Infinite families of non‐embeddable quasi‐residual Menon designs

A Menon design of order h² is a symmetric (4h²,2h²‐h,h²‐h)‐design. Quasi‐residual and quasi‐derived designs of a Menon design have parameters 2‐(2h² + h,h²,h²‐h) and 2‐(2h²‐h,h²‐h,h²‐h‐1), respectively. In this article, regular Hadamard matrices are used to construct non‐embeddable quasi‐residual and quasi‐derived Menon designs. As applications, we construct the first two new infinite families of non‐embeddable quasi‐residual and quasi‐derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009     

A Class of 2‐(3n7, 3n−17, (3n−17−1)/2) Designs

Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10¹⁷ nonisomorphic 2‐(63,21,10) designs. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 460–464, 2007     

Constructions for Circulant and Group‐Developed Generalized Weighing Matrices

An elementary construction yields a new class of circulant (so‐called “Butson‐type”) generalized weighing matrices, which have order and weight n², all of whose entries are nth roots of unity, for all positive integers , where . The idea is extended to a wider class of constructions giving various group‐developed generalized weighing matrices.     

New families of non‐embeddable quasi‐derived designs

The main result in this article is a method of constructing a non‐embeddable quasi‐derived design from a quasi‐derived design and an α‐resolvable design. This method is a generalization of techniques used by van Lint and Tonchev in 14 , 15 and Kageyama and Miao in 8 . As applications, we construct several new families of non‐embeddable quasi‐derived designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 263–275, 2008     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Turán problems for integer‐weighted graphs

A multigraph is (k,r)‐dense if every k‐set spans at most r edges. What is the maximum number of edges ex_?(n,k,r) in a (k,r)‐dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r>k³) by determining a constant m=m(k,r) and showing that ex_?(n,k,r)=m +O(n), thus giving a generalization of Turán's theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k,r)‐dense n‐vertex multigraphs with arbitrary integer weights with an O(n) error term. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 195–225, 2002     

Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

On quasi‐orthogonal cocycles

We introduce the notion of quasi‐orthogonal cocycle. This is motivated in part by the maximal determinant problem for square ‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi‐Hadamard groups, relative quasi‐difference sets, and certain partially balanced incomplete block designs, are proved.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Small non‐embeddable quasi‐derived designs

We establish the existence of non‐embeddable quasi‐derived 2‐designs with the parameters (13, 4, 3), (15, 6, 5), and (16, 6, 5). © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 364–372, 2008     

Computable analysis of a boundary‐value problem for the generalized KdV–Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV‐Burgers equation with initial‐boundary value problem. Here, the solution operator is a nonlinear map H^{3m ? 1}(R⁺) × H^m(0,T)→C([0,T];H^{3m ? 1}(R⁺)) from the initial‐boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer m ≥ 2, and computable real number T > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐standard Stokes and Navier–Stokes problems: existence and regularity in stationary case

This paper is devoted to Stokes and Navier–Stokes problems with non‐standard boundary conditions: we consider, in particular, the case where the pressure is given on a part of the boundary. These problems were studied by Bégue, Conca, Murat and Pironneau. They proved the existence of variational solutions, indicating that these were solutions of the initial non‐standard problems, if they are regular enough, but without specifying the conditions on the data which would imply this regularity. In this paper, first we show that the variational solutions, on supposing pressure on the boundary Γ₂ of regularity H^1/2 instead of H^?1/2, have their Laplacians in L² and, therefore, are solutions of non‐standard Stokes problem. Next, we give a result of regularity H², which we generalize, obtaining regularities W^{m, r}, m∈?, m?2, r?2. Finally, by a fixed‐point argument, we prove analogous results for the Navier–Stokes problem, in the case where the viscosity νis large compared to the data. Copyright © 2002 John Wiley & Sons, Ltd.     

An infinite series of regular edge‐ but not vertex‐transitive graphs

Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q?1, or n=2 and q odd, we construct a connected q‐regular edge‐but not vertex‐transitive graph of order 2qⁿ⁺¹. This graph is defined via a system of equations over the finite field of q elements. For n=2 and q=3, our graph is isomorphic to the Gray graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 249–258, 2002     

A class of group divisible 3‐designs and their applications

In this article, we first show that a group divisible 3‐design with block sizes from {4, 6}, index unity and group‐type 2^m exists for every integer m≥ 4 with the exception of m = 5. Such group divisible 3‐designs play an important role in our subsequent complete solution to the existence problem for directed H‐designs DH_λ(m, r, 4, 3)s. We also consider a way to construct optimal codes capable of correcting one deletion or insertion using the directed H‐designs. In this way, the optimal single‐deletion/insertion‐correcting codes of length 4 can be constructed for all even alphabet sizes. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 136–146, 2009     

A recursive construction of t‐wise uniform permutations

We present a recursive construction of a (2t + 1)‐wise uniform set of permutations on 2n objects using a combinatorial design, a t‐wise uniform set of permutations on n objects and a (2t + 1)‐wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t‐wise uniform set of permutations on n objects whose size is at most t²ⁿ, the first non‐trivial construction of an infinite family of t‐wise uniform sets for . If a non‐trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 531–540, 2015     

Johnson‐Lindenstrauss lemma for circulant matrices**

We prove a variant of a Johnson‐Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε^?2 log³ n) instead of the classical bound k = O(ε^?2 log n). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A lower bound on the number of Semi‐Boolean quadruple systems

A Steiner quadruple system of order 2ⁿ is Semi‐Boolean (SBQS(2ⁿ) in short) if all its derived triple systems are isomorphic to the point‐line design associated with the projective geometry PG(n?1, 2). We prove by means of explicit constructions that for any n, up to isomorphism, there exist at least 2^{? 3(n?4)/2?} regular and resolvable SBQS(2ⁿ). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 229–239, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10050     

Perturbation analysis for the generalized Schur complement of a positive semi‐definite matrix

Let and S=C?B^HA^?B be the generalized Schur complement of A?0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR‐95‐38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright © 2007 John Wiley & Sons, Ltd.