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.      

Strongly θ-b-continuous functions

Summary We introduce a new class of functions called strongly θ-b-continuous function which is a generalization of both strongly θ-precontinuous functions<span lang=EN-US style='font-size:10.0pt; mso-ansi-language:EN-US'>[16]and strongly θ-semicontinuous functions [7]. Some characterizations and several properties concerning strongly θ-b-continuous functions are obtained.     

Hypersurfaces of prescribed mean curvature enclosing a given body

Given a smooth domain Ω in ℝ ^m+1 with compact closure and a smooth integrable functionh: ℝ ^m+1→ℝ satisfyingh(x)≥H _∂Ω (x) on ∂Ω whereH _∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ ^m+1 with the properties andH _∂A=h on ∂A.     

A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound

Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ 24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, as the union of a λ₁-dimensional space, λ₂-dimensional space, ..., λ _h -dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove that a weighted -minihyper , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, is a sum of a λ₁-dimensional space, λ₂-dimensional space, ..., and λ _h -dimensional space. This research was supported by the Project Combined algorithmic and theoretical study of combinatorial structures between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Sciences. This research is also part of the FWO-Flanders project nr. G.0317.06 Linear codes and cryptography.     

On the Existence of Towers in Pseudo-Tree Algebras

A tower in a Boolean algebra (BA) is a strictly increasing sequence, of regular order type, of elements of the algebra different from 1 but with sum 1. A pseudo-tree is a partially ordered set T such that the set T↓t = {s ∈ T:s < t} is linearly ordered for every t ∈ T. If that set is well-ordered, then T is a tree. For any pseudo-tree T, the BA Treealg(T) is the algebra of subsets of T generated by all of the sets T↑t = {s ∈ T:t ≤ s}. The main theorem of this note is a characterization in tree terms of when Treealg(T) has a tower of order type κ (given in advance).     

Generalized subdifferential of the distance function

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space with norms on and locally uniformly convex.

     

Codebooks from almost difference sets

In direct spread CDMA systems, codebooks meeting the Welch bounds are used to distinguish among the signals of different users. Recently, constructions of codebooks with difference sets meeting Welch’s bound on the maximum cross-correlation amplitude were developed. In this paper, a generic construction of codebooks using almost difference sets is considered and several classes of codebooks nearly meeting the Welch bound are obtained. The parameters of the codebooks constructed in this paper are new.      

Convex Sets in Acyclic Digraphs

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with and . We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo.     

Tight Beltrami fields with symmetry

Let M be a compact orientable Seifered fibered 3-manifold without a boundary, and α an S ¹-invariant contact form on M. In a suitable adapted Riemannian metric to α, we provide a bound for the volume Vol(M) and the curvature, which implies the universal tightness of the contact structure ξ = ker α.      

Multifractal Decomposition of Statistically Self-Similar Sets

Let K be a statistically self-similar set defined by Graf. In this paper, we construct a random measure ρ which is supported by K and study the multifractal decomposition for K with ρ. Under such a decomposition, we obtain the expression of the spectrum function ƒ(α). Received July 18, 1997, Accepted October 13, 2000     

CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂ ⁿ , can be graphed over a convex subset in ℝ²ⁿ⁻¹, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂ ⁿ in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension. R. Dwilewicz is partially supported by the Polish Science Foundation (KBN) grant N201 019 32/805.     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

In this paper we consider the problem of bounding the Betti numbers, b _i (S), of a semi-algebraic set S⊂ℝ ^k defined by polynomial inequalities P ₁≥0,…,P _s ≥0, where P _i ∈ℝ[X ₁,…,X _k ], s<k, and deg (P _i )≤2, for 1≤i≤s. We prove that for 0≤i≤k−1,

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Computing the Top Betti Numbers of Semialgebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any ℓ>0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P ₁≤0,…,P _s ≤0, where each P _i ∈R[X ₁,…,X _k ] has degree≤2, and computes the top ℓ Betti numbers of S, b _k−1(S),…,b _k−ℓ (S), in polynomial time. The complexity of the algorithm, stated more precisely, is . For fixed ℓ, the complexity of the algorithm can be expressed as , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic sets in R ^k defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain, by letting ℓ=k, an algorithm for computing all the Betti numbers of S whose complexity is . An erratum to this article can be found at     

A note on recursive completions of the countable atomless Boolean algebras

The Baire category theorem for compact Hausdorff spaces is used to show that any recursive completion ℬ^ϕof the countable atomless Boolean algebra ℬ is ℬrecursively closed." Also, a theorem of R.D. Anderson on the simplicity of Aut(ℬ) is used to show that uncountably many automorphisms of ℬ extend to ℬ^ϕ. This revised version was published online in June 2006 with corrections to the Cover Date.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Transformational antiatomic characteristic of ordered sets

To any ordered set with a universally maximal element, a semigroup of its transformations with some natural properties that defines the ordered set up to an isomorphism is assigned. The system of such transformation semigroups is proved to be the minimal element in the set of all defining systems of transformation semigroups with respect to the following ordering: one system precedes another if for each ordered set from the class in question, the semigroup of its transformation belonging to the first system is contained in the semigroup of its transformation from the second system. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 112–119, July, 1999.     

Ascoli-Arzelà type theorem for rational iterations on the complex projective plane

For a dominant algebraically stable rational self-map of the complex projective plane of degree at least 2, we will consider three different definitions of the Fatou set and show the equivalence of them (Ascoli-Arzelà type theorem). As a corollary, it follows that all Fatou components are Stein. This is an improvement of an early result by Fornæss and Sibony.

     

Homomorphism-Homogeneous Partially Ordered Sets

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between partially ordered sets A and B is a mapping f : A →B satisfying ). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X ₅-dense locally bounded partially ordered sets. Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017.