Biharmonic extensions on trees without positive potentials

Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function, of the form f=βK+B+L, where β a constant, B is a biharmonic function on T, and L is a function, subject to certain normalization conditions, whose Laplacian is constant on all sectors sufficiently far from the root. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in Rn for n=2,3, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions biharmonic outside a finite set that have finite flux in this extended sense.     

A decomposition theorem on Euclidean Steiner minimal trees

The Euclidean Steiner minimal tree problem is known to be an NP-complete problem and current alogorithms cannot solve problems with more than 30 points. Thus decomposition theorems can be very helpful in extending the boundary of workable problems. There have been only two known decomposition theorems in the literature. This paper provides a 50% increase in the reservoir of decomposition theorems.     

锥中上调和函数的Riesz 分解定理及其应用

本文给出了锥中上调和函数的Riesz 分解定理. 同时, 得到了它在锥中无穷远点处的增长性质, 并且刻画了其例外集的几何性质. 作为应用, 我们证明了锥内次调和函数的Phragmén-Lindelöf 型定理.     

A Ramsey theorem for trees

We prove a Ramsey theorem for trees. The infinite version of this theorem can be stated: if T is a rooted tree of infinite height with each node of T having at least one but finitely many immediate successors, if n is a positive integer, and if the collection of all strongly embedded, height-n subtrees of T is partitioned into finitely many classes, then there must exist a strongly embedded subtree S of T with S having infinite height and with all the strongly embedded, height-n subtrees of S in the same class.     

Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy

The paper consists of two parts. In the first part, we prove a noncommutative analog of the Riesz(— Markov—Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the underlying space. In the second part, using this result, we prove a weak version of a Burnside type theorem on twisted conjugacy for arbitrary discrete groups.     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

A generalization of a theorem by F. Riesz

В работе получено нео бходимое и достаточн ое условие представимости функ ции в виде неопределенного инт еграла от функции огр аниченнойp-вариации в смысле Ви нера. Этот результат является о бобщением одной теор емы Ф. Рисса.     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

A decomposition theorem for neutrices

Neutrices are convex additive subgroups of the nonstandard space R^k, most of them are external sets. Because of the convexity and the invariance under some translations and multiplications, external neutrices are models for orders of magnitude. One dimensional neutrices have been applied to asymptotics, singular perturbations, and statistics. This paper shows that in R^k, with standard k, every neutrix is the direct sum of k neutrices of R. These components may be chosen to be orthogonal.     

A decomposition theorem form-convex sets

LetS be a closedm-convex subset of the plane,m≧2,Q the set of points of local nonconvexity ofS, with convQ ⊆S. If there is some pointp in [(bdryS) ∩ (kerS)] ∼Q, thenS is a union ofm−1 closed convex sets. The result is best possible for everym.     

A decomposition theorem for polymeasures

We prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a “discrete” polymeasure plus a “continuous” polymeasure. This generalizes a previous result of Saeki.     

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping of a metric tree with convex values has a selection for which for each . Here by we mean the Hausdroff distance. Many applications of this result are given.