A class of simple proper Bol loops

The existence of finite simple non-Moufang Bol loops has long been considered to be one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of simple proper Bol loops. This class contains finite and new infinite simple proper Bol loops. This paper was written during the author’s Marie Curie Fellowship MEIF-CT-2006-041105 at the University of Würzburg (Germany).     

Congruences on equivalence algebras

A certain class of atomic, semimodular, semisimple partition lattices is studied. It is shown that this class is precisely the class of congruence lattices of equivalence algebras. The first author is granted by project POCTI-ISFL-1-143 of the “Centro de álgebra da Universidade de Lisboa”, supported by FCT and FEDER.     

Avoidable structures, II: Finite distributive lattices and nicely structured ordered sets

Let be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in that are well-quasi-ordered by embeddability, and thus characterize the members of that belong to at least one infinite anti-chain in . While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

The product of two von Neumann n-frames, its characteristic, and modular fractal lattices

Let L be a bounded lattice. If for each a₁ < b₁ ∈ L and a₂ < b₂ ∈ L there is a lattice embedding ψ: [a₁, b₁] → [a₂, b₂] with ψ(a₁) = a₂ and ψ(b₁) = b₂, then we say that L is a quasifractal. If ψ can always be chosen to be an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals, then L will be called a fractal lattice. For a ring R with 1 let denote the lattice variety generated by the submodule lattices of R-modules. Varieties of this kind are completely described in [16]. The prime field of characteristic p will be denoted by F_p. Let be a lattice variety generated by a nondistributive modular quasifractal. The main theorem says that is neither too small nor too large in the following sense: there is a unique , a prime number or zero, such that and for any n ≥ 3 and any nontrivial (normalized von Neumann) n-frame of any lattice in , is of characteristic p. We do not know if in general; however we point out that, for any ring R with 1, implies . It will not be hard to show that is Arguesian. The main theorem does have a content, for it has been shown in [2] that each of the is generated by a single fractal lattice L_p; moreover we can stipulate either that L_p is a continuous geometry or that L_p is countable. The proof of the main theorem is based on the following result of the present paper: if is a nontrivial m-frame and is an n-frame of a modular lattice L with m, n ≥ 3 such that and , then these two frames have the same characteristic and, in addition, they determine a nontrivial mn-frame of the same characteristic in a canonical way, which we call the product frame. Presented by E. T. Schmidt.     

Spectral norm of random matrices

In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Füredi and Komlós. Research supported by an NSF CAREER award and by an Alfred P. Sloan fellowship.     

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Avoidable structures, I: Finite ordered sets, semilattices and lattices

We find all finite unavoidable ordered sets, finite unavoidable semilattices and finite unavoidable lattices. While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

On the measure of intersecting families,uniqueness and stability

Let t≥1 be an integer and let A be a family of subsets of {1,2,…,n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0,1} ⁿ we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the t-intersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable: any t-intersecting family that is close to attaining the maximal measure must in fact be close in structure to a genuine maximum family. This is stated precisely in Theorem 1.6. We deduce some similar results for the more classical case of Erdős-Ko-Rado type theorems where all the sets in the family are restricted to be of a fixed size. See Corollary 1.7. The main technique that we apply is spectral analysis of intersection matrices that encode the relevant combinatorial information concerning intersecting families. An interesting twist is that part of the linear algebra involved is done over certain polynomial rings and not in the traditional setting over the reals. A crucial tool that we use is a recent result of Kindler and Safra [22] concerning Boolean functions whose Fourier transforms are concentrated on small sets. Research supported in part by the Israel Science Foundation, grant no. 0329745.     

Embedding operators into strongly continuous semigroups

We study linear operators T on Banach spaces for which there exists a C₀-semigroup (T(t))_t≥0 such that T = T(1). We present a necessary condition in terms of the spectral value 0 and give classes of examples for which such a C₀-semigroup does or does not exist. Received: 22 December 2008, Revised: 7 April 2009     

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

Note edge-colourings of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n,n</Emphasis></Subscript> with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K _n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K _n,n with n colours having all two-coloured cycles of length ≤ 2q ², for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K _n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

Normalizers of p-subgroups in finite groups

In this paper we give a computational strategy for constructing the normalizer of a p-subgroup in a finite group. Received: 31 January 2008     

A simple example of two p-groups with the same automorphism group

It is an open problem whether there exist two different p-groups with the same automorphism group. In this paper, we give a positive answer to this problem by constructing a simple example. Received: 30 September 2008     

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

Sums, products and negations of contexts and complete lattices

In Formal Concept Analysis, one associates with every context its concept lattice , and conversely, with any complete lattice L the standard context L, constituted by the join-irreducible elements as ‘objects’, the meet-irreducible elements as ‘attributes’, and the incidence relation induced by the lattice order. We investigate the effect of the operators and on various (finite or infinite) sum and product constructions. The rules obtained confirm the ‘exponential’ behavior of and the ‘logarithmic’ behavior of with respect to cardinal operations but show a ‘linear’ behavior on ordinal sums. We use these results in order to establish several forms of De Morgan’s law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context. Received February 7, 2001; accepted in final form January 6, 2006.