Graph designs for the eight-edge five-vertex graphs

The existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order n exists exactly when and n≠16, with the possible exception of n=48. For the unique graph with vertex sequence (2,3,3,4,4), a graph design of order n exists exactly when , with the possible exceptions of n∈{32,48}.     

A -analogue of the 9- symbols and their orthogonality

We introduce a q-analogue of Wigner’s 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75-94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L²(m) (not only unitarily equivalent to it), then it has a cyclic vector in L^∞(m). We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L²(C,τ) is the norm-closure of the polynomials in z.     

Hilbert series of invariants, constant terms and Kostka-Foulkes polynomials

A problem that arose in the study of the mass of the neutrino led us to the evaluation of a constant term with a variety of ramifications into several areas from Invariant Theory, Representation Theory, the Theory of Symmetric Functions and Combinatorics. A significant by-product of our evaluation is the construction of a trigraded Cohen Macaulay basis for the Invariants under an action of SL_n(C) on a space of 2n+n² variables.     

Relations on generalized degree sequences

We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.     

Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E) be a finite, simple and non-empty (p,q)-graph of order p and size q. An (a,d)-vertex-antimagic total labeling is a bijection f from V(G)∪E(G) onto the set of consecutive integers 1,2,…,p+q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f(xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f(x). Such a labeling is called super if the smallest possible labels appear on the vertices.In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.     

The mathematics of the grand unified theory

This paper surveys the mathematics of GUT and highlights the author’s new real number system which is a continuum, non-Archimedean and non-Hausdorff, but its subspace of decimals is countably infinite, discrete, Archimedean and Hausdorff. The paper also proves Goldbach’s conjecture in and provides an overview of GUT and qualitative and computational models of many of the presently ill-defined physical concepts, e.g., gravity.     

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H, for every induced subgraph H of G. A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W₄, bull}-free, both superclasses of triangle-free graphs.