Description of periodic p-harmonic functions on Cayley tree

We show that any periodic (with respect to normal subgroups of finite index of the group representation of the Cayley tree) p-harmonic function on a Cayley tree is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic p-harmonic functions. We also prove that linear combinations of the p-harmonic functions described for normal subgroups of infinite index are also p-harmonic.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

Binary search tree recursions with harmonic toll functions

We study recursions that are traditional in the context of binary search trees and quicksort with the (non-standard) toll function H_n.     

A representation theorem of harmonic functions and its application to BMO on manifolds

Abstract. In this paper a representation theorem of harmonic functions on manifolds is set up.As an application, a characterization of BMO in terms of Carleson measures is obtained.     

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.     

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

We systematically expound the infodynamical method for analyzing lattice and grid systems. We establish the logic and algorithm for mapping given objects to coordination Cayley tree graphs and present their main properties. Tree graphs of grid systems are complicated objects, and the principle of cluster-type simplicial decomposition can be used to study them. Based on a simplicial decomposition, we construct the enumerating structures, from which we construct entropy-type functionals. We pose the percolation problem on Cayley tree graphs in a nonconventional sense, which may be considered for both enumerating structures and their entropies. The corresponding entropy percolational dependences and their critical indices can be considered sufficiently universal measures of order in lattice systems. The simpliciality also implies an analogy with the fractality principle. We introduce three types of fractal characteristics and give analytic expressions for fractal dimensions for the tangential and streamer representations and for the Mandelbrot shell.     

Integral representation and estimation of harmonic functions in the quaternionic half space

In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.     

Integral representation and asymptotic behavior of harmonic functions in half space

Using Carleman's formula of a harmonic function in the half space and Nevanlinna's representation of a harmonic function in the half sphere, we prove that a harmonic function, whose positive part satisfies a slowly growing condition, can be represented by a certain integral. This improves some classical Poisson integrals for harmonic functions.     

Moment vanishing properties of harmonic Bergman functions

We show that Poisson integrals belonging to certain weighted harmonic Bergman spaces b_δ^p on the upper half-space must have the moment vanishing properties. As an application, we show that b₀^p, p?1, contains a dense subspace whose members have the horizontal moment vanishing properties. Also, we derive related weighted norm inequalities for Poisson integrals. As a consequence, we obtain a characterization for Poisson integrals of continuous functions with compact support in order to belong to b_δ^p.     

A characterization of harmonic functions

Gauss' mean value characterization of harmonic functions involves circles (or spheres) centered at every point of the domain of the function. The present paper gives a criterion of this type which involves only one point of the domain; to make up for this, circles or spheres are replaced by a larger family of convex curves or surfaces which surround this point.     

Boundary properties of functions harmonic in a strip

We study in the L_p-norm, 1≤p≤∞, the boundary properties of the solution to the Dirichlet problem for the stripA ={(x, y):?∞0} and its dependence on the structural properties of the given boundary values (symmetric, antisymmetric). In particular, for the case of symmetric boundary values we obtain direct and inverse theorems on approximation in terms of the general modulus of continuity of second order.     

Some properties of harmonic functions generated by the bergman-whittaker operator

Summary We deduce properties concerning singular curves of harmonic functions generated by the Bergman-Whittaker operator and obtain an analogue of the Eisenstein theorem for a certain class of such harmonic functions. This research was supported in part by the Air Force Office of Scientific Research.     

On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n, let R₁(A,n), R₂(A,n) and R₃(A,n) denote the number of solutions of a+a^′=n with the additional restriction a,a^′∈A; a,a^′∈A,a<a^′ and a,a^′∈A,a≤a^′ respectively. In this paper, we specially focus on the monotonicity of R₃(A,n). Moreover, we show that there does not exist any set A⊂N such that R₂(A,n) or R₃(A,n) is eventually strictly increasing.