Graphensätze für topologische Räume

In generalising a closed graph theorem of Dektjarev [5] we show that each almost closed and uniformly almost continuous relation of a quasi-uniform space to a hypercomplete quasi-uniform space is uniformly continuous. The hypothesis of being almost closed is necessary and actually weaker than the requirement that the relation considered has a closed graph. Since each topological space can be quasi-uniformized, one obtains closed graph theorems for topological spaces.As consequences we get an analogous closed graph theorem for locally uniformly almost continuous relations and a theorem concerning the completeness of the range of each continuous uniformly almost open mapping of a hypercomplete space to a uniform space.Finally we prove a closed graph theorem for relations to locally compact spaces without referring to any quasi-uniformization and thus without assumptions and statements concerning uniformity.     

Singular hyperbolicity of star flows on surfaces

In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.     

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph’s binary cycle space, each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no non-trivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.     

Riemann-Roch-Grothendieck and torsion for foliations

In this article we prove a Riemann-Roch-Grothendieck theorem for the characteristic classes of a flat vector bundle over a foliation whose graph is Hausdorff. We assume that the strong foliation Novikov-Shubin invariants of the flat bundle are greater than three times the codimension of the foliation. Using transgression, we define a torsion form which in the odd acyclic case determines a Haefliger cohomology class which only depends on the foliation and the flat bundle. We construct examples where this torsion class is highly non-trivial.     

Strongly Mixing Convolution Operators on Fréchet Spaces of Holomorphic Functions

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on \({\mathbb{C}^n}\) are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy–Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.     

Krasnoselskii type fixed point theorems for multi-valued mappings with weakly sequentially closed graph

In this work, using an analogue of Sadovskii’s fixed point result for multi-valued mappings with weakly sequentially closed graph, we prove new multi-valued analogues of Krasnoselskii fixed point theorem for mappings with weakly sequentially closed graph and under weak topology features. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness.     

Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials

     

Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.     

The pluripolar hull of a graph and fine analytic continuation

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C² then the projection of its pluripolar hull contains a fine neighborhood of a point . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function ℱ which is defined on a fine neighborhood of a point and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of ℱ over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C^∞ functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.     

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186-209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253-267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown.We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time.Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.     

恰有k条非基本边的极小3连通图

设G是简单3连通图．G\e(删除边e)和G/e(收缩边e)都不是简单3连通图,则e称为G的基本边．对于3连通图中的非基本边．Tutte证明了：唯一没有非基本边的简单3连通图是轮．Oxley和Wu确定了至多有3条非基本边的所有极小3连通图以及恰有4条非基本的极小3连通图．Reid与Wu确定了至多有5条非基本边的极小3连通图．在本文中,我们在极小3连通图中定义了三种运算,然后通过轮利用这些运算的逆运算给出恰有k(k■2)条非基本边的极小3连通图的一种构造方法．     

Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.     

关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.     

New Korovkin Type Theorem for Non-Tensor Meyer–König and Zeller Operators

In this paper, we introduce a certain class of linear positive operators via a generating function, which includes the non-tensor MKZ operators and their non-trivial extension. In investigating the approximation properties, we prove a new Korovkin type approximation theorem by using appropriate test functions. We compute the rate of convergence of these operators by means of the modulus of continuity and the elements of modified Lipschitz class functions. Furthermore, we give functional partial differential equations for this class. Using the corresponding equations, we calculate the first few moments of the non-tensor MKZ operators and investigate their approximation properties. Finally, we state the multivariate versions of the results and obtain the convergence properties of the multivariate Meyer–König and Zeller operators.     

Harmonic functions on planar and almost planar graphs and manifolds,via circle packings

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If is a bounded circle packing in whose contacts graph is a bounded valence triangulation of a disk, then, with probability , the simple random walk on converges to a limit point. Moreover, in this situation any continuous function on the limit set of extends to a continuous harmonic function on the closure of the contacts graph of ; that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs and manifolds, and show that under the assumptions of transience and bounded local geometry these possess non constant, harmonic, bounded, Dirichlet functions. Let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph. Oblatum 4-I-1995 & 23-IV-1996     

Functions of event variables of a random system with complete connections

In applications it often occurs that the experimenter is faced with functions of random processes. Suppose, for instance, that he only can draw partial or incomplete information about the underlying process or that he has to classify events for the sake of efficiency. We assume that the underlying process is a random system with complete connections (which contains the Markovian case as a special one) satisfying some basic properties, and that a mapping operates on the event space. With these two elements we construct in Section 2 a new random system with complete connections which inherits the properties of the old one (Theorem 2.2.3). In Section 3 we prove a weak convergence theorem (Theorem 3.4.4) in the theoretical framework of the so-called distance diminishing models, which gives a straightforward application in Section 4 to conditional probabilities related to partially observed events (Theorems 4.1.3). Finally we prove a Shannon-McMillan-type theorem (Theorem 4.2.3) finding application to classification procedures.     

An inverse problem for an integro‐differential operator on a star‐shaped graph

A partial inverse problem for an integro‐differential Sturm‐Liouville operator on a star‐shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.     

On hyperholomorphic functions of the space variable

For quaternionic-differentiable functions of the space variable, we prove the theorem on the integral over a closed surface which is an analog of the Cauchy theorem from complex analysis.     

A Ramsey property for graph invariants

We consider the problem of which graph invariants have a certain property relating to Ramsey's theorem. Invariants which have this property are called Ramsey functions. We examine properties of chains of graphs associated with Ramsey functions. Methods are developed which enable one to prove that a given invariant is not a Ramsey function. Results for several familiar invariants are presented.