Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

Open covers and the square bracket partition relation

An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

     

On solutions of real analytic equations

Analyticity of solutions , of systems of real analytic equations , is studied. Sufficient conditions for and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by . In a special case when the 's are independent of , we prove that if a solution satisfies the condition , then is necessarily real analytic.

     

A sequential property of and a covering property of Hurewicz

has the monotonic sequence selection property if there is for each , and for every sequence where for each is a sequence converging pointwise monotonically to , a sequence such that for each is a term of , and converges pointwise to . We prove a theorem which implies for metric spaces that has the monotonic sequence selection property if, and only if, has a covering property of Hurewicz.

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations with continuous jointly in and of class . We also prove a bifurcation theorem for critical points of the function which is just continuous and changes at an isolated minimum (in ) to isolated maximum when passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.

     

Stable orders of stunted lens spaces mod

Let be the stunted lens space mod and its stable order. If , then was determined by H. Toda (1963). In this paper, we determine the number for .

     

muni de l'arithmétique faible de Penzin est décidable

We prove the decidability of the additive ordered group equipped with a predicate for , the multiplication restricted to and the -adic valuation ranging in .

     

Cellular filtration of K-theory and determinants of -algebras

In this note, we will disprove the following conjecture raised by Exel-Loring: Let be a -algebra with trace and let be a determinant associated to . If is a continuous family of homomorphisms and is the canonical matrix valued function on which represents the Bott element in , then . It should be noticed that the conjecture has been proved by Exel-Loring for the case that is a smooth family of homomorphisms.

     

Collapsing successors of singulars

Let be a singular cardinal in , and let be a model such that for some -cardinal with . We apply Shelah's pcf theory to study this situation, and prove the following results. 1) is not a -c.c generic extension of . 2) There is no ``good scale for ' in , so in particular weak forms of square must fail at . 3) If then and also . 4) If then .

     

Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.

     

Every local ring is dominated by a one-dimensional local ring

Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated -algebra.

     

A commutativity theorem for semibounded operators in hilbert space

Let and be semibounded (bounded from below) operators in a Hilbert space and a dense linear manifold contained in the domains of , , , and , and such that for all in . It is shown that if the restriction of to is essentially self-adjoint, then and are essentially self-adjoint and and commute, i.e. their spectral projections permute.

     

Note on faithful representations and a local property of Lie groups

Let be any analytic group, let be a maximal toroid of the radical of , and let be a maximal semisimple analytic subgroup of . If is the Lie algebra of , is the radical of , and is the center of , we show that has a faithful representation if and only if (i) , and (ii) has a faithful representation.

     

Local automorphisms and derivations on

Let be an algebra. A mapping is called a -local automorphism if for every there is an automorphism , depending on and , such that and (no linearity, surjectivity or continuity of is assumed). Let be an infinite-dimensional separable Hilbert space, and let be the algebra of all linear bounded operators on . Then every -local automorphism is an automorphism. An analogous result is obtained for derivations.

     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Let be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any Riemannian metric on , the topological entropy of the geodesic flow of is positive. It follows then, that there exist closed manifolds with arbitrary fundamental group, for which the geodesic flow of any Riemannian metric on has positive topological entropy.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Combinatorial aspects of F-sets

We discuss F filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F filter is the better known bounded evasion number . An application to -sets from trigonometric series is given by showing that if is an -set and has size less than , then is again an -set.