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.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Solvable block transitive automorphism groups of 2−(v,5,1) designs

Let G be a solvable block transitive automorphism group of a 2−(v,5,1) design and suppose that G is not flag transitive. We will prove that

(1) if G is point imprimitive, then v=21, and GZ₂₁:Z₆;

(2) if G is point primitive, then GAΓL(1,v) and v=p^a, where p is a prime number with p≡21 (mod 40), and a an odd integer.

     

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.     

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.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.     

The range of the p-Laplacian

We prove that for λ ≥ 0, p ≥ 3, there exists an open ball B L²(0,1) such that the problem

− (|u′|^p−2 u′)′ − λ|u|^p−2u = f, in (0,1)

, subject to certain separated boundary conditions on (0,1), has a solution for f B.     

A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B₀|x| = 0 has a nontrivial solution for some matrix B₀ of a very special form, |B₀| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.     

Inverse geophysical problems for some noncompactly supported inhomogeneities

Let[²+k²n(x₁,x₃)]u(x₁,x₂,x₃)=−δ(x₁,y₁δ(x₂,y₂)δ(x₃,y₃) in R³₊. Assume that u(x₁,x₂,x₃=0,y₁,y₂=0,y₃=0,k) is measured at the plane P {x:x₃=0} for all positions of the source on the line y = (y₁,y₂ = 0,y₃ = 0), -∞ < y₁ < ∞, and receiver on the plane(x₁,x₂,x₃ − <x₁,x₂ < ∞, and for low-frequencies 0 < k <k₀, k₀ > 0 is an arbitrary small wave number. Assume thatn(x₁,x₃) is an arbitrary bounded piecewise-continuous function. The basic result is: the above low-frequency surface data determinen(x₁,x₃)uniquely.     

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.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.     

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}.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Designs with no three mutually disjoint blocks

Let D(v,b,r,k,λ) be any quasi-symmetric block design with block intersection numbers 0 and y. Suppose D has no three mutually disjoint blocks. We show that for a given value of y, there are only finitely many parameter sets of such designs. Moreover, the ‘extremal’ designs D have one of the following parameter sets: (1) v = 4y, k = 2y, λ = 2y − 1 (y 2) (2) v = y(y²+3y+1), k = y(y+1), λ =y²+y−1(y2) (3) v = (y+1)(y²+2y−1), k = y(y+1), λ =y² (y2) A computer search revealed only three parameter sets in the range 1y199, which are not of the above types.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

Asymptotic behavior of a nonlinear delay difference equation

This paper considers a class of nonlinear difference equations

Δ₃y_n + ƒ(n, y_n, y_n−r) = 0, n N (n₀)

. A necessary and sufficient condition for the existence of a bounded nonoscillatory solution is given.     

Densest translational lattice packing of non-convex polygons

A translational lattice packing of k polygons P₁,P₂,P₃,…,P_k is a (non-overlapping) packing of the k polygons which is replicated without overlap at each point of a lattice i₀v₀+i₁v₁, where v₀ and v₁ are vectors generating the lattice and i₀ and i₁ range over all integers. A densest translational lattice packing is one which minimizes the area |v₀×v₁| of the fundamental parallelogram. An algorithm and implementation is given for densest translational lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.     

Elasticity of factorizations in integral domains

For an atomic integral domain R, define(R)=sup{mn|x₁x_m=y₁y_n, each x_i,y_jεR is irreducible}. We investigate (R), with emphasis for Krull domains R. When R is a Krull domain, we determine lower and upper bounds for (R); in particular,(R)≤max{|Cl(R)| 2, 1}. Moreover, we show that for any real numbers r≥1 or R=∞, there is a Dedekind domain R with torsion class group such that (R)=r.     

On the area of the intersection of disks in the plane

Let C₁,…, C_n and C′₁,…, C′_n be two collections of equal disks in the plane, with centers c₁,…, c_n and c′₁,…, c′_n. According to a well-known conjecture of Klee and Wagon (1991), if |c_i − c_j| ≥ |c′_i − c′_j| for all i, j, then Area(∩_i C_i) ≤ Area(∩_i C′_i).

We prove this statement in the special case when there is a continuous contraction of {c₁,…, c_n} onto {c′₁,…, c′_n}.