Scalar fields are used to represent physical quantities measured over a domain of interest. Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. This thesis proposes three methods to detect symmetry in scalar fields. The first method models symmetry detection as a subtree matching problem in the contour tree, which is a topological graph abstraction of the scalar field. The contour tree induces a hierarchical segmentation of features at different scales and hence this method can detect symmetry at different scales. The second method identifies symmetry by comparing distances between extrema from each symmetric region. The distance is computed robustly using a topological abstraction called the extremum graph. Hence, this method can detect symmetry even in the presence of significant noise. The above methods compare pairs of regions to identify symmetry instead of grouping the entire set of symmetric regions as a cluster. This motivates the third method which uses a clustering analysis for symmetry detection. In this method, the contours of a scalar field are mapped to points in a high-dimensional descriptor space such that points corresponding to similar contours lie in close proximity to each other. Symmetry is identified by clustering the points in the descriptor space. We show through experiments on real world data sets that these methods are robust in the presence of noise and can detect symmetry under different types of transformations. Extraction of symmetry information helps users in visualization and data analysis. We design novel applications that use symmetry information to enhance visualization of scalar field data and to facilitate their exploration.

This thesis investigates some of the properties of a variational approximation to scalar field theories: a trial wavefunctional which has a gaussian form is used as a ground state ansatz for an interacting scalar field theory - the expectation value of the Hamiltonian in this state is then minimized. This we call the Gaussian Approximation; the resulting effective potential we follow others by calling the Gaussian Effective Potential (GEP). An equivalent but more general finite temperature formalism is then reviewed and used for the calculations of the GEP in this thesis. Two scalar field theories are described: ϕ⁴ theory in four dimensions (ϕ⁴₄) and ϕ⁶ theory in three dimensions (ϕ⁶₃). After showing what the Gaussian Approximation does in terms of Feynman diagrams, renormalized GEP's are calculated for both theories. Dimensional Regularization is used in the renormalization and this this is especially convenient for the GEP in ϕ⁶₃ theory because it becomes trivially renor-malizable. It is noted that ϕ⁶₃ loses its infrared asymptotic freedom in the Gaussian Approximation. Finally, it is shown how a finite temperature GEP can be calculated by finding low and high temperature expansions of the temperature terms in ϕ⁶₃ theory. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This dissertation documents the results of two related efforts. Firstly, a model of nucleon-nucleon (NN) interactions is developed based on scalar field theory. Secondly, the relativistic 2-body Bethe-Salpeter equation (BSE) is generalized to handle inelastic processes in the ladder approximation. Scalar field theory describes the behavior of scalar particles, particles with spin 0. In the present work scalar field theory is used to describe NN interactions mediated by pion exchange. The scalar theory is applied to nucleons despite the fact that nucleons are fermions, spin 1/2 particles best described by fourcomponent Dirac spinor fields. Nevertheless, the scalar theory is shown to give a good fit to experiment for the total cross sections for several reactions [1]. The results are consistent with more elaborate spinor models involving one boson exchange (OBE). The results indicate that the spin and isospin of nucleons can to some extent be ignored under certain conditions. Being able to ignore spin and isospin greatly reduces the complexity of the model. A limitation of the scalar theory is that it does not distinguish between particle and anti-particle. Consequently one must decide how to interpret the s-channel diagrams generated by the theory, diagrams which involve particle creation and annihilation. The issue is resolved by extending the scalar theory to include electric charge, and formulating NN interactions in terms of complex scalar fields, which are able to describe both particles and anti-particles. A generalized Bethe-Salpeter equation (GBSE) is developed to handle inelastic processes in the ladder approximation. The GBSE, formulated using the scalar theory, is new, and introduces a systematic method for analyzing families of coupled reactions. A formalism is developed centered around the amplitude matrix M' defined for a given Lagrangian. M' gives the amplitudes of a family of reactions that arise from the Lagrangian. The formalism demonstrates how these amplitudes, to 2nd order, segregate into independent groups of coupled BSE's. The GBSE formalism is applied to the coupled BSE (CBSE) of Faassen and Tjon (FT) [2] for the reaction N+N->N+Delta, showing that the CBSE is missing a coupling channel, and in the expansion, under counts ladder diagrams. A proof is given of the equivalence of the series of ladder diagrams generated by M' and the S-matrix. A section on future work discusses several projects for further development and application of the GBSE.

