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).     

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.     

On particle creation by black holes

Hawking's analysis of particle creation by black holes is extended by explicitly obtaining the expression for the quantum mechanical state vector ψ which results from particle creation starting from the vacuum during gravitational collapse. (Hawking calculated only the expected number of particles in each mode for this state.) We first discuss the quantum field theory of a Hermitian scalar field in an external potential or in a curved but asymptotically flat spacetime with no horizon present. In agreement with previously known results, we find that we are led to a unique quantum scattering theory which is completely well behaved mathematically provided a certain condition is satisfied by the operators which describe the scattering of classical positive frequency solutions. In terms of these operators we derive the expression for the state vector describing particle creation from the vacuum, and we prove that S-matrix is unitary. Making the necessary modification for the case when a horizon is present, we apply this theory for a massless Hermitian scalar field to get the state vector describing the steady state emission at late times for particle creation during gravitational collapse to a Schwarzschild black hole. There is some ambiguity in the theory in this case arising from freedom involved in defining what one means by “positive frequency” at the future event horizon. However, it is proven that the expression for the density matrix formed from ψ describing the emission of particles to infinity is independent of this choice, and thus unambiguous predictions for the results of all possible measurements at infinity are obtained. We find that the state vector describing particle creation from the vacuum decomposes into a simple product of state vectors for each individual mode. The density matrix describing emission of particles to infinity by this particle creation process is found to be identical to that of black body emission. Thus, black hole emission agrees in complete detail (i.e., not only in expected number of particles) with black body emission.