A note on a conjecture of Ryser

A special case of a conjecture of Ryser states that if a 3-partite 3-uniform hypergraph has at mostv pairwise disjoint edges then there is a set of vertices of cardinality at most 2v meeting all edges of the hypergraph. The best known upper bound for the size of such a set is (8/3)v, given by Tuza [7]. In this note we improve this to (5/2)v.     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

A second note on the equational theory of modular ortholattices

We show how to alter the material in [4] to prove that every variety of modular ortholattices is generated by its simple members. Supported by NSERC Operating Grant 0041702.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

A note on best constants in discrete inequalities

In this paper, optimization theory is used for the determination of best constants in certain discrete inequalities. An example is given.     

Hypergeometric approach to Weideman’s conjecture

By applying the derivative operator to Dixon’s formula, we prove several harmonic number identities including one of the hardest challenge identities conjectured by Weideman (2003). Received: 28 October 2005     

A note onR 1-topological spaces

The purpose of the present note is to give a number of characterizations of theR ₁-axiom and to show that theR ₁-axiom is equivalent to the weakly Hausdorff axiom introduced byB. Banaschewski andJ. M. Maranda [2]. In anR ₁-space it is shown that the locally compactness property is also open hereditary and that the closure of an almost compact set is the union of the closures of its points. A necessary and sufficient condition is obtained under which a locally compact set dense in anR ₁-space is open. Finally a variant of a well-known theorem regarding two continuous functions of a topological space into aT ₂-space is formulated forR ₁-spaces.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Finite digraphs with given regular automorphism groups

With five exceptions, every finite regular permutation group occurs as the automorphism group of a digraph.One of the corollaries: given a finite groupG of ordern, there is a commutative semigroupS of order 2n+2 such that AutSG. The problem whether a latticeL of order Cn with AutLG exists (for some constantC), remains open.     

A note on the solvability of finite Moufang loops

In this short note we show that finite Moufang loops with nilpotent inner mapping groups are solvable. Received: 24 May 2006     

Belousov equations on ternary quasigroups

Summary The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krape and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no longer than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krape and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ ₂.In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no longer than any member of the system or the system is equivalent to a pair of equations each with three variables.     

Homological symbols and the Quillen conjecture

We formulate a “correct” version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory “homological symbols algebra”. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.     

Sur un problème au sujet des homomorphismes

Sans résumé

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Dédié à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire avec les meilleurs vœux.     

A combinatorial approach for Keller's conjecture

The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a counterexample for this conjecture if and only if the following graphs _n has a 2 ⁿ size clique. The 4 ⁿ vertices of _n aren-tuples of integers 0, 1, 2, and 3. A pair of thesen-tuples are adjacent if there is a position at which the difference of the corresponding components is 2 modulo 4 and if there is a further position at which the corresponding components are different. We will give the size of the maximal cliques of _n forn5.     

A note on 1-factors in graphs

Conditions on a graphG are presented which are sufficient to guarantee thatG–Z contains a 1-factor, whereZ is a set of edges ofG of restricted cardinality. These conditions provide generalizations of several known results and, further, establish the result that ifG is anr-regular, (r–2)-edge-connected graph (r2) of even order andz is an integer with 0zr–1 such thatG contains fewer thanr–z edge cut sets of cardinalityr–2, thenG–Z has a 1-factor for each setZ ofz edges ofG.     

Rational congruence for uniform multiplicative designs

A multiplicative design is a family ofn subsets of ann-set whose incidence matrixA satisfiesA ^T A=D+ ^T where is a positive real vector andD is a positive diagonal matrix. This is a generalization of-designs where is a constant vector and of (v, k, )-designs whereD is additionally required to be scalar. In a uniform design we only require thatD be scalar, so the equation isA ^T A=dI+ ^T .One of the basic results on (v, k, )-designs is the Bruck-Ryser-Chowla Theorem which says thatk– must be a square ifv is even and thatz ²=(k–)x ²+y ²(–1)^(v–1)/2 must have a nontrivial integral solution ifv is odd. This can be proved with or without reference to the theory of rational congruences.The purpose of this paper is to investigate the implications of the theory of rational congruences for the existence of uniform multiplicative designs. The Hass-Minkowski Theorem provides the main line of attack. The main result gives a finite set of equations, suitable for programming on a computer, which must be satisfied if there is a rational matrix satisfying the equationA ^T A=dI+ ^T for a uniform design.     

On a conjecture about orders of products of elements in the symmetric group

For any $a,b,c,n\in \mathbb{N}$ with $1<a,b,c\le n?2$, then there exist $\alpha ,\beta \in S_n$ such that $o(\alpha )=a$, $o(\beta )=b$ and $o(\alpha \beta )=c$. This is a conjecture of Stefan Kohl and which is closely related to problem on covers of the complex projective line. In this note we prove the conjecture is true.