On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x ₁,…,x _n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I _n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I _n,d )=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I _n,d )≤⌊(n−d)/(d+1)⌋+d.     

Gersgorin variations II: On themes of Fan and Gudkov

Assume F={f₁,. . .,f_n} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |a_ii|>f_i (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(a_ij), and all i, f_i≥∑_j|a_ij|{x_j}/x_i. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship     

Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms are permutable.     

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

Boundary Harnack principle for second order elliptic equations with unbounded drift

We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a _ij D _ij u + b _i D _i u = 0 in a bounded domain Ω ⊂  \mathbb R {\mathbb R} ⁿ . We assume that b _i ∈ L ⁿ (Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1]. Based on this result, we derive the H?lder regularity of u/v for uniform domains. Bibliography: 27 titles.     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

Elliptic equation with singular potential

We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R ⁿ , n ≥ 3:

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

LOWER SEMICONTINUITY OF A CLASS OF FUNCTIONALS IN SBVH

In this paper lower semicontinuity of the functional I(u)=∫ _Ω f(x,u,Δ _Hu)dx is investigated for f being a Carathéodory function defined on H ⁿ × R × R²ⁿ and for u∈SBV _H (Ω), where H ⁿ is the Heisenberg group with dimension 2n+1, Ω∩H ⁿ is an open set and ∇ Hu denotes the approximate derivative of the absolute continuous part D ^a _Hu with respect to D _Hu. In addition, a Lusin type approximation theorem for a SBV _H function is proved.