Matrices and set intersections

The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AA^T=(k?λ)I+λJ, where A^T denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved.     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Incidence matrices of t-designs

We point out a generalization of the matrix equation NN^T=(r? λ)I+λJ to t-designs with t>2 and derive extensions of Fisher's, Connor's, and Mann's inequalities for block designs.     

The Ostaszewski square and homogeneous Souslin trees

Assume GCH and let λ denote an uncountable cardinal. We prove that if □_λ holds, then this may be witnessed by a coherent sequence 〈C _α|α < λ⁺〉 with the following remarkable guessing property For every sequence 〈A _i | i < λ〉 of unbounded subsets of λ ⁺, and every limit θ < λ, there exists some α < λ ⁺ such that otp(C _α)=θ and the (i + 1) _th -element of C _α is a member of A _i , for all i < θ. As an application, we construct a homogeneous λ ⁺-Souslin tree from □_λ + CH_λ, for every singular cardinal λ. In addition, as a by-product, a theorem of Farah and Veli?kovi?, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.     

Nonlinearity and nontrivial solutions of fourth order semilinear elliptic equations

We consider the existence of nontrivial solutions of a fourth order semilinear elliptic boundary value problem with Dirichlet boundary condition, Δ²u+cΔu=b₁[(u+1)⁺−1]+b₂u⁺ in Ω, where Ω is a bounded open set in R^N with smooth boundary ∂Ω. The variation of linking theorem is useful to investigate them. We investigate them in six regions of (b₁,b₂) when λ₁<c<λ₂.     

On the spectral radius of (0,1)-matrices

We determine the maximum spectral radius for (0,1)-matrices with k² andk²+1 1's, respectively, and for symmetric (0,1)-matrices with zero trace and $\text{e=}\text{k}\text{2}$1's (graphs with e edges). In all cases, equality is characterized.     

A relation between Bernoulli numbers

We give a p-adic proof of a certain new relation between the Bernoulli numbers B_k, similar to Euler's formula Σ_k=2^m?2(_k^m)B_kB_m?k = ?(m+1)B_m, m ≥ 4.     

On Schmerl's effective version of Brooks' theorem

A new proof is given of Schmerl's recent result that a highly recursive graph G with χ(G) ≤ k according to Brooks' theorem, has a recursive k-colouring.     

On Abel—Poisson type and Riesz means

В работе исследуются ядра методов суммиро вания типа Абеля—Пуассона и Рис са, применяемых к кратны м интегралам Фурье. Вы ясняются условия на параметры, определяющие эти методы, при которы х их ядра неотрицател ьны. Полученные результа ты можно сформулировать в тер минах положительной определенности неко торых функций. Наприм ер, функция exp(? ¦x¦^α) при 0<а ≦2 является, а при α>2 не является положитель но определенной в евклидовом простра нствеE _N размерностиN (N=1, 2, ...). Далее, еслиt ₊=max (t, 0), то при любом натураль номN на интервале 0<λ<2 существует неубываю щая непрерывная функцияk _N (λ) такая, что функция (1 ? ¦х¦^λ) ₊ ^k приk≧k _N (λ) является, а приk<k _N (λ) не является положительно опреде ленной в пространств еE _N . При этом $$k_N (1) = \frac{{N + 1}}{2}, k_N (2 - 0) = + \infty , k_N (\lambda ) \geqq \lambda + \frac{{N - 1}}{2}.$$ Если же λ≧2, то функция (1?¦x¦^λ) ₊ ^k ни при каком значении параметраk не является положите льно определенной в прост ранствеE _N ,N=1, 2, .... Кроме того, исследует ся порядок приближен ия функцийN переменных класса Н икольскогоk _P ^α , 1≦р<∞, 0<а<2, операторам и типа Абеля—Пуассон а в метрикеL _p (E _N ).     

Cayley's anticipation of a generalised Cayley-Hamilton theorem

The principal result of Cayley's famouus memoir on matrices of 1858 is his contribution to what is now known as ‘the Cayley-Hamilton theorem’. We discuss this theorem and show that prior to its publication Cayley was aware of a more general theorem, a result that he left unpublished. This theorem is associated with the binary algebraic form det (μP ? λQ) analogous to the standard characteristic polynomial det (A ? λI).     

