Investigations on asymptotic safety of metric, tetrad and Einstein-Cartan gravity

Among the different approaches for a construction of a fundamental quantum theory of gravity the Asymptotic Safety scenario conjectures that quantum gravity can be defined within the framework of conventional quantum field theory, but only non-perturbatively. In this case its high energy behavior is controlled by a non-Gaussian fixed point of the renormalization group flow, such that its infinite cutoff limit can be taken in a well defined way. A theory of this kind is referred to as non-perturbatively renormalizable. In the last decade a considerable amount of evidence has been collected that in four dimensional metric gravity such a fixed point, suitable for the Asymptotic Safety construction, indeed exists. This thesis extends the Asymptotic Safety program of quantum gravity by three independent studies that differ in the fundamental field variables the investigated quantum theory is based on, but all exhibit a gauge group of equivalent semi-direct product structure. It allows for the first time for a direct comparison of three asymptotically safe theories of gravity constructed from different field variables. The first study investigates metric gravity coupled to SU(N) Yang-Mills theory. In particular the gravitational effects to the running of the gauge coupling are analyzed and its implications for QED and the Standard Model are discussed. The second analysis amounts to the first investigation on an asymptotically safe theory of gravity in a pure tetrad formulation. Its renormalization group flow is compared to the corresponding approximation of the metric theory and the influence of its enlarged gauge group on the UV behavior of the theory is analyzed. The third study explores Asymptotic Safety of gravity in the Einstein-Cartan setting. Here, besides the tetrad, the spin connection is considered a second fundamental field. The larger number of independent field components and the enlarged gauge group render any RG analysis of this system much more difficult than the analog metric analysis. In order to reduce the complexity of this task a novel functional renormalization group equation is proposed, that allows for an evaluation of the flow in a purely algebraic manner. As a first example of its suitability it is applied to a three dimensional truncation of the form of the Holst action, with the Newton constant, the cosmological constant and the Immirzi parameter as its running couplings. A detailed comparison of the resulting renormalization group flow to a previous study of the same system demonstrates the reliability of the new equation and suggests its use for future studies of extended truncations in this framework.

Representation theorems for indefinite quadratic forms and applications

Diese Arbeit widmet sich den Darstellungssätzen für symmetrische indefinite (das heißt nicht-halbbeschränkte) Sesquilinearformen und deren Anwendungen. Insbesondere betrachten wir den Fall, dass der zur Form assoziierte Operator keine Spektrallücke um Null besitzt. Desweiteren untersuchen wir die Beziehung zwischen reduzierenden Graphräumen, Lösungen von Operator-Riccati-Gleichungen und der Block-Diagonalisierung für diagonaldominante Block-Operator-Matrizen. Mit Hilfe der Darstellungssätze wird eine entsprechende Beziehung zwischen Operatoren, die zu indefiniten Formen assoziiert sind, und Form-Riccati-Gleichungen erreicht. In diesem Rahmen wird eine explizite Block-Diagonalisierung und eine Spektralzerlegung für den Stokes Operator sowie eine Darstellung für dessen Kern erreicht. Wir wenden die Darstellungssätze auf durch (grad u, h() grad v) gegebene Formen an, wobei Vorzeichen-indefinite Koeffzienten-Matrizen h() zugelassen sind. Als ein Resultat werden selbstadjungierte indefinite Differentialoperatoren div h() grad mit homogenen Dirichlet oder Neumann Randbedingungen konstruiert. Beispiele solcher Art sind Operatoren die in der Modellierung von optischen Metamaterialien auftauchen und links-indefinite Sturm-Liouville Operatoren.

