Complexity of projected images of convex subdivisions

Let S be a subdivision of ^d into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d − 1, we construct a subdivision whose projected image has Ω(n^(3d−2)/2) complexity, which is tight when d 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane.     

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.     

On certain incomplete statistics

The foundations of the incomplete statistics recently proposed by Wang is rediscussed in the context of the canonical statistical ensemble. It is found that the incomplete normalization condition, ∑p^q_i=1 (i=1,…,w), where p_i is the probability of a given microstate, is not compatible with the entropic non-extensive formula proposed by Tsallis. It is proved that the entropic function proposed by Wang must be written as S_q=−k_B∑p_i^2q−1ln_qp_i, whereas the form proposed by Tsallis namely, S_q=−k_B∑p_i^qln_qp_i, is directly associated with the standard normalization condition (∑_ip_i=1).     

Extending shelling orders and a hierarchy of functions of unimodal simple polytopes

Let P be a simplicial d-polytope with n facets ((d − 1)-dimensional faces) in R^d. A shelling of P is an ordering of the facets of P such that the intersection of each facet F with the union of all facets that precede it the ordering is a nonempty union of (d − 2)-faces of F. The following open question was raised by Tverberg and is recorded in [4]. Suppose for some k < n, there is an ordering of k of the facets of P so that the intersection of each of these facets with the union of all of the facets that precede it in the ordering is a nonempty union of (d − 2)-faces. Can this initial “segment” be extended to a shelling of all the facets? This question is open even in the case that P is the dual of the d-dimensional hypercube. The question in this case has resurfaced several times since G. Danaraj and V. Klee (1978) in a variety of forms. It is related to the hierarchies of completely unimodal pseudo-Boolean functions studied in P.L. Hammer et al. (1988), the author (1988) and D. Wiedemann (1986). (A pseudo-Boolean function is a function mapping the vertices of the d-dimensional hypercube into the reals). In this paper, the hierarchies are compared and combined. This hierarchy is then extended to general simple polytopes, and the relationship to the above open question is explained.     

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.     

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.     

ContributionsSign-balanced covering matrices

A q × n array with entries from 0, 1,…,q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,… q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1]^t, a λq × n array will be said to form a (t, q, λ, Δ) sign-balanced matrix if for each choice C₁, C₂,…, C_t of t columns and for each choice = (₁,…,_t) Δ of signs, the linear combination ∑_j=1^t _jC_j contains (mod q) each entry of [0, 1,…, q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k(n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice (₁, …, _t) of signs in δ, the linear combination ∑_j=1^t _jC_j contains each entry of [0, 1,…, q t- 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k(n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.     

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).     

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.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Maximum number of edges in a critically k-connected graph

A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any vV(G). We show that if n,k are integers with k4 and nk+2, and G is a critically k-connected graph of order n, then |E(G)|n(n−1)/2−p(n−k)+p²/2, where p=(n/k)+1 if n/k is an odd integer and p=n/k otherwise. We also characterize extremal graphs.     

A Helly-type theorem for higher-dimensional transversals

We prove that a collection of compact convex sets of bounded diameters in that is unbounded in k independent directions has a k-flat transversal for k<d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

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 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.     

q-distributions and Markov processes

We consider a sequence of integer-valued random variables X_n, n 1, representing a special Markov process with transition probability λ_{n, l}, satisfying P_{n, l} = (1 − λ_{n, l}) P_{n−1, l} + λ_{n, l−1} P_{n−1, l−1}. Whenever the transition probability is given by λ_{n, l} = q^{n + βl + γ} and λ_{n, l} = 1 − q^n+βl+γ, we can find closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q-binomial coefficients and the q-Stirling numbers. In general, it turns out that the q-notation, up to now mainly used in the theory of q-hypergeometrical series, represents a powerful tool to deal with these kinds of problems. In this context we speak therefore about q-distributions. Finally, we present some possible, mainly graph theoretical interpretations of these random variables for special choices of , β and γ.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

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}.     

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.     

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.