Measurable Categories and 2-Groups

Using the theory of measurable categories developed in [10], we provide a notion of representations of 2-groups better suited to physically and geometrically interesting examples than that using 2-VECT (cf. [8]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [6], and suggest state-integral constructions for 4-manifold invariants.     

Measurable dynamics of maps on profinite groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on _p.     

Bender's generalizedq-binomial Vandermonde convolution

A simple arithmetical proof and a generalization of Bender's generalizedq-binomial Vandermonde convolution are given.     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

Multiepireflective subcategories

Y. Diers has defined multireflective subcategories as a generalization of reflective subcategories. In this paper, the related concepts of multiepireflective and monomultireflective subcategories are defined and investigated. It is proved that, for categories with appropriate (E,M) factorization structures, every multireflection can be expressed as the composition of an epireflection followed by a multiepireflection. Characterizations of multi-E-reflective subcategories are also given for categories with (E,M)-factorization structures. Finally, a list of subcategories of Top which are: multireflective in Top, multiepireflective in Top₂ and {initial-monosources}-multireflective in CRog T₂ is given.     

Uniform ergodicities and perturbation bounds of Markov chains on base norm spaces

It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on base norm spaces have been studied. In this paper, we consider uniformly mean ergodic and asymptotically stable Markov operators on such spaces. In terms of the ergodicity coefficient, we establish uniform mean ergodicity criterion. Moreover, we develop the perturbation theory for uniformly asymptotically stable Markov chains on base norm spaces. In particularly, main results open new perspectives in the perturbation theory for quantum Markov processes defined on von Neumann algebras.     

A structure theorem for right invariant right holoids

Communicated by Boris M. Schein     

A generalization of group difference sets and the matrix equationA m =dI+λJ

LetG be a finite group. In this paper, we will study the-group matrices forG which satisfy the matrixA ^m =dI+J and we will show that the existence of such a solution is equivalent to the existence of a combinatorial structure inG which is a generalization of group difference sets.     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

Solving equations in groups: A survey of Frobenius' theorem

A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationx ⁿ=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.LetG be a finite group of order |G|. Leto(g) denote the order ofg( G). LetH(s, k)={xG:k|o(x)| sk} wherea/b meansa dividesb and leth(s,k)=|H(s,k)|. Using this notation the simplest of Frobenius' results states ifn/|G|, then/h(n, 1). The minimum value ofh(n, 1) is discussed in the first section. Various conditions are known to insure thath(n, 1)=n. A long standing conjecture of Frobenius states ifn=h(n, 1) thenH(n, 1) is a subgroup (where of coursen/|G|). This conjecture is valid for solvable groups, as well as for various arithmetic conditions.In the second section other divisibility conditions arising from Frobenius' Theorem are discussed. One direction covers more general arithmetic divisibility condition. Another direction has a much wider scope, involving a finite number of equations of an unspecified form and is mainly due to P. Hall. Recently some divisibility conditions involving all groups of a given order have been obtained. Divisibility conditions also hold in infinite groups, and for automorphism analogues of element order. In the next section generalizations to group characters relating back to Frobenius are given. Some of these expressions are used in analyzing properties of group representations and have applications in quantum theory. In the last section clear evidence is established for the combinatorial rather than group-theoretic nature of these results. In particular, some recent work of Snapper links the counting of solutions of equations with the cycle indices in combinatorial theory. Counting solutions of equations in the symmetric groups is also discussed.     

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

Chaines maximales de préordres

Résumé La famille des préordres sur un ensemble fixé constitue un treillis pour l'inclusion. Répondant à une question rencontrée par S. Eilenberg dans l'étude des automates non déterministes on établit une propriété des chaînes maximales de préordres sur un ensemble fini.On en déduit que si l'ensemble a n éléments, de telles chaînes contiennent au plus [n(n + 1)]/2 préordres.

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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