Perfect Mendelsohn Packing Designs with Block Size Five

A (v, k, 1) perfect Mendelsohn packing design (briefly (v, k, 1)-PMPD) is a pair (X, A) where X is a v-set (of points) and A is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair of points of X appears t-apart in at most one block of A for all t = 1, 2,..., k-1. If no other such packing has more blocks, the packing is said to be maximum and the number of blocks in a maximum packing is called the packing number, denoted by P(v, k, 1). The values of the function P(v, 5, 1) are determined here for all v 5 with a few possible exceptions. This result is established by means of a result on incomplete perfect Mendelsohn designs which is of interest in its own right.     

q‐pseudoconvex and q‐holomorphically convex domains

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q‐pseudoconvexity and q‐holomorphic convexity: we prove that any open subset $\mathrm{\Omega }?{\mathbb{C}}^n$ with smooth boundary, strictly q‐pseudoconvex, is $(q+1)$‐holomorphically convex; moreover, assuming that Ω verifies an additional assumption, we prove that it is q‐holomorphically convex. We also prove that any open subset of ${\mathbb{C}}^n$ is n‐holomorphically convex.     

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

q‐Voronovskaya type theorems for q‐Baskakov operators

In the present paper, we prove quantitative q‐Voronovskaya type theorems for q‐Baskakov operators in terms of weighted modulus of continuity. We also present a new form of Voronovskaya theorem, that is, q‐Grüss‐Voronovskaya type theorem for q‐Baskakov operators in quantitative mean. Hence, we describe the rate of convergence and upper bound for the error of approximation, simultaneously. Our results are valid for the subspace of continuous functions although classical ones is valid for differentiable functions. Copyright © 2015 John Wiley & Sons, Ltd.     

On the eigenvectors of the q‐Bernstein operators

In this article, both the eigenvectors and the eigenvalues of the q‐Bernstein operators have been studied. Explicit formulae are presented for the eigenvectors, whose limit behavior is determined both in the case 0 < q < 1 and in the case q > 1. Because the classical case, where q = 1, was investigated exhaustively by S. Cooper and S. Waldron back in 2000, the present article also discusses the related similarities and distinctions with the results in the classical case. Copyright © 2013 John Wiley & Sons, Ltd.     

Higher order limit q‐Bernstein operators

In this paper we give the estimates of the central moments for the limit q‐Bernstein operators. We introduce the higher order generalization of the limit q‐Bernstein operators and using the moment estimations study the approximation properties of these newly defined operators. It is shown that the higher order limit q‐Bernstein operators faster than the q‐Bernstein operators for the smooth functions defined on [0, 1]. Copyright © 2011 John Wiley & Sons, Ltd.     

A Note on q‐Covering Designs in PG(5, q)

A construction of q‐covering designs in PG(5, q) is given, providing an improvement on the upper bound of the q‐covering number .     

Packing Arrays and Packing Designs

A packing array is a b × k array, A with entriesa _i,j from a g-ary alphabet such that given any two columns,i and j, and for all ordered pairs of elements from a g-ary alphabet,(g ₁, g ₂), there is at most one row, r, such thata _r,i = g ₁ anda _r,j = g ₂. Further, there is a set of at leastn rows that pairwise differ in each column: they are disjoint. A central question is to determine, forgiven g, k and n, the maximum possible b. We examine the implications whenn is close to g. We give a brief analysis of the case n = g and showthat 2g rows is always achievable whenever more than g exist. We give an upper bound derivedfrom design packing numbers when n = g – 1. When g + 1 k then this bound is always at least as good as the modified Plotkin bound of [12]. When theassociated packing has as many points as blocks and has reasonably uniform replication numbers, we show thatthis bound is tight. In particular, finite geometries imply the existence of a family of optimal or near optimalpacking arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when .     