Scientific phenomena are often studied through collections of related scalar fields such as data generated by simulation experiments that are parameter or time dependent . Exploration of such data requires robust measures to compare them in a feature aware and intuitive manner. Topological data analysis is a growing area that has had success in analyzing and visualizing scalar fields in a feature aware manner based on the topological features. Various data structures such as contour and merge trees, Morse-Smale complexes and extremum graphs have been developed to study scalar fields. The extremum graph is a topological data structure based on either the maxima or the minima of a scalar field. It preserves local geometrical structure by maintaining relative locations of extrema and their neighborhoods. It provides a suitable abstraction to study a collection of datasets where features are expressed by descending or ascending manifolds and their proximity is of importance. In this thesis, we design a measure to understand the similarity between scalar fields based on the extremum graph abstraction. We propose a topological structure called the complete extremum graph and define a distance measure on it that compares scalar fields in a feature aware manner. We design an algorithm for computing the distance and show its applications in analyzing time varying data such as understanding periodicity, feature correspondence and tracking, and identifying key frames.

In the first part of the thesis, we carried out direct two-dimensional (2D) multipoint measurements of the velocity fields in a turbulent Rayleigh-Benard convection cell to study the properties of small-scale convective turbulence. The local homogeneity and isotropy of the velocity field are tested using a number of criteria and are found to hold to an excellent degree. The properties of velocity circulation Gammar are also studied. The results show that the circulation appears to be more effective to capture the effect of local anisotropy than the velocity field itself. The distribution of Gammar is found to depend on the scale r, reflecting strong intermittency. It is further found that velocity circulation has the same anomalous scaling exponents as the longitudinal and transverse structure functions for low-order moments (p ≲ 5). Whereas, for high-order moments (p ≳ 5), the anomalous scaling exponents for circulation are found to be systematically smaller than the scaling exponents of the longitudinal and transverse structure functions. / In the second part of the thesis, the simultaneous visualization of the temperature and velocity fields was used to study the properties of thermal plumes. Our visualization reveals the process of the morphological evolution between sheetlike and mushroomlike plumes, which were also quantified by the height dependence of plume numbers and of vorticity fluctuations. A direct connection between the heat transport and coherent structures, i.e. thermal plumes, was established, which shows that it is plume number that primarily determines the Nu-Ra scaling relation. Individual plumes were extracted and their statistical and geometric properties were studied. It is found that the log-normal distribution is universal for thermal plumes and the log-normal statistics may be used to model them. In addition, both our quantitative characteristic and direct 3D spatial visualizations indicate that the previously-believed sheetlike plumes should be reconsidered to be only one-dimensional structures. / In the third part of the thesis, the planar laser-induced fluorescence technique was induced to study the 2D passive scalar mixing in high-Schmidt-number buoyancy-driven turbulence. The passive scalar mixing evolution was studied and various geometric properties, such as shape complexity, fractal dimension and local curvature, were used to characterize the isoconcentration contours of the 2D passive scalar fields. It is found that when the flow gets more turbulent the shape of passive scalar packets becomes closer to a circular shape and the passive scalar mixing becomes more isotropic, indicating the increased mixing and stirring of the turbulent flow. / The objective of this thesis is to address the following three key issues in turbulent thermal convection, i.e. turbulent fluctuations in small scales, coherent structures and passive scalar mixing in buoyancy-driven turbulence. / Zhou, Quan = 對流熱湍流中能量級串, 相干結構和標量場混合的实验研究 / 周全. / Adviser: Ke-Qing Xia. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3576. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 103-117). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307. / Zhou, Quan = Dui liu re tuan liu zhong neng liang ji chuan, xiang gan jie gou he biao liang chang hun he de shi yan yan jiu / Zhou Quan.

