Convex 4-valent polytopes

Let {p_k}_k≥3 be a sequence of nonnegative integers which satisfies 8 + Σ_k≥3 (k-4) p_k = 0 and p₄ ≥ p₃. Then there is a convex 4-valent polytope P in E³ such that P has exactly p_k k-gons as faces. The inequality p₄ ≥ p₃ is the best possible in the sense that for c < 1 there exist sequences that are not 4-realizable that satisfy both 8 + Σ_{k ≥3} (k - 4) p_k = 0 and p₄ > cp₃. When Σ_{k ≥ 5} p_k ≠ 1, one can make the stronger statement that the sequence {p_k} is 4-reliazable if it satisfies 8 + Σ_{k ≥ 3} (k - 4) p_k = 0 and p₄ ≥ 2Σ_{k ≥ 5} p_k + max{k ¦ p_k ≠ 0}.     

Extremal regular graphs for the achromatic number

The problem of constructing (m, n) cages suggests the following class of problems. For a graph parameter θ, determine the minimum or maximum value of p for which there exists a k-regular graph on p points having a given value of θ. The minimization problem is solved here when θ is the achromatic number, denoted by ψ. This result follows from the following main theorem. Let M(p, k) be the maximum value of ψ(G) over all k-regular graphs G with p points, let {x} be the least integer of size at least x, and let be given by ω(k) = {i(ik+1)+1:1i<∞}. Define the function ƒ(p, k) by

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. Then for fixed k2 we have M(p, K=ƒ(p, k) if pω(k) and M(p, k)=ƒ(p,k-1 if pε ω(k) for all p sufficiently large with respect to k.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

On the number of edge-disjoint one factors and the existence of k-factors in complete multipartite graphs

In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1), …, p(n)), conditions that are reminiscent of the Erdös-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number of edge-disjoint 1-factors in K(p(1), …, p(n)), settling the problem in the case where this number is greater than δ - p(2), where p(1) p(2) … p(n).     

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.     

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.     

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.     

Comparison theorems for the oscillation of higher order difference equations with deviating arguments

The difference equation Δy + δp(k)f (y (g (k))) = 0, where p(k) is positive, is classified into four cases according to is odd or even and δ is 1 or −1. In each case, we shall offer comparison theorems for the oscillation of the difference equation. Examples are also included to illustrate the importance of the results obtained.     

The location of zeros of polynomials

Let A be a complex n×n matrix. p an equilibrated vectonal norm and x(A) the spectrial abscissa of A. Then, it is known [5] x(A)≤x(γ_p(A)) where γp is the matricial logarithmic derivative induced by p. We will make use of the above inequality to obtain regions in the plane which contain the zeros of complex polynomials.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.     

On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann–Robin boundary conditions

We consider boundary value problem

where   0, λ > 0 are parameters and f  C²[0, ∞) such that f(0) < 0. In this paper we study for the cases p  (0, β) and p  (β, θ) (p is the value of the solution at x = 0 and β, θ are such that f(β) = 0, , the relation between λ and the number of interior critical points of the positive solutions of the above system.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

New constructions of divisible designs

A construction is given for a (p^2a(p+1),p²,p^2a+1(p+1),p^2a+1,p^2a(p+1)) (p a prime) divisible difference set in the group H×Z²_pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ₁≠0, and those are fairly rare. We also give a construction for a (p^a−1+p^a−2+…+p+2,p^a+2, p^a(p^a+p^a−1+…+p+1), p^a(p^a−1+…+p+1), p^a−1(p^a+…+p²+2)) divisible difference set in the group H×Z_p2×Z^a_p. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ₁=λ₂, and this corresponds to the parameters for the ordinary Menon difference sets.     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

A special k-coloring for a connected k-chromatic graph

For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices x_o and x_p in any color class there exist vertices x₁, x₂, …, x_p-1 of the same class with dist(x_i, x_i+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.

We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1.     

The product of the independent domination numbers of a graph and its complement

It is shown that the maximum value over p vertex graphs of the product of the independent domination numbers of a graph and its complement is at most min {(p + 3)²/8, (p + 8)²/10.8}.     

One-factors and the existence of affine designs

Köhler (1982) defines a graph G_p for p a prime which he relates to affine designs. He shows that if the graph has a 1-factor then there exists an affine 3-(p,4,λ) design with λ satisfying the usual necessary conditions. Here we generalize Köhler's results to include finite fields.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.