On the bounded domination number of tournaments

In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γ_k(G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γ_k(T)=n/(k+1) when T is a tournament with n14k lg k vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lg n−lglg n+2 vertices.     

Divisible difference families from Galois rings GR(4,n) and Hadamard matrices

We give a new construction of difference families generalizing Szekeres’s difference families Szekeres (Enseignment Math 15:269–278, 1969). As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring \(GR(4,n)\) . Furthermore, we obtain a new construction method of symmetric Hadamard matrices by using divisible difference families and a new array.     

A constructive solution to a problem of ranking tournaments

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd?s and J. W. Moon. By probabilistic methods, the existence of ?? ?? unrankable” tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.     

On the Williamson type j matrices of orders 4·29, 4·41, and 4·37

Hadamard matrices of the Williamson type invariant under an automorphism of order 2 are considered. A new Hadamard matrix of order 148 of this type is obtained.     

经典Hadamard不等式的高维推广

在n维Euclid空间利用多重积分的一般Stokes公式,将一元凸函数的经典H adam ard不等式在高维空间一般凸区域上进行了推广,得到了相应的高维Hadamard型不等式.这个结果蕴涵了经典的H adam ard不等式以及几个特殊凸体上的H adam ard型不等式.     

Order Types of Convex Bodies

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies.     

On the classification of Hadamard matrices of order 32

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13, 680, 757 Hadamard matrices of one type and 26, 369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant. © 2009 Wiley Periodicals, Inc. J Combin Designs 18:328–336, 2010     

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+A^t = J ? I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).     

Hadamard matrices with few distinct types

The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. In this paper, we investigate Hadamard matrices with few distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types.     

Hadamard matrices of order 764 exist

Two Hadamard matrices of order 764 of Goethals-Seidel type are constructed. The author was supported by an NSERC Discovery Grant.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

An infinite class of Williamson matrices

Only a finite number of Hadamard matrices of Williamson type are known so far; it has been conjectured that one such exists of any order 4t. An infinite family is constructed here, and as a corollary it is shown that an Hadamard matrix of order 6(q + 1) exists if q is a prime power ≡ 1 (mod 4).     

Analysis of sensitivity of reciprocal matrices

We develop a method based on the Hadamard Product of matrices to analyze the sensitivity of reciprocal matrices. We show that this type of matrices can be decomposed into the Hadamard Product of a consistent matrix and an inconsistent matrix. The consistent matrix has the same principal eigenvector as the original matrix, and the inconsistent matrix has the same principal eigenvalue as the original one. We use this decomposition in the analysis of sensitivity to compute the principle eigenvector of a perturbed reciprocal matrix.     

On the shielding effect of the Helmholtz equation

Hadamard‐type instability has been known for over a century as a cause of ill‐posedness of the Cauchy problem for elliptic PDEs. This ill‐posedness manifests itself as evanescent modes growing exponentially when propagated in the reverse direction. Since every oscillating mode of the Laplace equation is evanescent, the ill‐posedness of its Cauchy problem is solely due to Hadamard‐type instability. The presence of the propagating modes and beams for the Helmholtz equation gives rise to an entirely different type of ill‐posedness, hitherto unknown to the practice, and untreated by the theory, of inverse scattering. We will present this fundamental phenomenon of ill‐posedness for the Helmholtz equation. © 2007 Wiley Periodicals, Inc.     

Hadamard matrices,Baumert-Hall units,four-symbol sequences,pulse compression,and surface wave encodings

If a Williamson matrix of order 4w exists and a special type of design, a set of Baumert-Hall units of order 4t, exists, then there exists a Hadamard matrix of order 4tw. A number of special Baumert-Hall sets of units, including an infinite class, are constructed here; these give the densest known classes of Hadamard matrices. The constructions relate to various topics such as pulse compression and image encodings.     

Irreducible characters of Hadamard algebras

The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classicalHadamard matrix, which corresponds to the case of commutative algebras. The algebras admitting Hadamard decompositions are called Hadamard algebras. A relation for the values of an irreducible character of a Hadamard algebra on the products of involutions forming an orthogonal basis of the algebra is obtained. This relation is then applied to describe the Hadamard decompositions in an algebra of dimension 8.     

On Erdős-Szekeres-type problems

In this paper, we will present some of our recent results concerning the classical Erdős – Szekeres problem in combinatorial geometry     

Unified Approaches to Well-Posedness with Some Applications

We present unified approaches to Hadamard and Tykhonov well-posedness. As applications, we deduce Tykhonov well- posedness for optimization problems, Nash equilibrium point problems and fixed point problems etc. Especially, by applying such approaches, we deal with the well- posedness as stated in (Lignola and Morgan (2000), Journal of Global Optimization 16, 57–67) in which Lignola and Morgan investigated directly and intensively Tykhonov types of well- posedness for optimization problems with constraints defined by variational inequalities, namely, generalized well- posedness and strong well- posedness. We give some sufficient conditions for Hadamard well- posedness of such problems and deduce relations between Hadamard type and Tykhonov type of well- posedness. Finally, as corollaries, we derive generalized well- posedness and strong well- posedness for these problems.     

On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property

Further progress is achieved for the growth conjecture for Hadamard matrices. It is proved that the leading principal minors of a CP Hadamard matrix form an increasing sequence. Bounds for the sixth and seventh pivot of any CP Hadamard matrix are given. A new proof demonstrating that the growth of a Hadamard matrix of order 12 is 12, is presented. Moreover, a new notion of good pivots is introduced and its importance for the study of the growth problem for CP Hadamard matrices is examined. We establish that CP Hadamard matrices with good pivots satisfy Cryer’s growth conjecture with equality, namely their growth factor is equal to their order. A construction of an infinite class of Hadamard matrices is proposed.     

Szasz不等式在亚正定矩阵和拟广义正定矩阵上的推广

实对称正定矩阵的Szasz不等式是Hadamard不等式的加细;本文将Szasz不等式推广到一类亚正定矩阵和拟广义正定矩阵上去,从而推广了关于实对称正定矩阵的Szasz不等式和Hadamard不等式.