The topological theory of current graphs

W. Gustin's introduction of combinatorial current graphs as a device for obtaining orientable imbeddings of Cayley “color” graphs was fundamental to the solution of the Heawood map-coloring problem by G. Ringel, J. W. T. Youngs, C. M. Terry, and L. R. Welch. The topological current graphs of this paper lead to a construction that generalizes the method of Gustin and its augmentation to “vortex” graphs by Youngs, extending the scope of current graph theory from Cayley graphs alone to the much larger class of graphs that are covering spaces.     

Fuzzy closure spaces

Fuzzy topological spaces do not constitute a natural boundary for the validity of theorems, but many results can be extended to what are called fuzzy closure spaces (or fcs's, for short). The notions of a subspace, a sum, and a product are extended to fcs's. The hereditary, additivity, and productivity behaviour of compactness in fcs's is investigated and some weak forms of compactness and fuzzy continuous functions in fcs's are introduced. The interaction between fuzzy proximity spaces and fcs's is investigated; A necessary background is included for completeness.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

Critical perfect graphs and perfect 3-chromatic graphs

This paper builds on results based on D. R. Fulkerson's antiblocking polyhedra approach to perfect graphs to obtain information about critical perfect graphs and related clique-generated graphs. Then we prove that Berge's Strong Perfect Graph Conjecture is valid for 3-chromatic graphs.     

Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces

It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar) quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the indecomposable orthoscalar representations are classified up to unitary equivalence.     

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.     

Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs

This paper presents a new proof of Whitney's theorem on edge-isomorphisms of graphs and extends Whitney's theorem to hypergraphs. Whitney's theorem asserts that any two edge-isomorphic graphs of order at least 5 have their edge-isomorphism induced by a node-isomorphism isomorphism. Previous results of Gardner and of Berge and Rado are used.     

Doob's inequality,Burkholder-Gundy inequality and martingale transforms on martingale Morrey spaces

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob's inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob's inequality on martingale block spaces, we obtain the Davis' decompositions for martingale Morrey spaces.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

The enumeration of bipartite graphs

The standard construction of graphs with n connected components is modified here for bicolored graphs by letting S_n × H act on the function space $\text{Y}{}^1\text{where}\text{X={1,2,…,n}, Y}$ is the set of connected bicolored graphs, and H is the group that interchanges the vertex colors. Then DeBruijn's Generalization of Polya's Theorem is applied to arrive at a direct algebraic relationship between the generating functions for bicolored and connected bicolored graphs. As the former generating function is easily computable, this relationship gives us the latter generating function which is precisely the generating function for connected bipartite graphs.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

Inversion of dynamic programs and its applications to allocation processes

We consider three types of discrete time deterministic dynamic programs (DP's) on one-dimensional state spaces whose reward functions depend on both state and action, namely, type I: finite-stage DP's with invertible terminal reward function, type II:finite-stage DP's without terminal reward function, and type III: infinite-stage DP's with additive reward function. Types I and II have a general objective function, which is backwards recursively generated by stage-wise reward functions. Given a (main) DP, an inverse DP yielding a new expression is defined. The inverse DP has additive expression of objective function. Deriving recursive formulae, we establish Inverse Theorems between main and inverse DP's. Each Inverse Theorem is applied to Bellman's multi-stage allocation process. Uniqueness of solution to an inverse functional equation is proved.     

Intersection theory for graphs

An intersection theory developed by the author for matroids embedded in uniform geometries is applied to the case when the ambient geometry is the lattice of partitions of a finite set so that the matroid is a graph. General embedding theorems when applied to graphs give new interpretations to such invariants as the dichromate of Tutte. A polynomial in n + 1 variables, the polychromate, is defined for graphs with n vertices. This invariant is shown to be strictly stronger than the dichromate, it is edge-reconstructible and can be calculated for proper graphs from the polychromate of the complementary graph. By using Tutte's construction for codichromatic graphs (J. Combinatorial Theory 16 (1974), 168–174), copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper.     

A construction of geodetic graphs based on pulling subgraphs homeomorphic to complete graphs

Generalizing Cook and Pryce's construction procedures for geodetic blocks, an operation in geodetic graphs is discussed consisting of “pulling” a subgraph homeomorphic to a complete graph. This unifies and generalizes several of the known constructions of geodetic graphs.     

On chaotic graphs and applications in physics and biology

El Naschie’s ε^∞ theory in Quantum space time is given and discussed geometrically and topologically as a category of fuzzy spaces, these fuzzy categories in which lines are fuzzy fractal lines. In this paper, we represent the chaotic graphs as many fuzzy fractal lines up to ∞. We will describe them by chaotic matrices. Many fuzzy systems (chaotic systems) are described and applied in [8], [9], [10], [11], [12]. This article introduces some operations on the chaotic graphs such as the union and the intersection; also both of the chaotic incidence matrices and the chaotic adjacency matrices representing the chaotic graphs induced from these operations will be studied. Theorems governing these studies are obtained. Some applications on chaotic graphs are given [18], [19], [20], [21].     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.