A New Method for Constructing Williamson Matrices

For every prime power q 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x²+4y² for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q².The research supported by NSF of China (No. 10071029).     

AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

Hamiltonian decomposition of Cayley graphs of ordersp 2 andpq

In this paper, it is proved that any connected Cayley graph on an abelian group of order pq orp ² has a hamiltonian decomposition, wherep andq are odd primes. This result answers partially a conjecture of Alspach concerning hamiltonian decomposition of 2k-regular connected Cayley graphs on abelian groups.     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

A Duality Method in Prediction Theory of Multivariate Stationary Sequences

Let W be an integrable positive Hermitian q × q–matrix valued function on the dual group of a discrete abelian group G such that W^–1 is integrable. Generalizing results of T. Nakazi [N] and of A. G. Miamee and M. Pourahmadi [MiP] for q = 1 we establish a correspondence between trigonometric approximation problems in L²(W) and certain approximation problems in L²(W^–1). The result is applied to prediction problems for q–variate stationary processes over G , inparticular, to the case G = ℤ.     

On R*-sequenceability and symmetric harmoniousness of groups

We introduce a special harmoniousness called symmetric harmoniousness of groups and extend the R^*-sequenceability of abelian groups to nonabelian groups. We prove that the direct product of an R^*-sequenceable group of even order with a symmetric harmonious group of odd order is R^*-sequenceable. Examples of nonabelian R^*-sequenceable groups and nonabelian symmetric harmonious groups are given. It is shown that the nonabelian groups of order 3q (q prime) are symmetric harmonious. © 1994 John Wiley & Sons, Inc.     

Q-matrices and boundedness of solutions to linear complementarity problems

This paper is concerned with the existence and boundedness of the solutions to the linear complementarity problemw=Mz+q,w0,z0,w ^T z=0, for eachq ⁿ . It has been previously established that, ifM is copositive plus, then the solution set is nonempty and bounded for eachq ⁿ iffM is aQ-matrix. This result is shown to be valid also forL ₂-matrices,P ₀-matrices, nonnegative matrices, andZ-matrices.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

Actions of GLq (2,C) on C(1,3) and its four dimensional representations

A complete classification is given of all inner actions on the Clifford algebra C(l,3) defined by representations of the quantum group GL_q (2,C)q^m ≠1, which are not reduced to representations of two commuting “q-spinors”. As a consequence of this classification it is shown that the space of invariants of every GL_q (2,C)-action of this type, which is not an action of SL_q (2,C), is generatedby 1 and the value of the quantum determinant for the given representation.     

On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qⁿ⁻¹ (q—a prime power, n2) over GF(q) are related to symmetric nets in affine 2-(qⁿ,qⁿ⁻¹,(qⁿ⁻¹−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q).     

Tightening Turyn’s bound for Hadamard difference sets

This work examines the existence of (4q ²,2q ²−q,q ²−q) difference sets, for q=p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q ² which has a normal subgroup K of order q such that G/K ≅ C _q ×C ₂×C ₂, where C _q ,C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q ²,2q ²−q,q ²−q) difference sets.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

Permutation polynomials of the formx r f(x q−1)/d) and their group structure

The object of this paper is to give a systematic treatment of permutation polynomials (over a finite fieldF _q ) of the formx ^r f(x ^q–1)/d). In particular, a criterion is obtained for such a polynomial to be a permutation polynomial and it is proved that all such permutation polynomials form a group isomorphic to a generalized wreath product of certain abelian groups.     

Abelian Difference Sets Without Self-conjugacy

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=q^d+1( q^d+ q^d-1 +...+ q+2) ,k=q^d( q^d+q^d-1+...+q+1) , = q^d ( q^(d-1)+q^(d-2)+...+q+1), where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k- = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich's table with answer no.     

Magnetic impurity effect on superconductivity in systems with comparable Fermi and Debye energies

We obtain the basic system of equations for a superconductivity theory of a system with a chaotically distributed paramagnetic substitution impurity in which the Migdal theorem is violated (the condition ω_D ≪E _F cannot be assumed). The electron-phonon and impurity diagrams and also the additional diagrams corresponding to intersections of the electron-phonon and electron-impurity lines are taken into account. In the weak electron-phonon coupling limit, we obtain an equation for the superconducting transition temperature T_C that differs from the corresponding equation for the usual superconductors by renormalizations of T_C0 and of the impurity scattering parameter, ρ. These quantities depend essentially on the Migdal parameter ω_D/E _F and on the transferred momentum q_c. We show that the decrease of T_C with the increase of the impurity concentration is slowed, as compared with usual superconductors, to an extent determined by m and q_c. We also evaluate the isotopic coefficient α, whose behavior as a function of the impurity concentration depends on m and q_c. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp 455–474, June, 1999     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

Walker's Cancellation Theorem

Walker's cancellation theorem says that, if B ⊕ Z is isomorphic to C ⊕ Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).