Primitive symmetric designs with prime power number of points

In this paper we either prove the non‐existence or give explicit construction of primitive symmetric (v, k, λ) designs with v=p^m<2500, p prime and m>1. The method of design construction is based on an automorphism group action; non‐existence results additionally include the theory of difference sets, multiplier theorems in particular. The research involves programming and wide‐range computations. We make use of software package GAP and the library of primitive groups which it contains. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 141–154, 2010     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Geometric groups. I

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.

     

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time–frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory. The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms, and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames. In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory.

     

Extension and Restriction Principles for the HRT Conjecture

The HRT (Heil–Ramanathan–Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on \(\mathbb {R}\) is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following. Suppose that the HRT is true for a given set of N points and a given function. We identify the set of all new points such that the conjecture remains true for the same function and the set of \(N+1\) points obtained by adding one of these new points to the original set. To achieve this we introduce a real-valued function whose global maximizers describe when the HRT is true. To motivate this new approach we re-derive a special case of the HRT for sets of 3 points. Subsequently, we establish new results for points in (1, n) configurations, and for a family of symmetric (2, 3) configurations. Furthermore, we use these results and the refinements of other known ones to prove that the HRT holds for certain families of 4 points.

     

The Maximal Closed Classes of Unary Functions in p-Valued Logic

In many-valued logic the decision of functional completeness is a basic and important problem, and the thorough solution to this problem depends on determining all maximal closed sets in the set of many-valued logic functions. It includes three famous problems, i.e., to determine all maximal closed sets in the set of the total, of the partial and of the unary many-valued logic functions, respectively. The first two problems have been completely solved ([1], [2], [8]), and the solution to the third problem boils down to determining all maximal subgroups in the k-degree symmetric group S_k, which is an open problem in the finite group theory. In this paper, all maximal closed sets in the set of unary p-valued logic functions are determined, where p is a prime. Mathematics Subject Classification: 03B50, 20B35.     

A very special nonexistence theorem for abelian difference sets

It is wellknown that the technique of character sums together with the tools of algebraic number theory is the adequate method for the study of difference sets in abelian groups, compare for instance Ott [5] or Turyn [6]. In this paper we use this method to prove a new non-existence theorem for certain difference sets in abelian groups of order rp^a rp^a , where r 1 2 r \neq 2 and p are distinct primes.     

The configurations 123 revisited

As is known, there are 229 symmetric configurations 12₃, (Daublebsky von Sterneck in Monatshefte Math Phys 5:223–255, 1895; Gropp in J Comb Inf Syst Sci 15:34–48, 1990). We use tactical decompositions by automorphism group (TDA) to study these configurations in detail. In (Daublebsky von Sterneck in Monatshefte Math Phys 14, 254–260, 1903) the automorphism groups of the configurations were determined. We find some errors there and correct them. For the configurations with a rather big automorphism group, we give models which display the structure of the group.     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

On the structure of pseudo-Riemannian symmetric spaces

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semisimple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a triple consisting of:

The lie n-engel property in group rings

Let FGbe the group ring of a group Gover a field Fwhose characteristic is p≠ 2 Let ? denote the involution on FGwhich sends each group element to its inverse. Let (FG)⁺and (FG)^–denote, respectively, the sets of symmetric and skew elements with respect to ?.The conditions under which the group ring is Lie n-Engel for some nare known.We show that if either (FG)⁺or (FG)^–- is Lie n-Engel, and Gis devoid of 2-elements, then FGis Lie m-Engel for some m. Furthermore, we completely classify the remaining groups for which (FG)⁺is Lie n-Engel.     

On the Loewy Length and the Noetherian Dimension of Artinian Modules

The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions.     

On level sets of smooth functions

Based on the established earlier general estimation method of the lengths of level sets of real functions, the paper proves a theorem which is an analog of the second fundamental theorem of the theory of Gamma-lines which, in its turn, is an analog of the second fundamental theorem of R. Nevanlinna.     

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO (n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO (n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

The Round Functions of Cryptosystem PGM Generate the Symmetric Group

S. S. Magliveras et al. have described symmetric and public key cryptosystems based on logarithmic signatures (also known as group bases) for finite permutation groups. In this paper we show that if G is a nontrivial finite group which is not cyclic of order a prime, or the square of a prime, then the round (or encryption) functions of these systems, that are the permutations of G induced by the exact-transversal logarithmic signatures (also known as transversal group bases), generate the full symmetric group on G. This answers a question of S. S. Magliveras, D. R. Stinson and Tran van Trung. AMS Classification:94A60, 20B15, 20B20     

Diophantine geometry over groups V1: Quantifier elimination I

This paper is the first part (out of two) of the fifth paper in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of anAE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra ofAE sets defined over a free group is invariant under projections, and hence show that every elementary set defined over a free group is in the Boolean algebra ofAE sets. The procedures we use for quantifier elimination, presented in this paper and its successor, enable us to answer affirmatively some of Tarski's questions on the elementary theory of a free group in the sixth paper of this sequence. Partially supported by an Israel Academy of Sciences Fellowship.     

Diophantine geometry over groups V2: quantifier elimination II

This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of an AE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra of AE sets defined over a free group is invariant under projections, hence, show that every elementary set defined over a free group is in the Boolean algebra of AE sets. The procedures we use for quantifier elimination, presented in this paper, enable us to answer affirmatively some of Tarski’s questions on the elementary theory of a free group in the last paper of this sequence. Received (resubmission): January 2004 Revision: November 2005 Accepted: March 2006 Partially supported by an Israel Academy of Sciences Fellowship.