Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Orthogonal Polynomials on the Ball and the Simplex for Weight Functions with Reflection Symmetries

Generalized classical orthogonal polynomials on the unit ball B ^d and the standard simplex T ^d are orthogonal with respect to weight functions that are reflection-invariant on B ^d and, after a composition, on T ^d , respectively. They are also eigenfunctions of a second-order differential—difference operator that is closely related to Dunkl's h -Laplacian for the reflection groups. Under a proper limit, the generalized classical orthogonal polynomials on B ^d converge to the generalized Hermite polynomials on R ^d , and those on T ^d converge to the generalized Laguerre polynomials on R ^d ₊ . The latter two are related to the Calogero—Sutherland models associated to the Weyl groups of type A and type B . February 14, 2000. Date revised: July 26, 2000. Date accepted: August 4, 2000.     

COMPARATIVE PROBABILITY ON THE C* ALGEBRA OF COMPACT OPERATORS

Abstract

Let A be a von Neumann algebra on a Hilbert space H and let P(A) denote the projections of A. A comparative probability (CP) on A (or more correctly on P(A)) is a preorder ? on P(A) satisfying:

0 ? P ? P ε P(A) with Q ≠ 0 for some Q ε P(A).

If P, Q ε P(A) then either P ? Q or Q ? P.

If P, Q and R are all in P(A) and P⊥R, Q⊥R, then P ? Q ? P + R ? Q + R.

Let τ be any of the usual locally convex topologies on A. We say ? is τ continuous if the interval topology induced on P(A) by ? is weaker than the τ topology on P(A). If μ an additive (completely additive) measure on P(A) then μ induces a uniformly (weakly) continuous CP ?μ on P(A) given by P ?_μ Q if μ(P) ? μ(Q). We show that if A is the C* algebra C(H) of compact operators on an infinite dimensional Hilbert space H, the converse is true under an extra boundedness condition on the CP which is automatically satisfied whenever the identity is present in A = P(C(H)).     

Universal Overconvergence of Polynomial Expansions of Harmonic Functions

For each compact subset K of ^N let (K) denote the space of functions that are harmonic on some neighbourhood of K. The space (K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of ^N such that 0Ω and ^N\Ω is connected. It is shown that there exists a series ∑H_n, where H_n is a homogeneous harmonic polynomial of degree n on ^N, such that (i) ∑H_n converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑H_n are dense in (K) for every compact subset K of ^N\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

High dimensional limit theorems and matrix decompositions on the Stiefel manifold

The main purpose of this paper is to investigate high dimensional limiting behaviors, as m becomes infinite (m → ∞), of matrix statistics on the Stiefel manifold V_{k, m}, which consists of m × k (m ≥ k) matrices X such that X′X = I_k. The results extend those of Watson. Let X be a random matrix on V_{k, m}. We present a matrix decomposition of X as the sum of mutually orthogonal singular value decompositions of the projections P X and P X, where and are each a subspace of R^m of dimension p and their orthogonal compliment, respectively (p ≥ k and m ≥ k + p). Based on this decomposition of X, the invariant measure on V_{k, m} is expressed as the product of the measures on the component subspaces. Some distributions related to these decompositions are obtained for some population distributions on V_{k, m}. We show the limiting normalities, as m → ∞, of some matrix statistics derived from the uniform distribution and the distributions having densities of the general forms f(P X) and f(m^1/2P X) on V_{k, m}. Subsequently, applications of these high dimensional limit theorems are considered in some testing problems.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Simultaneous approximation on two subsets of an open Riemann surface

Given functions f₁ and f₂ meromorphic, respectively, on subsets E₁ and E₂ of a Riemann surface R, we seek a function meromorphic on all of R, which simultaneously approximates f₁ on E₁ and f₂ on E₂.     

