Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.     

The Classification of Symmetric Transversal Designs <Emphasis Type="Italic">STD</Emphasis><Subscript>4</Subscript>[12; 3]’s

In this article we prove that there is only one symmetric transversal design STD₄[12;3] up to isomorphism. We also show that the order of the full automorphism group of STD₄[12; 3] is 2⁵· 3³ and Aut STD₄[12;3] has a regular subgroup as a permutation group on the point set. We used a computer for our research.Communicated by: C.J. Colbourn     

Some new (s,k,λ)-translation transversal designs with non-abelian translation group

For >1 and many values of s andk, we give a construction of (s,k,)-partitions of finite non-abelian p-groups and of Frobenius groups with non-abelian kernel. These groups are associated with translation transversl designs of the same parameters.     

Perfect Mendelsohn Packing Designs with Block Size Five

A (v, k, 1) perfect Mendelsohn packing design (briefly (v, k, 1)-PMPD) is a pair (X, A) where X is a v-set (of points) and A is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair of points of X appears t-apart in at most one block of A for all t = 1, 2,..., k-1. If no other such packing has more blocks, the packing is said to be maximum and the number of blocks in a maximum packing is called the packing number, denoted by P(v, k, 1). The values of the function P(v, 5, 1) are determined here for all v 5 with a few possible exceptions. This result is established by means of a result on incomplete perfect Mendelsohn designs which is of interest in its own right.     

Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Cocyclic matrices have the form where G is a finite group, C is a finite abelian group and : G × G C is a (two-dimensional) cocycle; that is,

λ-Automorphisms of a Riemannian foliation

We study geometric properties of -automorphisms of a Riemannian foliationF which is not harmonic. This notion was first introduced in [KTT] for the case whereF is harmonic. Transversal Killing, affine, conformal, projective fields are all examples of -automorphisms. We derive several general identities for a -automorphism. In particular, we extend the results on the transversal conformal and Killing fields obtained in [PrY], [NY1,2]. Furthermore, we analyse the geometric meaning of the condition appearing in our results.The present studies were supported (in part) by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1404     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

Rainbow matchings and Hamilton cycles in random graphs

Let be drawn uniformly from all m‐edge, k‐uniform, k‐partite hypergraphs where each part of the partition is a disjoint copy of . We let be an edge colored version, where we color each edge randomly from one of colors. We show that if and where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in . Here denotes a random edge coloring of with n colors. When n is odd, our proof requires for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016     

Permanents and Determinants of Latin Squares

Let L be a latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 latin subsquares in L can be determined from both and per(L). More generally, we can diagnose from or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic latin squares with equal permanents and equal determinants exist for all orders that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.