Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.     

Spectral theory for tensor products of Hermitian holomorphic line bundles

Let L be a quasipositive line bundle over a compact complex Hermitian manifold. It has been conjectured that the smallest positive eigenvalue of the Laplacian, acting on the kth tensor power of L, is uniformly bounded independent of k. Various special cases of the conjecture are proved, but a counterexample is given to the general conjecture. This research was partially supported by NSF DMS 0203070.Mathematics Subject Classification (2000):58G25     

Inequalities for <Emphasis Type="Bold">cd</Emphasis>-Indices of Joins and Products of Polytopes

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For polytopes U and V, we determine explicit recurrences for computing the cd-index of the free join and the cd-index of the Cartesian product U x V. As an application of these recurrences, we prove the inequality involving the cd-indices of three polytopes.     

Variational representations of Varadhan functionals

Motivated by the theory of large deviations, we introduce a class of non-negative non-linear functionals that have a variational ``rate function" representation.

     

GO-spaces with -closed discrete dense subsets

In this paper we study the question ``When does a perfect generalized ordered space have a -closed-discrete dense subset?' and we characterize such spaces in terms of their subspace structure, -mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a -closed-discrete dense set and a weak monotone ortho-base. That metrization theorem cannot be proved in ZFC for perfect GO-spaces because if there is a Souslin line, then there is a non-metrizable, perfect, linearly ordered topological space that has a weak monotone ortho-base.

     

Directional fractional kinetics

Kinetic equations used to describe systems with dynamical chaos may contain fractional derivatives of an order alpha in space and beta in time in order to represent processes of stickiness, intermittency, and so on. We demonstrate for a simple example that the kinetics is anisotropic not only in the angular dependence of the diffusion constant, but also in the angular dependence of the exponents alpha and beta. A theory of such kinetic processes has been developed on the basis of integral representation and asymptotic solutions for different cases have been obtained. The results show the existence of self-similar solutions as well as possible logarithmic deviations. (c) 2001 American Institute of Physics.     

Statistical energy analysis by finite elements for middle frequency vibration

Statistical energy analysis (SEA) is commonly used in industry to predict high-frequency vibrational response of structures. Since only local modes are used in SEA, only high-frequency responses can be predicted. This study extends SEA to the middle-frequency region by additionally using global modes. Methods using impedance matrices that can be found by NASTRAN are developed. Then the results are post processed to determine coupling loss factors.     

Prion protein detection in serum using micromechanical resonator arrays

Prion proteins that have transformed from their normal cellular counterparts (PrP^c) into infectious form (PrP^res) are responsible for causing progressive neurodegenerative diseases in numerous species, such as bovine spongiform encephalopathy (BSE) in cattle (also known as mad cow disease), scrapie in sheep, and Creutzfeldt-Jakob disease (CJD) in humans. Due to a possible link between BSE and CJD it is highly desirable to develop non-invasive and ante mortem tests for the detection of prion proteins in bovine samples. Such ante mortem tests of all cows prior to slaughter will help to prevent the introduction of PrP^res into the human food supply. Furthermore, detection of PrP^res in donated blood will also help to prevent the transmission of CJD among humans through blood transfusion. In this study, we have continued development of a micromechanical resonator array that is capable of detecting PrP^c in bovine blood serum. The sensitivity of the resonators for the detection of PrP^c is further enhanced by the use of secondary mass labels. A pair of antibodies is used in a sandwich immunoassay format to immobilize PrP^c on the surface of resonators and attach nanoparticles as secondary mass labels to PrP^c. Secondary mass labeling is optimized in terms of incubation time to maximize the frequency shifts that correspond to the presence of PrP^c on the surface of resonators. Our results show that a minimum of 200 pg mL⁻¹ of PrP^c in blood serum can be detected using micromechanical resonator arrays.     

Die Elementaranalyse mittels einer Kalorimeterbombe

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