Isomorphic factorization of the complete graph into Cayley digraphs

Let Z_p denote the cyclic group of order p where p is a prime number. Let X = X(Z_p, H) denote the Cayley digraph of Z_p with respect to the symbol H. We obtain a necessary and sufficient condition on H so that the complete graph on p vertices can be edge‐partitioned into three copies of Cayley digraphs of the same group Z_p each isomorphic to X. Based on this condition on H, we then enumerate all such Cayley graphs and digraphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 243–256, 2006     

Pseudodifferential Operators on Prehomogeneous Vector Spaces

ABSTRACT

Let G be a connected, linear algebraic group defined over ?, acting regularly on a finite dimensional vector space V over ? with ?-structure V _?. Assume that V possesses a Zariski-dense orbit, so that (G, ?, V) becomes a prehomogeneous vector space over ?. We consider the left regular representation π of the group of ?-rational points G _? on the Banach space C₀(V _?) of continuous functions on V _? vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G _?. Denote the complement of the dense orbit by S, and put S _? = S ∩ V _?. It turns out that, on V _? ? S _?, π(f) is a smooth operator. If S _? = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V _? ? {0}. Its nonunique extension to V _? can then be regarded as a trace of π(f). If G is reductive, and S and S _? are irreducible hypersurfaces, π(f) corresponds, on each connected component of V _? ? S _?, to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V _? ? S _? given by some power |p(m)| ^s of a relative invariant p(m) of (G, ?, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V _?, and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.     

Rationality for Subgroups of S 6

For a transitive subgroup G ≤ S ₆ which contain C ₃ × C ₃ as subgroup, we prove that K(x ₁,…, x ₆) ^G is rational over K, where K is any field, and G acts naturally on K(x ₁,…, x ₆) by permutations on the variables. We also give an application on construction of generic polynomials.     

Criteria for farthest points on convex surfaces

We provide a sharp, sufficient condition to decide if a point y on a convex surface S is a farthest point (i.e., is at maximal intrinsic distance from some point) on S, involving a lower bound π on the total curvature ω_y at y, ω_y ≥ π. Further consequences are obtained when ω_y > π, and sufficient conditions are derived to guarantee that a convex cap contains at least one farthest point. A connection between simple closed quasigeodesics O of S, points y ∈ S\O with ω_y > π, and the set ?? of all farthest points on S, is also investigated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formally Real Involutions on Central Simple Algebras

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r ^# r where r ? R is nonzero. Suppose that R is a central simple algebra (i.e., R = M _n (D) for some integer n and central division algebra D) and # is an involution on R of the form r ^# = a ^?1 r? a, where ? is some transpose involution on R and a is an invertible matrix such that a? = ±a. In Section 1 we characterize formal reality of # in terms of a and ?| _D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ? _F K ? M _n (K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ? tr(x ^# x) is not positive semidefinite.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Lazarsfeld-Mukai reflexive sheaves and their stability

Consider an ample and globally generated line bundle L on a smooth projective variety X of dimension N≥2 over ?. Let D be a smooth divisor in the complete linear system of L. We construct reflexive sheaves on X by an elementary transformation of a trivial bundle on X along certain globally generated torsion-free sheaves on D. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the μ_L-(semi)stability of such reflexive sheaves under certain conditions.     

Fourier Inversion of Distributions Supported by a Hypersurface

Let μ _Σ be the natural measure on R ^N (N≥3) supported by a compact oriented analytic hypersurface Σ, ψ a smooth function on R ^N and P(D) a differential operator in N variables of order m. We determine a sufficient condition on the number λ such that the Fourier integral of the distribution P(D)ψ μ _Σ be summable by Cesàro means of order λ to zero in a point outside the hypersurface. This condition depends on m and on the position of the point with respect to the caustic of the hypersurface.     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

   Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

Global italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G = (V, E) is a function f: V → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1. The weight of an IDF f is the value ∑_v∈V(G) f (v). The Italian domination number of a graph G, denoted by γ_I (G), is the minimum weight of an IDF on G. An IDF f on G is called a global Italian dominating function (GIDF) on G if f is also an IDF on the complement ? of G. The global Italian domination number of G, denoted by γ_gI (G), is the minimum weight of a GIDF on G. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree T of order n ≥ 4, γ_gI (T) ≤ γ_I (T) + 2 and we characterize all trees with γ_gI (T) = γ_I (T) + 2 and γ_gI (T) = γ_I (T) + 1.