A Metric to Evaluate and Synthesize Distributed Compliant Mechanisms

Compliant mechanisms with evenly distributed stresses have better load-bearing ability and larger range of motion than mechanisms with compliance and stresses lumped at flexural hinges. In this paper, we present a metric to quantify how uniformly the strain energy of deformation and thus the stresses are distributed throughout the mechanism topology. The resulting metric is used to optimize cross-sections of conceptual compliant topologies leading to designs with maximal stress distribution. This optimization framework is demonstrated for both single-port mechanisms and single-input single-output mechanisms. It is observed that the optimized designs have lower stresses than their nonoptimized counterparts, which implies an ability for single-port mechanisms to store larger strain energy, and single-input single-output mechanisms to perform larger output work before failure.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

A g-factor metric for k-t-GRAPPA- and PEAK-GRAPPA-based parallel imaging.

PURPOSE The aim of this work is to derive a theoretical framework for quantitative noise and temporal fidelity analysis of time-resolved k-space-based parallel imaging methods. THEORY An analytical formalism of noise distribution is derived extending the existing g-factor formulation for nontime-resolved generalized autocalibrating partially parallel acquisition (GRAPPA) to time-resolved k-space-based methods. The noise analysis considers temporal noise correlations and is further accompanied by a temporal filtering analysis. METHODS All methods are derived and presented for k-t-GRAPPA and PEAK-GRAPPA. A sliding window reconstruction and nontime-resolved GRAPPA are taken as a reference. Statistical validation is based on series of pseudoreplica images. The analysis is demonstrated on a short-axis cardiac CINE dataset. RESULTS The superior signal-to-noise performance of time-resolved over nontime-resolved parallel imaging methods at the expense of temporal frequency filtering is analytically confirmed. Further, different temporal frequency filter characteristics of k-t-GRAPPA, PEAK-GRAPPA, and sliding window are revealed. CONCLUSION The proposed analysis of noise behavior and temporal fidelity establishes a theoretical basis for a quantitative evaluation of time-resolved reconstruction methods. Therefore, the presented theory allows for comparison between time-resolved parallel imaging methods and also nontime-resolved methods. Magn Reson Med, 2014. © 2014 Wiley Periodicals, Inc.

VIRTOPSY--scientific documentation, reconstruction and animation in forensic: individual and real 3D data based geo-metric approach including optical body/object surface and radiological CT/MRI scanning.

Until today, most of the documentation of forensic relevant medical findings is limited to traditional 2D photography, 2D conventional radiographs, sketches and verbal description. There are still some limitations of the classic documentation in forensic science especially if a 3D documentation is necessary. The goal of this paper is to demonstrate new 3D real data based geo-metric technology approaches. This paper present approaches to a 3D geo-metric documentation of injuries on the body surface and internal injuries in the living and deceased cases. Using modern imaging methods such as photogrammetry, optical surface and radiological CT/MRI scanning in combination it could be demonstrated that a real, full 3D data based individual documentation of the body surface and internal structures is possible in a non-invasive and non-destructive manner. Using the data merging/fusing and animation possibilities, it is possible to answer reconstructive questions of the dynamic development of patterned injuries (morphologic imprints) and to evaluate the possibility, that they are matchable or linkable to suspected injury-causing instruments. For the first time, to our knowledge, the method of optical and radiological 3D scanning was used to document the forensic relevant injuries of human body in combination with vehicle damages. By this complementary documentation approach, individual forensic real data based analysis and animation were possible linking body injuries to vehicle deformations or damages. These data allow conclusions to be drawn for automobile accident research, optimization of vehicle safety (pedestrian and passenger) and for further development of crash dummies. Real 3D data based documentation opens a new horizon for scientific reconstruction and animation by bringing added value and a real quality improvement in forensic science.

The selection of preferred metric values for design and construction /

Sponsored by Naval Facilities Engineering Command, Department of the Navy.

Metric conversion in the construction industries--technical issues and status /

Mode of access: Internet.

Set theory and metric spaces. Math. 261, autumn, 1950.

Cover title.

Units of weight and measure (United States customary and metric). Definitions and tables of equivalents.

Vol. for 1914 has supplement (publ. 1917).

Providing a metric option : can laws and regulations be amended in a timely manner?

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