Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

A non-Hausdorff model for the complement of a complexified hyperplane arrangement

Given a hyperplane arrangement in a real vector space , we introduce a real algebraic prevariety , and exhibit the complement of in the complexification of as the total space of an affine bundle over with fibers modeled on the dual vector space .

     

Turing incomparability in Scott sets

For every Scott set and every nonrecursive set in , there is a such that and are Turing incomparable.

     

Decomposable form equations without the finiteness property

Let be a finitely generated (but not necessarily algebraic) extension field of . Let be a form (homogeneous polynomial) in variables with coefficients in , and suppose that is decomposable (i.e., that it factorizes into linear factors over some finite extension of ). We say that has the finiteness property over if for every (here denotes the set of non-zero elements in ) and for every subring of which is finitely generated over , the equation

has only finitely many solutions. This paper proves the following result: Let be a decomposable form in variables with coefficients in , which factorizes into linear factors over . Let denote a maximal set of pairwise linearly independent linear factors of . If has the finiteness property over , then 2(m-1)$">.

     

Metrically generated theories

Many examples are known of natural functors describing the transition from categories of generalized metric spaces to the ``metrizable" objects in some given topological construct . If preserves initial morphisms and if is initially dense in , then we say that is -metrically generated. Our main theorem proves that is -metrically generated if and only if can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

An extremal function for the Chang-Marshall inequality over the Beurling functions

S.-Y. A. Chang and D. E. Marshall showed that the functional is bounded on the unit ball of the space of analytic functions in the unit disk with and Dirichlet integral not exceeding one. Andreev and Matheson conjectured that the identity function is a global maximum on for the functional . We prove that attains its maximum at over a subset of determined by kernel functions, which provides a positive answer to a conjecture of Cima and Matheson.

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

Hodge structures on posets

Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a Hodge structure on which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

We show that if has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of into Hodge pieces.

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .

     

Relations approximated by continuous functions

Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

Multiplicatively spectrum-preserving maps of function algebras

Let be a compact Hausdorff space and a function algebra. Assume that is the maximal ideal space of . Denoting by the spectrum of an , which in this case coincides with the range of , a result of Molnár is generalized by our Main Theorem: If is a surjective map with the property for every pair of functions , then there exists a homeomorphism such that

for every and every with .

     

On the Betti numbers of sign conditions

Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that

This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by

making the bound more precise.

     

Julia sets converging to the unit disk

We consider the family of rational maps , where and is small. If is equal to 0, the limiting map is and the Julia set is the unit circle. We investigate the behavior of the Julia sets of when tends to 0, obtaining two very different cases depending on and . The first case occurs when ; here the Julia sets of converge as sets to the closed unit disk. In the second case, when one of or is larger than , there is always an annulus of some fixed size in the complement of the Julia set, no matter how small is.

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

Stable constant mean curvature hypersurfaces

Let be a Riemannian manifold with sectional curvatures uniformly bounded from below. When we prove that there are no complete (strongly) stable -hypersurfaces, without boundary, provided is large enough. In particular, we prove that there are no complete strongly stable -hypersurfaces in without boundary,

     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.