On the rational cuspidal subgroup and the rational torsion points of

For two distinct prime numbers , , we compute the rational cuspidal subgroup of and determine the -primary part of the rational torsion subgroup of the old subvariety of for most primes . Some results of Berkovic on the nontriviality of the Mordell-Weil group of some Eisenstein factors of are also refined.

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

     

On divisibility of the class number

In this paper, criteria of divisibility of the class number of the real cyclotomic field of a prime conductor and of a prime degree by primes the order modulo of which is , are given. A corollary of these criteria is the possibility to make a computational proof that a given does not divide for any (conductor) such that both are primes. Note that on the basis of Schinzel's hypothesis there are infinitely many such primes .

     

On a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

Upper bounds for the number of facets of a simplicial complex

Here we study the maximal dimension of the annihilator ideals
of artinian graded rings with a given Hilbert function, where is the polynomial ring in the variables over a field with each , is a graded ideal of , and is the graded maximal ideal of . As an application to combinatorics, we introduce the notion of -facets and obtain some informations on the number of -facets of simplicial complexes with a given -vector.

     

Induction theorems on the stable rationality of the center of the ring of generic matrices

Following Procesi and Formanek, the center of the division ring of generic matrices over the complex numbers is stably equivalent to the fixed field under the action of , of the function field of the group algebra of a -lattice, denoted by . We study the question of the stable rationality of the center over the complex numbers when is a prime, in this module theoretic setting. Let be the normalizer of an -sylow subgroup of . Let be a -lattice. We show that under certain conditions on , induction-restriction from to does not affect the stable type of the corresponding field. In particular, and are stably isomorphic and the isomorphism preserves the -action. We further reduce the problem to the study of the localization of at the prime ; all other primes behave well. We also present new simple proofs for the stable rationality of over , in the cases and .

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

On the group of homotopy equivalences of a manifold

We consider the group of homotopy equivalences of a simply connected manifold which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group for special manifolds , for example if is a connected sum of products of spheres. In particular the group is determined completely. Also the connection of with the group of isotopy classes of diffeomorphisms of is studied.

     

A Dauns-Hofmann theorem for TAF-algebras

Let be a TAF-algebra, the centre of the ideal lattice of , and the space of meet-irreducible elements of , equipped with the hull-kernel topology. It is shown that is a compact, locally compact, second countable, -space, that is an algebraic lattice isomorphic to the lattice of open subsets of , and that is isomorphic to the algebra of continuous, complex functions on . If is semisimple, then is isomorphic to the algebra of continuous, complex functions on , the primitive ideal space of . If is strongly maximal, then the sum of two closed ideals of is closed.

     

On the lengths of closed geodesics on a two-sphere

Let be an isolated closed geodesic of length on a compact Riemannian manifold which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that has a sequence , , of prime closed geodesics of length where and . The hypotheses hold in particular when is a two-sphere and the ``shortest' Lusternik-Schnirelmann closed geodesic is isolated and ``nonrotating'.

     

An infinite series of Kronecker conjugate polynomials

Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two non-constant polynomials are said to be Kronecker conjugate if an element of fixes a root of if and only if it fixes a root of . If is a number field, and where is the ring of integers of , then and are Kronecker conjugate if and only if the value set equals modulo all but finitely many non-zero prime ideals of . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials and differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

Structure of a Hecke algebra quotient

Let be a Coxeter group with Coxeter graph . Let be the associated Hecke algebra. We define a certain ideal in and study the quotient algebra . We show that when is one of the infinite series of graphs of type , the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where is of type , , or .

     

Factorization of an integrally closed ideal in two-dimensional regular local rings

Let be a two-dimensional regular local ring with algebraically closed residue field and be an -primary integrally closed ideal in . Let be the set of Rees valuations of and be the residue field of the valuation ring associated with . Assume that is any minimal reduction of . We show that if is the product of the distinct simple -primary integrally closed ideals in , then is generated by the image of over for all , and the converse of this is also true.

     

On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal differentiable functions and such that the Zalcwasser rank and the Kechris-Woodin rank of are but the Denjoy rank of is 2 and the Denjoy rank and the Kechris-Woodin rank of are but the Zalcwasser rank of is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.

     

Factorisation in nest algebras. II

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator for the existence of an operator in the nest algebra of a nest satisfying (resp. . In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator has the property that there exists for every nest an operator in satisfying (resp. ) if and only if is a Fredholm operator. In Section 4 we show that for a given operator in there exists an operator in satisfying if and only if the range of is equal to the range of some operator in . We also determine the algebraic structure of the set of ranges of operators in . Let be the set of positive operators for which there exists an operator in satisfying . In Section 5 we obtain information about this set. In particular we discuss the following question: Assume and are positive operators such that and belongs to . Which further conditions permit us to conclude that belongs to ?

     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.

     

A characterization for spaces of sections

The space of smooth sections of a bundle over a compact smooth manifold can be equipped with a manifold structure, called an -manifold, where represents the Fréchet algebra of real valued smooth functions on . We prove that the -manifold structure characterizes the spaces of sections of bundles over and its open subspaces. We also describe the -maps between -manifolds.