If the Hodge *-operator on the L2-cohomology of Kuga fiber varieties is algebraic, then the Hodge conjecture is true for all abelian varieties.

In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or miliary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear dipohantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution exists, and exhibiting it if it does.

This paper considers Hilbert C*-bimodules, a slight generalization of imprimitivity bimodules which were introduced by Rieffel [20]. Brown, Green, and Rieffel [7] showed that every imprimitivity bimodule X can be embedded into a certain C*-algebra L, called the linking algebra of X. We consider arbitrary embeddings of Hilbert C*-bimodules into C*-algebras; i.e. we describe the relative position of two arbitrary hereditary C*-algebras of a C*-algebra, in an analogy with Dixmier's description [10] of the relative position of two subspaces of a Hilbert space.

The main result of this paper (Theorem 4.3) is taken from the doctoral dissertation of the third author [22], although the proof here follows a different approach. In Section 1 we set out the definitions and basic properties (mostly folklore) of Hilbert C*-bimodules. In Section 2 we show how every quasi-multiplier gives rise to an embedding of a bimodule. In Section 3 we show that , the enveloping C*-algebra of the C*-algebraA with its product perturbed by a positive quasi-multiplier , is isomorphic to the closure (Proposition 3.1). Section 4 contains the main theorem (4.3), and in Section 5 we explain the analogy with the relative position of two subspaces of a Hilbert spaces and present some complements.

Peano continua which are images of the unit interval [0,1] or the circle S under a continuous and irreducible map are investigated. Necessary conditions for a space to be the irreducible image of [0,1] are given, and it is conjectured that these conditions are sufficient as well. Also, various results on irreducible images of [0,1] and S are given within some classes of regular curves. Some of them involve inverse limits of inverse sequences of Euler graphs with monotone bonding maps.

The purpose of this paper is to examine which classes of functions from can be topologized in a sense that there exist topologies τ1 and τ2 on and respectively, such that is equal to the class C(τ1 , τ2) of all continuous functions . We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functions for which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈ f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T on such that the class L of all linear functions g(x) = ax + b coincides with . This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the class of all polynomials, the class of all analytic functions and the class of all harmonic functions.

We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞ functions, differentiable functions, Darboux functions and derivatives.

We determine abstract characterizations of the fixed point subalgebras of the rotation algebra under the automorphisms and .

We approximate the exit distribution of a Markov jump process into a set of forbidden states and we apply these general results to an ATM multiplexor. In this case the forbidden states represent an overloaded multiplexor. Statistics for this overload or busy period are difficult to obtain since this is such a rare event. Starting from the approximate exit distribution, one may simulate the busy period without wasting simulation time waiting for the overload to occur.

Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.

Suppose that Gis a locally compact group with modular function Δ and that p, q, r are three numbers in the interval (l,∞) satisfying . If cp,q(G) is the smallest constant c such that for all functions f, g ∈ Cc(G) (here the convolution product is with respect to left Haar measure and is the exponent which is conjugate to p) then Young's inequality asserts that cp,q(G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then cp,q(G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant cp,q < 1 such that if G does not contain a compact open subgroup then c<(G) ≤ c≤. Secondly, Beckner's calculation of is used to obtain the value of cp,q(G) for all simply-connected solvable Lie groups and all nilpotent Lie groups. And thirdly, it is shown that for a nilpotent Lie group the set is not contained in the union of the spaces Ls(G), .

In this paper, we shall give a method in constructing dihedral Galois covering with prescribed branch locus. As an application, we shall look into dihedral Galois covering of P2, where torsion elements of the Mordell-Weil group of an elliptic surface play key roles in constructing coverings

If μk,n(α,β) denotes the relative extrema of the Jacobi polynomial P(α,β)n(x), ordered so that μk+1,n(α,β) lies to the left of μk,n(α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μk,n(α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.