Martinaxiom und die Beschreibung gewisser Homomorphismen in der Theorie der ℵ1-freien abelschen Gruppen

P(I) denotes the abelian group of integer-valuated functions defined on the set I, P_κ (I) its subgroup consisting of all elements, which support has size less than the infinite cardinal κ and Q_κ (I) the quotient P(I)/P_κ(I). We formulate a statement I(κ) in the following way: For an arbitrary set I you can lift every homomorphism from P (?) to Q_κ(I) to a homomorphism from P(?) to P(I). Using Martin's Axiom we show, that L(?) is independent of ZFC.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

A new method for solving perturbation equations

A new, generally applicable technique is described for constructing solutions of a coupled linear system of partial differential equations when a decoupled equation has been derived. This method has already yielded an extremely simple derivation of perturbation formulas given by Cohen and Kegeles and by Chrzanowski as well as new formulas for the complete solutions of the coupled Einstein-Maxwell equations describing Reissner-Nordström perturbations. It is hoped that it will prove valuable for many other applications.This essay received the third award from the Gravity Research Foundation for the year 1979-Ed.     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.     

Determining the Elastic Modulus of Compliant Thin Films Supported on Substrates from Flat Punch Indentation Measurements

Instrumented indentation is a technique that can be used to measure the elastic properties of soft thin films supported on stiffer substrates, including polymer films, cellulosic sheets, and thin layers of biological materials. When measuring thin film properties using indentation, the effect of the substrate must be considered. Most existing models for determining the properties of thin films from indentation measurements were developed for metal and dielectric films bonded to semiconductor substrates and have been applied to systems with film-substrate modulus ratios between 0.1 and 10. In the present work, flat punch indentation of a thin film either bonded to or in contact with a substrate is examined using finite element modeling. A broad range of film-substrate modulus ratios from 0.0001 to 1 are investigated. As the substrate is effectively rigid compared to the film when the film-substrate modulus ratio is less than 0.0001, the results are also useful for understanding systems with lower film-substrate modulus ratios. The effects of the contact radius, film thickness, elastic properties, and friction between the film and the substrate on the measured stiffness were quantified using finite element modeling in order to understand how the elastic properties of the film can be extracted from indentation measurements. A semi-analytical model was developed to describe the finite element modeling results and facilitate the use of the results to analyze experimental measurements. The model was validated through analysis of indentation measurements of thin polyethylene sheets that were supported on substrates of various stiffness.     

Three-dimensional magnetic resonance spectroscopic imaging of histologically confirmed brain tumors

The goal of this study was to determine whether presurgical metabolite levels measured by 3D MR Spectroscopic Imaging (MRSI) can accurately detect viable cancer within human brain tumor masses. A total of 31 patients (33 exams, 39 pathology correlations) with brain tumors were studied prior to surgical biopsy and/or resection. The 3D MRSI was obtained with a spatial resolution of 0.2 to 1 cc throughout the majority of the mass and adjacent brain tissue using PRESS-CSI localization. Levels of choline, creatine and NAA were estimated from the locations of the resected tissue and normalized to normal appearing brain tissue. The data were correlated with subsequent histologic analysis of the biopsy tissue samples. Although there were large variations in the metabolite ratios, all regions of confirmed cancer demonstrated significant choline levels and a mean choline/NAA ratio of 5.84 + 2.58 with the lowest value being 1.3. This lowest value is greater than 4 standard deviations above the mean (0.52 +/- 0.13) found in 8 normal volunteers. The choline signal intensities in confirmed cancers were significantly elevated compared to normal appearing brain tissue with a mean ratio of 1.71 +/- 0.69. Spectra with no significant metabolite levels were observed in the non-enhancing necrotic core of the tumor masses. The results of this study indicate that 3D MRSI of brain tumors can detect abnormal metabolite levels in regions of viable cancer and grades and can differentiate cancer from necrosis and/or normal brain tissue.