On the number of irreducible coverings by edges of complete bipartite graphs

In this paper it is proved that the exponential generating function of the numbers, denoted by N(p, q), of irreducible coverings by edges of the vertices of complete bipartite graphs K_p,q equals exp(xe^y + ye^x − x − y − xy) − 1.     

Orthochronous subgroups of O(p,q)

The matrices if O(p,q) whose upper left p × p corners have positive determinant form a subgroup of O(p,q).     

Approximating geometric bottleneck shortest paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π/(3cos(π/6)) times the Euclidean distance |pq| between p and q, and all of whose edges have length at most |pq|.     

Cohomology of the Universal Enveloping Algebras of Certain Bigraded Lie Algebras

Let p be an odd prime and q = 2(p-1).Up to total degree t-s max{(5p~3+ 6p~2+ 6 p +4)q-10,p~4q},the generators of H~(s,t)(U(L)),the cohomology of the universal enveloping algebra of a bigraded Lie algebra L,are determined and their convergence is also verified.Furthermore our results reveal that this cohomology satisfies an analogous Poinare duality property.This largely generalizes an earlier classical results due to J.P.May.     

Factorizations of root-based polynomial compositions

Let denote the finite field of order q=p^r, p a prime and r a positive integer, and let f(x) and g(x) denote monic polynomials in of degrees m and n, respectively. Brawley and Carlitz (Discrete Math. 65 (1987) 115–139) introduce a general notion of root-based polynomial composition which they call the composed product and denote by fg. They prove that fg is irreducible over if and only if f and g are irreducible with gcd(m,n)=1. In this paper, we extend Brawley and Carlitz's work by examining polynomials which are composed products of irreducibles of non-coprime degrees. We give an upper bound on the number of distinct factors of fg, and we determine the possible degrees that the factors of fg can assume. We also determine when the bound on the number of factors of fg is met.     

On some integrals arising in mathematical physics

This paper presents in the first section the exact evaluation of three single integrals relating to the dielectric behavior of two-dimensional electron plasmas. In the second section we present a procedure for reducing 3d-dimensional integrals of the form: ∫∫∫dqdpdkD(q)(p+k+q)ƒ(p)[1−ƒ(p+q)]ƒ(k)[1−ƒ(k+q)], where the vectors lie in d-dimensional space and ƒ denotes the Fermi function, to tractable form. The second-order exchange integral for a d-dimensional electron gas is taken as an example and is evaluated in closed form as a function of d.     

Möbius Regular Maps of Order pq

M?bius regular maps are surface embeddings of graphs with doubled edges such that(i)the automorphism group of the embedding acts regularly on flags and(ii) each doubled edge is a center of a M?bius band on the surface. In this paper, we classify M?bius regular maps of order pq for any two primes p and q, where p≠q.     

The order-interval hypergraph of a finite poset and the König property

We study the hypergraph H(P) whose vertices are the points of a finite poset and whose edges are the maximal intervals in P (i.e. sets of the form I = {{v ε P: p ν q}}, p minimal, q maximal). We mention resp. show that the problems of the determination of the independence number , the point covering number τ, the matching number v and the edge covering number p are NP-complete. For interval orders we describe polynomial algorithms and prove the König property (v = τ) and the dual König property (a = p). Finally we show that the (dual) König property is preserved by product.     

Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

Near vector spaces over GF(q) and (v,q+1,1)-BIBD''s

The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v₁,q+1,1)−BIBD P₁,B₁ and a (v₂,q+1,1)−BIBD P₂,B₂, there is a ((q−1)v₁v₂+v₁+v₂,q+1,1)−BIBD P₃,B₃ containing P₁,B₁ and P₂,B₂ as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p²× p² matrices if p is an odd prime.     

Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li [A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng [B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge [A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].     

Maximal partial spreads and the modular n-queen problem

We prove that for any integer n in the interval there is a maximal partial spread of size n in PG (3, q) where q is odd and q7. We also prove that there are maximal partial spreads of size (q²+3)/2 when gcd(q+1,24)=2 or 4 and of size (q²+5)/2 when gcd(q+1,24)=4.     

Representation of elliptic Ding-Iohara algebra

We define a vector representation V(u)of elliptic Ding-Iohara algebra u(q,t,p).Furthermore,we construct the tensor products of the vector representations and the Fock modulesF(u)by taking the inductive limit of certain subspaces in the finite tensor products of vector representations.     

Constructions of (q, K, λ, t, Q) almost difference families

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.     

Normal Sylow subgroups and monomial Brauer characters

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G : ker φ|p' = φ(1)p' for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p' is the p'-part of φ(1).     

Kernel structure of Toeplitz-plus-Hankel matrices

The structure of the kernel of block Toeplitz-plus-Hankel matrices R=[a_j−k+b_j+k], where a_j and b_j are the given p×q blocks with entries from a given field, is investigated. It is shown that R corresponds to two systems of at most p+q vector polynomials from which a basis of the kernel of R and all other Toeplitz-plus-Hankel matrices with the same parameters a_j and b_j can be built. The main result is an analogue of a known kernel structure theorem for block Toeplitz and block Hankel matrices.     

A characterization for the boundedness of geometric mean operator

We give a characterization for the geometric mean inequality

to hold for the case 0 < q < p ≤ ∞, p > 1, where f is positive a.e. on (0, ∞), and C > 0 independent of f.     

On oscillation of second order neutral type delay differential equations

Oscillation criteria are obtained by using the so called H-method for the second order neutral type delay differential equations of the form

(r(t)ψ(x(t))z^′(t))^′+q(t)f(x(σ(t)))=0, tt₀,

where z(t)=x(t)+p(t)x(τ(t)), r, p, q, τ, σ, C([t₀,∞),R) and f,ψC(R,R).

The results of the paper contains several results obtained previously as special cases. Furthermore, we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations.     

On the number of flats spanned by a set of points in PG(d, q)

It is shown that for fixed 1 r s < d and > 0, if X PG(d, q) contains (1 + )q^s points, then the number of r-flats spanned by X is at least c()q^{(r+1)(s+1−r)}, i.e. a positive fraction of the number of r-flats in PG(s + 1,q).