Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

Mapping spaces and homology isomorphisms

Let denote the space of pointed continuous maps from a finite cell complex to a space . Let be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on and , will send an -isomorphism in either variable to a map that is monic in homology. Interesting examples arise by letting be -theory, the finite complex be a sphere, and the map in the variable be an exotic unstable Adams map between Moore spaces.

     

Dynamical systems method (DSM) for unbounded operators

Let be an unbounded linear operator in a real Hilbert space , a generator of a semigroup, and let be a nonlinear map. The DSM (dynamical systems method) for solving equation consists of solving the Cauchy problem , , where is a suitable operator, and proving that i) 0$">, ii) , and iii) .

Conditions on and are given which allow one to choose such that i), ii), and iii) hold.

     

for commutative rings with identity

Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.

     

Hyperelliptic surfaces are Loewner

We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to away from Weierstrass points. The loops are then transplanted to , and surgered to obtain a Loewner loop on . In higher genus, we exploit M. Gromov's area estimates for -regular metrics on .

     

Eventual arm and leg widths in cocharacters of P. I. Algebras

Given a p.i. algebra , we study which partitions correspond to characters with non-zero multiplicities in the cocharacter sequence of . We define the , the eventual arm width to be the maximal so that such can have parts arbitrarily large, and to be the maximum so that the conjugate could have arbitrarily large parts. Our main result is that for any , .

     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

The power of the tangent bundle of the real projective space, its complexification and extendibility

We establish the formulas on the power of the tangent bundle of the real projective -space and its complexification , and apply the formulas to the problem of extendibility and stable extendiblity of and .

     

Uniformly complete quotient space

The uniformly complete quotient space of a locally compact group is introduced. It is shown that the operator space dual is a completely contractive Banach algebra, which contains the completely bounded Fourier multiplier algebra as a completely contractively complemented Banach subalgebra. A natural completely isometric representation of on is studied and some equivalent amenability conditions associated with are proved.

     

Borel sets with countable sections for nonseparable spaces

We prove that every (extended) Borel subset of , where is a complete metric and is Polish, can be covered by countably many extended Borel graphs of mappings from to if the sections , , are countable. This is a nonseparable version of a classical theorem of Luzin and Novikov.

     

Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups

In this paper we show that if is an amenable locally compact group and if is a closed subgroup, then the ideal has an approximate identity of norm If is not open, this bound is the best possible.

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.

     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

Failure of Krull-Schmidt for invertible lattices over a discrete valuation ring

Let be a prime greater than , and let be the semi-direct product of a group of order by a cyclic group of order , which acts faithfully on . Let be the localization of at . We show that the Krull-Schmidt Theorem fails for the category of invertible -lattices.

     

Maps preserving numerical ranges of operator products

Let be a complex Hilbert space, the algebra of all bounded linear operators on and the real linear space of all self-adjoint operators on . We characterize the surjective maps on or that preserve the numerical ranges of products or Jordan triple-products of operators.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator is hypercyclic if there is an such that the orbit is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces does contain such an operator. We prove that for any infinite-dimensional Banach space with a shrinking symmetric basis, such as and any , there is an operator , where both and are hypercyclic.

     

Compact perturbations of -accretive operators in Banach spaces

This paper continues a discussion that arose twenty years ago, concerning the perturbation of an -accretive operator by a compact mapping in Banach spaces. Indeed, if is -accretive and is compact, then the boundary condition for and implies that is in the closure of the range of . Perhaps the most interesting aspect of this result is the proof itself, which does not appeal to the classical degree theory argument used for this type of problem.

     

Witt kernels of bilinear forms for algebraic extensions in characteristic

Let be a field of characteristic and let be a purely inseparable extension of exponent . We determine the kernel of the natural restriction map between the Witt rings of bilinear forms of and , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension . Based on this result, we will determine for a wide class of finite extensions which are not necessarily purely inseparable.

     

Functional calculus and -regularity of a class of Banach algebras

Suppose that is a -dynamical system such that is of polynomial growth. If is finite dimensional, we show that any element in has slow growth and that is -regular. Furthermore, if is discrete and is a ``nice representation' of , we define a new Banach -algebra which coincides with when is finite dimensional. We also show that any element in has slow growth and is -regular.