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Student Number : 9900131F - MSc (Eng) dissertation - School of Mechanical Engineering - Faculty of Engineering / Recent high-resolution numerical studies of weak shock reflections have shown that a complex flow structure exists behind the triple point which consists of multiple shocks, expansion fans and triple points. This region had not been detected earlier in experimental observations or numerical studies of weak shock reflections due to the small size of this region. New components were designed and built to modify an existing large-scale shock tube in order to obtain experimental observations to validate the numerical results. The shock tube produced a large, expanding cylindrical incident wave which was reflected off a 15° corner on the roof of the section to produce a weak shock Mach reflection with a large Mach stem in the test section. The shock tube was equipped with PCB high-speed pressure transducers and digital scope for data acquisition, and a schlieren optical system to visualise the region behind the triple point. The tests were conducted over a range of incident wave Mach numbers (M12 = 1.060-1.094) and produced Mach stems of between 694 mm and 850 mm in length. The schlieren photographs clearly show an expansion fan centered on the triple point in all the successful tests conducted. In some of the more resolved images, a shocklet can be seen terminating the expansion fan, and in others a second expansion fan and/or shocklet can be seen. A ‘von Neumann reflection’ was not visualised experimentally, and hence it has been proposed that the four-wave reflection found in these tests be named a ‘Guderley reflection’. The experimental validation of Hunter & Tesdall’s (2002) work resolves the ‘von Neumann Paradox’.

by Lo Chui-sim. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 91-93). / Abstract --- p.i / Introcution --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Preliminaries on von-Neumann algebra --- p.3 / Chapter 1.2 --- Modular theory --- p.6 / Chapter 2 --- Abstract Lp Spaces --- p.10 / Chapter 2.1 --- "Preliminaries on dual action, dual weights and extended positive part" --- p.10 / Chapter 2.2 --- Abstract LP spaces associated with von-Neumann algebras --- p.20 / Chapter 2.3 --- "LP(M) is a Banach space for p E [1, ∞ ]" --- p.25 / Chapter 2.4 --- Independence of the choice of ψ --- p.32 / Chapter 3 --- Spatial Lp Spaces --- p.34 / Chapter 3.1 --- Definition and elementary properties of spatial derivative --- p.35 / Chapter 3.2 --- Modular properties of spatial derivatives --- p.47 / Chapter 3.3 --- Spatial Lp spaces --- p.51 / Chapter 4 --- LP Spaces constructed by using complex interpolation method --- p.60 / Chapter 4.1 --- The complex interpolation space --- p.60 / Chapter 4.2 --- LP space with respect to a faithful normal state --- p.71 / Chapter 4.3 --- LP spaces with respect to a normal faithful semifinite weight . . --- p.78 / Chapter 4.4 --- Equivalence to spatial LP spaces --- p.87 / Bibliography --- p.91

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The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and in particular, type III factors. We replace traces with weights, and for a von Neumann algebra $\mathcal{M}$ which admits a weight $\phi$, we show the existence of an automorphic action $\sigma^\phi:\mathbb{R}\rightarrow\text{Aut}(\mathcal{M})$. After showing the existence of these actions we can discuss the crossed product construction, which will then allow us to study the structure of the algebra.

An outcome is ambiguous if it is an incomplete description of the probability distribution over consequences. An `incomplete description' is identified with the set of probabilities that satisfy the incomplete description. A choice problem is uncertain if the decision maker is choosing between distributions, and is ambiguous if the decision maker is choosing between sets of probabilities. The von Neumann/Morgenstern approach to uncertain choice problems uses a continuous linear function on probabilities. This paper develops the theory of ambiguous choice problems as a continuous, linear functions on closed convex sets of probabilities. This delivers: a framework encompassing most of the extant ambiguity averse preferences; a complete separation of attitudes towards risk and attitudes toward ambiguity; and generalizations of rst and second order stochastic dominance rankings to ambiguous decision problem. Quasi-concave preferences on sets that satisfy a restricted betweenness property capture variational preferences. / text

The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and in particular, type III factors. We replace traces with weights, and for a von Neumann algebra $\mathcal{M}$ which admits a weight $\phi$, we show the existence of an automorphic action $\sigma^\phi:\mathbb{R}\rightarrow\text{Aut}(\mathcal{M})$. After showing the existence of these actions we can discuss the crossed product construction, which will then allow us to study the structure of the algebra.

Spectral flow, in its simplest incarnation, counts the net number of eigenvalues which change sign as one traverses a path of self-adjoint Fredholm operators in the set of of bounded operators B(H) on a Hilbert space. A generalization of this idea changes the setting to a semifinite von Neumann algebra N and uses the trace τ to measure the amount of spectrum which changes from negative to positive along a path; the operators are still self-adjoint, but the Fredholm requirement is replaced by its von Neumann algebras counterpart, Breuer-Fredholm. Our work is ensconced in this semifinite von Neumann algebra setting. We prove a uniqueness result in the case when N is a factor. In the case when the operators under consideration are bounded perturbations of a fixed unbounded operator with τ-compact resolvents, we give a different proof of a p-summable integral formula which calculates spectral flow, and fill in some of the gaps in the proof that spectral flow can be viewed as an intersection number if N = B(H). / Graduate / 0280

We investigate some properties of (geometric) fields in toposes of sheaves over Boolean spaces and establish the internal validity of a number of classical theorems from Algebraic Geometry and the theory of ordered fields. We then use our results to obtain, via sheaf representations, some know theorems about (von Neumann) regular rings as well as some new theorems for regular f-rings. By contrast with previous investigations in these last two subjects (Saracino and Weispfenning {39} and van den Dries {42}) a more natural approach, inspired by work of Macintyre {30}, Loullis {29}, Bunge-Reyes {7} and Bunge {4},{5} is employed here. In addition to sheaf theoretic methods we use a variety of logical methods from geometric logic, infinitary intuitionistic logic and model theory. We also prove some new theorems on the transfer of subobjects along certain morphisms and a "lifting theorem" taking truth from statements about global sections to their internal validity.

Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering. / Typescript bound. Includes bibliographical references: (leaves 174-180) and index. Also available online.