О раэложении мероморфных функций специального вида на простейщие дроби

The paper is devoted to study the entire functions L(λ) with simple real zeros λ_k, k = 1, 2, ..., that admit an expansion of Krein’s type: $$\frac{1}{{\mathcal{L}(\lambda )}} = \sum\limits_{k = 1}^\infty {\frac{{c_k }}{{\lambda - \lambda _k }}} ,\sum\limits_{k = 1}^\infty {\left| {c_k } \right| < \infty } .$$ We present a criterion for these expansions in terms of the sequence {L′ (λ _k )} _k=1 ^∞ . We show that this criterion is applicable to certain classes of meromorphic functions and make more precise a theorem of Sedletski? on the annihilating property in L ² systems of exponents.     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

Simple continued fractions for some irrational numbers,II

In this paper we prove a theorem allowing us to determine the continued fraction expansion for Σ_k=0^∞u^?c(k), where c(k) is any sequence of positive integers that grows sufficiently quickly. As an application, we determine the continued fraction expansion for Liouville's famous transcendental number Σ_k=0^∞m^{?(k + 1)!}.     

Decompositions of complete graphs into regular bichromatic factors

The proof of the following theorem is given: A complete graph with n vertices can be decomposed into r regular bichromatic factors if and only if n is even and greater than 4 and there exists a natural number k with the properties that k ≤ r and 2^{k ? 1} < n ≤ 2^k.     

A construction for PBIBD(2)'s

This note gives a new construction for PBIBD(2)'s that generalizes a construction of Hall's for finite projective planes, and that leads to a new PBIBD(2) with parameters (v, b, k, r, λ₁, λ₂) = (36, 60, 10, 0, 2).     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

Some limit theorems for walsh-harmonizable dyadic stationary sequences

This paper deals with a Walsh-harmonizable dyadic stationary sequence {X(k): k=0, 1, 2,…} which is represented as $\text{X(k)=}\text{∫}\text{0}\text{1}\text{ψ}{}_{\mathrm{k}}\text{(λ) dζ(λ)}$, where ψ_k(λ) is the k-th Walsh function and ζ(λ) is a second-order process with orthogonal increments. One of the aims is to express the process {ζ(λ): λ?[0, 1)} in terms of the Walsh–Stieltjes series ∑ X(k)ψ_k(λ) of the original sequence X(k). In order to do this a Littlewood's Tauberian theorem for a series of random variables is introduced. A finite Walsh series expression of X(k) is derived by introducing an approximate Walsh series of X(k). Also derived is a strong law of large numbers for the dyadic stationary sequences.     

Incidence graphs and subdesigns of semisymmetric designs

A semisymmetric design is a connected incidence structure satisfying; two points (blocks) are on 0 or λ blocks (points). Every block (point) is incident with k points (blocks). Properties of the incidence graph of these structures are investigated, leading to bounds on its diameter (d?k if λ = 2, d?[2k/(λ + 1)]+ 1 if λ > 2), and the number of points of these structures (υ?2^k-1 if λ = 2, υ?k2^{[2k/(λ + 1)]} if λ > 2). Bounds are also found for semisymmetric designs containing a subdesign. We give characterizations of semisymmetric designs with λ = 2 (semibiplanes) which contain a subdesign and achieve the bounds. This leads to a construction for a semibiplane with parameters υ = 2^r-1 (q²?1), k = q+q₁+?+q_r, where q_r is aprime power, q_i = q²_i+1 and q=q²₁.     

On the chromaticity of complete multipartite graphs with certain edges added

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t−1)×p,p+k) by adding a set S of s edges to the partite set of size p+k such that 〈S〉 is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K⁺((t−1)×p,p+k). In this paper, we shall prove that for k=0,1 and p+k≥s+2, each graph G∈K(s,t,p,k) is chromatically unique if and only if 〈S〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K⁺((t−1)×p,p+k) is chromatically unique for k=0,1 and p+k≥3.