Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

We prove that a certain binary linear code associated with the incidence matrix of a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self‐dual code of length 40. Using the classification of extremal doubly even self‐dual codes of length 40, we show that a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 does not exist.     

The existence of extremal self-dual [50,25,10] codes and quasi-symmetric 2-(49,9,6) designs

All extremal binary self-dual [50,25,10] codes with an automorphism of order 7 are enumerated. Up to equivalence, there are four such codes, three with full automorphism group of order 21, and one code with full group of order 7. The minimum weight codewords yield quasi-symmetric 2-(49,9,6) designs.Research supported in part by NSA grant MDA904-95-H-1019.     

Mutually Disjoint 5‐Designs and 5‐Spontaneous Emission Error Designs from Extremal Ternary Self‐Dual Codes

It is known that extremal ternary self‐dual codes of length mod 12) yield 5‐designs. Previously, mutually disjoint 5‐designs were constructed by using single known generator matrix of bordered double circulant ternary self‐dual codes (see [1, 2]). In this paper, a number of generator matrices of bordered double circulant extremal ternary self‐dual codes are searched with the aid of computer. Using these codes we give many mutually disjoint 5‐designs. As a consequence, a list of 5‐spontaneous emission error designs are obtained.     

Pseudo quasi‐3 designs and their applications to coding theory

We define a pseudo quasi‐3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi‐symmetric. Quasi‐symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi‐3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 411–418, 2009     

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     

Some Hadamard matrices of order 32 and their binary codes

It is known that all doubly‐even self‐dual codes of lengths 8 or 16, and the extended Golay code, can be constructed from some binary Hadamard matrix of orders 8, 16, and 24, respectively. In this note, we demonstrate that every extremal doubly‐even self‐dual [32,16,8] code can be constructed from some binary Hadamard matrix of order 32. © 2004 Wiley Periodicals, Inc.     

Orthogonal designs,self‐dual codes,and the Leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self‐dual codes. Orthogonal designs are shown to be useful in construction of self‐dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self‐dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.     

On the self‐dual 𝔽5‐codes constructed from Hadamard matrices of order 24

There are exactly 60 inequivalent Hadamard matrices of order 24. In this note, we give a classification of the self‐dual ??₅‐codes of length 48 constructed from the Hadamard matrices of order 24. © 2004 Wiley Periodicals, Inc.     

A technique for constructing non‐embeddable quasi‐residual designs

We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3^m,3^m?1,(3^m?1?1)/2)‐design, and, if r_m=(3^m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900     

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     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A note on triangle‐free quasi‐symmetric designs

Triangle‐free quasi‐symmetric 2‐ (v, k, λ) designs with intersection numbers x, y; 0<x<y<kand λ>1, are investigated. It is proved that λ?2y ? x ? 3. As a consequence it is seen that for fixed λ, there are finitely many triangle‐free quasi‐symmetric designs. It is also proved that: k?y(y ? x) + x. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:422‐426, 2011     

Nontopological N‐vortex condensates for the self‐dual Chern‐Simons theory

We prove the existence of nontopological N‐vortex solutions for an arbitrary number N of vortex points for the self‐dual Chern‐Simons‐Higgs theory with 't Hooft “periodic” boundary conditions. We use a shadowing‐type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the Liouville equation. © 2003 Wiley Periodicals, Inc.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The multipliers‐free domain decomposition methods

This article concludes the development and summarizes a new approach to dual‐primal domain decomposition methods (DDM), generally referred to as “the multipliers‐free dual‐primal method.” Contrary to standard approaches, these new dual‐primal methods are formulated without recourse to Lagrange‐multipliers. In this manner, simple and unified matrix‐expressions, which include the most important dual‐primal methods that exist at present are obtained, which can be effectively applied to floating subdomains, as well. The derivation of such general matrix‐formulas is independent of the partial differential equations that originate them and of the number of dimensions of the problem. This yields robust and easy‐to‐construct computer codes. In particular, 2D codes can be easily transformed into 3D codes. The systematic use of the average and jump matrices, which are introduced in this approach as generalizations of the “average” and “jump” of a function, can be effectively applied not only at internal‐boundary‐nodes but also at edges and corners. Their use yields significant advantages because of their superior algebraic and computational properties. Furthermore, it is shown that some well‐known difficulties that occur when primal nodes are introduced are efficiently handled by the multipliers‐free dual‐primal method. The concept of the Steklov–Poincaré operator for matrices is revised by our theory and a new version of it, which has clear advantages over standard definitions, is given. Extensive numerical experiments that confirm the efficiency of the multipliers‐free dual‐primal methods are also reported here. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

6‐regular 4‐critical graph

P. Erd?s conjectured in [2] that r‐regular 4‐critical graphs exist for every r ≥ 3 and noted that no such graphs are known for r ≥ 6. This article contains the first example of a 6‐regular 4‐critical graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 286–291, 2002     

Nonlinear Stability of Expanding Star Solutions of the Radially Symmetric Mass‐Critical Euler‐Poisson System

We prove nonlinear stability of compactly supported expanding star solutions of the mass‐critical gravitational Euler‐Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expansion rate of such solutions can be either self‐similar or non‐self‐similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global‐in‐time radially symmetric solutions, which are not homologous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free‐boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass‐critical with respect to an invariant rescaling and the analysis is carried out in similarity variables. © 2017 Wiley Periodicals, Inc.     

New extremal doubly-even [64, 32, 12] codes

In this paper, we consider a general construction of doubly-even self-dual codes. From three symmetric 2-(31, 10, 3) designs, we construct at least 3228 inequivalent extremal doubly-even [64, 32, 12] codes. These codes are distinguished by their K-matrices.     

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Extremal Self-Dual [50, 25, 10] Codes with Automorphisms of Order 3 and Quasi-Symmetric 2-(49, 9,6) Designs

Five non-isomorphic quasi-symmetric 2-(49, 9, 6) designs are known. They arise from extremal self-dual [50, 25, 10] codes with a certain weight enumerator. Four of them have an automorphism of order 3 fixing two points. In this paper, it is shown that there are exactly 48 inequivalent extremal self-dual [50, 25, 10] code with this weight enumerator and an automorphism of order 3 fixing two points. 44 new quasi-symmetric 2-(49, 9, 6) designs with an automorphism of order 3 are constructed from these codes.