheaves? This note announces the following solutions: (a) Any E carries invariant Brelot harmonic sheaves; (b) Any invariant elliptic Bauer harmonic elliptic sheaf is a Brelot sheaf if and only if E is a finite dimensional Lie group. These results are obtained by studying product diffusions on infinite products of manifolds. e.g.. compacts Lie groups.     

广义拟容量的C-右连续性

本文探讨广义容量的性质,得到如下结果：（a）具有C-右连续,即关于紧集连续的凸拟容量是凸容量;（b）一大类广义容量具有C-右连续性;（c）凸拟容量的准上积分是凸Choquet容量．此外,给出具有C-右连续性的弱拟容量的可容性定理．本文还利用上述结果研究调和空间上正超调和函数的缩减函数和扫除函数．     

Distributional boundary values of harmonic and analytic functions

A theory for distributional boundary values of harmonic and analytic functions is presented. In this analysis there arise several indicators that measure the growth of these functions near the boundaries. An extension of the Phragmén-Lindelöf maximum principle is derived. Furthermore, the algebraic properties of the space of real periodic distributions are studied. By introducing a new product, the harmonic product, the boundary conditions involving harmonic functions are transformed into ordinary differential equations.     

Symmetric Harmonic Sheaves Possessing Bipotentials

Let be the family of sheaves H of continuous functions on a Brelot harmonic space with a countable base such that locally the Dirichlet problem with respect to H is solvable, H satisfies Harnack inequalities and also H has a symmetry property. Defining the notions of H-biharmonic functions, H-biharmonic Green functions, H-bipotentials, H-biharmonic extensions, etc. we study the interrelation between them and exhibit various classifications of the family of sheaves.     

Continuous linear functionals on certain function spaces satisfying a system of two singular second order differential equations

In this paper, we study a system of two singular second order differential equations, which arises from the theory of harmonic analysis on complex symmetric spaces. First of all, the distributional solutions on an neighborhood of zero in R² are determined. Next, some new function spaces are introduced and the system is solved in the duals of these new spaces.     

On a large class of symmetric systems of linear PDEs,for tensor functions,useful in mathematical physics

We study a class of symmetric systems of linear partial differential equations which involve tensor functions relating tensor spaces on a three dimensional vector space, on the real field, equipped with an inner product. These systems arise by coupling certain simpler symmetric systems studied in a previous paper. In order to investigate some questions, related to constitutive equations for bodies of the differential type, certain classes of physically privileged solutions are characterized for some of the aforementioned systems.This work has been performed within the activity of the Consiglio Nazionale delle Ricerche, Group n. 3, in the academic years 1988/89 and 1989/90.     

On the heritability of the property <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">π</Emphasis></Subscript> by subgroups

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.     

On Positivity Conditions for the Cauchy Function of Functional-Differential Equations

We study how the statements on estimates of solutions to linear functional-differential equations, analogous to the Chaplygin differential inequality theorem, are connected with the positivity of the Cauchy function and the fundamental solution. We prove a comparison theorem for the Cauchy functions and the fundamental solutions to two functional-differential equations. In the theorem, it is assumed that the difference of the operators corresponding to the equations (and acting from the space of absolutely continuous functions to the space of summable ones) is a monotone totally continuous Volterra operator. We also obtain the positivity conditions for the Cauchy function and the fundamental solution to some equations with delay as long as those of neutral type.     

超空时调和函数与一类非线性抛物方程

本文对超扩散过程定义了超空时调和函数并讨论了它们的某些性质，在此基础上建立了一类非线性抛物方程的正解与超空时调和函数之间的对应关系．     

Evolution equations for abstract differential operators

We study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces.     

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

Functions of shift operator and their applications to difference equations

We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.     

An operational calculus for the euclidean motion group with applications in robotics and polymer science

In this article we develop analytical and computational tools arising from harmonic analysis on the motion group of three-dimensional Euclidean space. We demonstrate these tools in the context of applications in robotics and polymer science. To this end, we review the theory of unitary representations of the motion group of three dimensional Euclidean space. The matrix elements of the irreducible unitary representations are calculated and the Fourier transform of functions on the motion group is defined. New symmetry and operational properties of the Fourier transform are derived. A technique for the solution of convolution equations arising in robotics is presented and the corresponding regularized problem is solved explicity for particular functions. A partial differential equation from polymer science is shown to be solvable using the operational properties of the Euclidean-group Fourier transform.