Some existence and construction results of polygonal designs

This paper revisits the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so we employ several construction methods — some conventional and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated ‘λ-fold triple systems’. Finally, we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for ‘perfect grouping systems’ which we introduce.     

The evaluation of irreducible polynomial representations of the general linear groups and of the unitary groups over fields of characteristic 0

We describe an efficient method for the computer evaluation of the ordinary irreducible polynomial representations of general linear groups using an integral form of the ordinary irreducible representations of symmetric groups. In order to do this, we first give an algebraic explanation of D. E. Littlewood's modification of I. Schur's construction. Then we derive a formula for the entries of the representing matrices which is much more concise and adapted to the effective use of computer calculations. Finally, we describe how one obtains — using this time an orthogonal form of the ordinary irreducible representations of symmetric groups — a version which yields a unitary representation when it is restricted to the unitary subgroup. In this way we adapt D. B. Hunter's results which heavily rely on Littlewood's methods, and boson polynomials come into the play so that we also meet the needs of applications to physics.     

Two-level fractional factorial designs in blocks of size two for the orthogonal estimation of all main effects and two-factor interactions

In this paper we consider the problem of constructing two-level fractional factorial designs in blocks of size two that allow for the orthogonal estimation of all main effects and two-factor interactions (after adjusting for blocks). This problem has been considered in the literature, e.g., see Yang and Draper (2003), Wang (2004) and Kerr (2006). In this paper we give two systematic methods for the construction of such designs. The first construction method gives in many situations designs requiring fewer runs than those designs previously given whereas the second method gives a systematic method for constructing designs analogous to those illustrated in Yang and Draper (2003) by example.     

Semi-regular divisible designs and Frobenius groups

We study and characterize semi-regular (s, k, λ₁, λ₂)-divisible designs which admit a Frobenius group as their translation group. Moreover, we give a construction method for such designs by generalized admissible triads.     

Designs and Codes: An update

Since the publication in 1992 of ``Designsand their Codes" significant progress has been made in the generalarea of codes coming from designs. This article reviews thisprogress and presents some of the results — including confirmationof certain conjectures made and answers to some of the questionsraised in the book.     

MDS Self-Dual Codes over Large Prime Fields

Combinatorial designs have been used widely in the construction of self-dual codes. Recently a new method of constructing self-dual codes was established using orthogonal designs. This method has led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we generalize this method by using generalized orthogonal designs, and we give another new method that creates and solves Diophantine equations over GF(p) in order to find suitable generator matrices for self-dual codes. We show that under the necessary conditions these methods can be applied as well to small and large fields. We apply these two methods to study self-dual codes over GF(31) and GF(37). Using these methods we obtain some new maximum distance separable self-dual codes of small orders.     

Price caps

Price-cap regulation of AT&T, which became effective on July 1, 1989, is an example of an idea that made its way from economic theory to institutional practice; in this case the process took about seven years. We describe both price-cap regulation and its predecessor — rate-of-return regulation — with particular regard to their incentive properties. We then give the assumptions and conclusions of a theoretical study (in fact a principal-agent model) that bears on the likely effectiveness of price-cap regulation. In the concluding section we describe some aspects of the progress of this idea from theory to practice, and draw tentative conclusions about the conditions that made it possible for this progress to be successfully completed.The views expressed here are the authors', and not necessarily those of AT&T Bell Laboratories.     

Szegö-Kolmogorov-Krein theorems on weighted trigonometrical approximation and Cárleman-type relations

We consider the well-known Szegö — Kolmogorov — Krein theorems on weighted approximations by functions with semibounded spectra defined on a circle or on a line and suggest an efficient construction that realizes these approximations. This construction is based on relations similar to the Cárleman formula for reconstructing analytic functions in terms of their traces on the boundary of their domains of definition.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 100–127, January–February. 1994.     

Paley type group schemes and planar Dembowski–Ostrom polynomials

In this paper we give some necessary and sufficient conditions for Dembowski–Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter–Matthews and Ding–Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha–Kyureghyan–Wang commutative semifields. We then give a generalization of Feng’s construction of Paley type group schemes in extra-special p-groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.     

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Supported by DFG.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 158–191, July, 1995.     

Some remarks on Maltsev and Goursat categories

Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.     

Variations on seven points: An introduction to the scope and methods of coding theory and finite geometries

We use a simple example (the projective plane on seven points) to give an introductory survey on the problems and methods in finite geometries — an area of mathematics related to geometry, combinatorial theory, algebra, group theory and number theory as well as to applied mathematics (e.g., coding theory, information theory, statistical design of experiments, tomography, cryptography, etc.). As this list already indicates, finite geometries is — both from the point of view of pure mathematics and from that of applications related to computer science and communication engineering — one of the most interesting and active fields of mathematics. It is the aim of this paper to introduce the nonspecialist to some of these aspects.To Professor Günter Pickert on the occasion of his 65th birthday     

Balinski—Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces

This paper shows the relationship between degeneracy degrees and multiple solutions in linear programming (LP) models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for LP-models for which the feasible region contains at least one vertex it holds that the dimension of the optimal face is equal to the degeneracy degree of the optimal face of the corresponding dual model. This result is obtained by means of the so-called Balinski—Tucker (B—T) Simplex Tableaus. Furthermore, we give a strong polynomial algorithm for constructing such a B—T Simplex Tableau when a solution in the relative interior of the optimal face is known. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Ebene absolute geometrie über Euklidischen Körpern

We give a set of seven postulates for plane absolute geometry based on the geometric notions points and lines and show that every absolute plane is — up to isomorphisms — either a Euclidean or a hyperbolic plane over a Euclidean field.     

Double Darboux Transformation for AKNS hierarchy

In this paper we construct a double Darboux Transformation for AKNS hierarchy and give its decomposition theorem. A remarkable characteristic of the double Darboux Transformation — incommutativity is proved.The Project Supported by National Natural Science Foundation of China.     

Zur Konstruktion eindimensionaler homogener Markoffscher Prozesse

Summary The aim of this paper is the construction of a class of semi-groups of homogeneous onedimensional Markov processes with respect to a given infinitesimal operator. This is done namely by the method of stochastic integration represented in the book of Skorokhod. The Lipschitzian conditions — needed there — are weakened to uniform continuity by a method of approximation of semi-groups. Also to these semi-groups we can construct Markov processes as solutions of stochastic integral equations. The representation is connected with the result of Meyer, Watanabe, Motoo.     

Partially balanced incomplete block designs from weakly divisible nearrings

In [[6] Riv. Mat. Univ. Parma 11 (2) (1970) 79-96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

The boundary characteristic and the volume of lattice polyhedra

The boundary characteristic — introduced by Ding and Reay — is a functional defined for a given planar tiling which associates with a given lattice figure, some integer. It appeared to be a very useful parameter to determine the area of lattice figures in the planar tilings with congruent regular polygons. The purpose of this paper is to extend the notion of the boundary characteristic to lattice polyhedra inR³. Studying some of its properties we show, in particular, that it can be applied to determine the volume of lattice polyhedra.     

The ternary rings of Desarguesian and Pappian planes — A simple proof using perspectivities

The aim of this paper is to give a direct, simple proof of the well-known theorems — that the Hall ternary ring (R, T) of a Pappian projective plane is a linear ternary ring over a field, and that of a Desarguesian plane is a linear one over a skew field — by making repeated application of the perspectivity theorem in a Pappian plane and the characterization of Desarguesian planes in terms of perspectivities.This is a revised version of the paper The Ternary Ring of a Pappian Plane — A Simple Proof presented at the 49th Conference of the Indian Mathematical Society, held at Madras, December 27–29, 1983.