Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

The exponent three class group problem for some real cyclic cubic number fields

We determine all the simplest cubic fields whose ideal class groups have exponent dividing , thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number and the determination by D. Byeon of all all the simplest cubic fields with class number . We prove that there are simplest cubic fields with ideal class groups of exponent (and simplest cubic fields with ideal class groups of exponent , i.e. with class number one).

     

The stable signature of a regular cyclic action

Let be an odd prime and a smooth map of order . Suppose that the cyclic action defined by is regular and has fixed point set . If the -signature Sign is a rational integer and , then there exists a choice of orientations such that Sign Sign .     

On the comparison of the spaces

The notion of -variation and the space arise in the study of regularity properties of solutions to perturbed conservation laws. In this article we show that this notion is equivalent to variation in the regular sense, and therefore the space is the same as the space in the sense of Cesari-Tonelli. We also point out some connection between the space and the Favard classes for translation semigroups.

     

On classes of maps which preserve finitisticness

We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.

     

Linearity of dimension functions for semilinear -spheres

In this paper, we show that the dimension function of every semilinear -sphere is equal to that of a linear -sphere for finite nilpotent groups of order , where , are primes. We also show that there exists a semilinear -sphere whose dimension function is not virtually linear for an arbitrary nonsolvable compact Lie group .

     

Fréchet-Urysohn spaces in free topological groups

Let and be respectively the free topological group and the free Abelian topological group on a Tychonoff space . For every natural number we denote by () the subset of () consisting of all words of reduced length . It is well known that if a space is not discrete, then neither nor is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both and are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space . In this paper, we prove first that for a metrizable space , () is Fréchet-Urysohn if and only if the set of all non-isolated points of is compact and is Fréchet-Urysohn if and only if is compact or discrete. As applications, we characterize the metrizable space such that is Fréchet-Urysohn for each and is Fréchet-Urysohn for each except for . In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace of () which is not contained in any (). We shall show that if such a space is first countable, then it has a special form in (). On the other hand, we give an example showing that if the space is Fréchet-Urysohn, then it need not have the form.

     

On images of Borel measures under Borel mappings

Let and be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure on , under all Borel mappings , form a closed set in the space of tight Borel probability measures on with the weak-topology. In contrast, the set of images of under all continuous mappings from to may not be closed. We also characterize completely the set of tight images of under Borel mappings. For example, if is non-atomic, then all tight Borel probability measures on can be obtained as images of , and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.

     

Averaging distances in finite dimensional normed spaces and John's ellipsoid

A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all in the unit sphere of there is some in the unit sphere of such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , then . This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of .

     

Convexity numbers of closed sets in

For 2$"> let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each 2$"> a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the convexity number of .

For every non-increasing function \aleph_0\}$"> we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Proof of the prime power conjecture for projective planes of order

Let be an abelian collineation group of order of a projective plane of order . We show that must be a prime power, and that the -rank of is at least if for an odd prime .

     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.

     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

On the class number of certain imaginary quadratic fields

Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .

     

On the predictability of discrete dynamical systems

Let be a metric space. A function is said to be non-sensitive at a point if for every 0$"> there is a 0$"> such that for any choice of points , , , we have that for every . Let be the set of all homeomorphisms from onto endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces , ``most' functions in are non-sensitive at ``most' points of .