Self-Complementary Balanced Codes and Quasi-Symmetric Designs

An upper bound for self-complementary balanced codes is presented in this paper. We give a characterization for self-complementary balanced codes meeting this upper bound. We show that the existence of certain quasi-symmetric designs implies the existence of such optimal self-complementary balanced codes.     

Statistically q‐deformed and tau‐deformed systems

Two classes of statistically deformed systems are known in literature. They are, respectively, the q‐deformed systems (Lavagno and Narayana Swamy, Phys Rev E 2002, 65, 036101) and the κ‐deformed systems (Kaniadakis and Scarfone, Physica A 2002, 305, 69). In this article, a new class, i.e., the tau‐deformed systems, is introduced. For each of these systems, a consistent thermodynamics may be developed. A summary of the main similarities between the thermodynamic properties of q‐deformed and tau‐deformed systems is presented. The deformation outlined in this article is radically different from the nonextensive Tsallis statistics, where the structure of the entropy is rather arbitrary deformed via the logarithmic function. In contrast, the theory of tau‐deformed systems is developed on a purely physical basis. However, one finally shows that the tau‐systems may be described by using a new form of deformed logarithmic function. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

Asymptotical stability analysis of Riemann‐Liouville q‐fractional neutral systems with mixed delays

This paper investigates the asymptotical stability of Riemann‐Liouville q‐fractional neutral systems with mixed delays (constant time delay and distributed delay). By constructing some appropriate Lyapunov‐Kravsovskii functionals, some sufficient conditions on delay‐dependent and delay‐independent asymptotical stability are obtained in terms of linear matrix inequality (LMI). Our employed method is based on the direct calculation of quantum derivatives of the Lyapunov‐Kravsovskii functionals. Finally, two examples are presented to demonstrate the availability of our obtained results.     

On the image of the limit q‐Bernstein operator

The limit q‐Bernstein operator B_q emerges naturally as an analogue to the Szász–Mirakyan operator related to the Euler distribution. Alternatively, B_q comes out as a limit for a sequence of q‐Bernstein polynomials in the case 0<q<1. Lately, different properties of the limit q‐Bernstein operator and its iterates have been studied by a number of authors. In particular, it has been shown that B_q is a positive shape‐preserving linear operator on C[0, 1] with ∥B_q∥=1, which possesses the following remarkable property: in general, it improves the analytic properties of a function. In this paper, new results on the properties of the image of B_q are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Jacobi Polynomials, Type II Codes, and Designs

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.     

On the Enumeration of ‐Optimal and Minimax‐Optimal k‐Circulant Supersaturated Designs

In recent years, several methods have been proposed for constructing ‐optimal and minimax‐optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, ‐optimal and minimax‐optimal k‐circulant SSDs with 6, 10, 14, 18, 22, and 26 runs, factors and are enumerated in a computer search. We have also enumerated all ‐optimal and minimax‐optimal k‐circulant SSDs with (mod 4) and . The computer search utilizes the fact that theses designs are equivalent to certain 1‐rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.     

Packing of graphic n‐tuples

An n‐tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v₁, …, v_n} in which the degree of v_i is the ith entry of π. Graphic n‐tuples (d, …, d) and (d, …, d) pack if there are edge‐disjoint n‐vertex graphs G₁ and G₂ such that d(v_i) = d and d(v_i) = d for all i. We prove that graphic n‐tuples π₁ and π₂ pack if , where Δand δdenote the largest and smallest entries in π₁ + π₂ (strict inequality when δ = 1); also, the bound is sharp. Kundu and Lovász independently proved that a graphic n‐tuple π is realized by a graph with a k‐factor if the n‐tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge‐disjoint 1‐factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k?3. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Super-simple (ν, 5, 5) Designs

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (ν, 5, 5) balanced incomplete block design and show that such a design exists if and only if ν ≡ 1 (mod 4) and ν ≥ 17 except possibly when ν = 21. Applications of the results to optical orthogonal codes are also mentioned. Research supported by NSERC grant 239135-01.     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.