This thesis develops novel analytic models of scalar-tensor theories with quadratic coupling. In this framework, the coupling strength between scalar and matter is regulated in a way that allows the vacuum expectation value to vanish for low matter densities while becoming non-vanishingly large in the high-density regime. This results in significant deviations from the predictions of General Relativity in the strong-gravity regime. In astrophysics, we addressed the core-collapse supernova problem to account for the apparently missing energy required to explain the observed powerful explosions. We assumed a small, massless scalar gravitational field, thus allowing General Relativity to be recovered in the weak-gravity asymptotic limit. The non-trivial effects coming from the coupling function in the presence of a high-density field were analyzed at the instant of neutron star formation. Our results show that the scalar gravitational field evolves from a cosmological value to a new equilibrium via a Higgs-like mechanism. Additionally, the calculations associated with the gravitational binding energy shift and relevant relaxation timescale are explicitly shown. The full theory space of the model was also investigated for positive values of the coupling parameter. We studied a mechanism to address the stalled shock issue in core-collapse scenarios, which involved the application of sufficiently large positive values to the coupling parameter. Our results show that pulsating neutron stars act like optical cavities in which resonant scalar waves are parametrically amplified. It implies that the surface of a neutron star acts like an anti-phase reflector, releasing traveling scalar gravitational waves similar to an optical laser. In cosmology, the same framework was applied to a generic Friedman-Robertson-Walker universe involving general metric coupling and scalar potential functions. In cosmology, the same framework was applied to a generic Friedman-Robertson-Walker universe involving general metric coupling and scalar potential functions. We developed a mechanism which allowed the scalar field to be dynamically trapped, thus generating a potential capable of driving primordial inflation. Our results show that a trapped scalar field produces non-trivial dynamical consequences when applied to standard cosmology. Additionally, our analytic solutions for the generic inflationary behaviour, produce acceptable duration and e-foldings, thus recovering the Hubble parameter which is consistent with the present-day value. A feature of our cosmological model is that the universe can undergo several accelerating or decelerating phases, even though the scalar potential and metric coupling are monotonic functions overall. As this is important for the current dark energy problem, the quasi-static motion of the gravitational field induced by the scalar potential in the early universe, is investigated for a small value of the scalar field with normalized metric at the present time. Our results show that a variable Lambda Cold Dark Matter universe emerges naturally from the quadratic model.

This study analyzes a turbulent passive scalar field using two-point and three-point correlations of the fluctuating scalar field. Multipoint correlation functions are investigated because they retain scaling property information and simultaneously probe the concentration field for the spatial structure of the scalar filaments. Thus, multipoint correlation functions provide unique information about the spatial properties of the concentration filaments. The concentration field is created by the iso-kinetic release of a high Schmidt number dye into a fully developed turbulent boundary layer of an open channel flow. The concentration fields were previously measured using the planar laser-induced fluorescence technique. The two-point correlations of the fluctuating scalar field indicate that as the scalar field evolves downstream, the anisotropic influence of the tracer injection method diminishes, and the scalar field becomes dominated by the mean velocity shear. As the scalar filaments align with the mean velocity gradient, the elliptical shape associated with the contours of the correlation function tilts in the direction of the mean velocity gradient. As a result, the two-point correlation contours of the concentration fluctuations indicate that anisotropic conditions (i.e. the tilted, asymmetric, elliptical shape) develop as a consequence of the mean velocity shear. Three-point correlations of the fluctuating scalar field are calculated based on configuration geometries defined by previous researchers. The first configuration follows Mydlarski and Warhaft (1998), which employs two cold-wire measurements and Taylor's frozen turbulence hypothesis. The three-point correlation contours of the concentration fluctuations associated with the cold-wire measurements exhibit a symmetric characteristic V-shape. Similar symmetric properties are observed in the current study. The second set of configurations follows on recent theoretical predictions, which indicate that the three-point correlation of the fluctuating scalar field is dependent on the size, shape, and orientation of the triangle created by the three points. The current study analyzes two geometric configurations (isosceles and collinear). The geometric configurations are defined to ensure that the influence of the shape remains constant as the configuration is rotated, translated, and dilated. Additionally, the scaling exponent in the inertial-convective regime is calculated to determine the dependence of the correlation function on the size of the triangle pattern.

Not available

Errata inserted opposite p.177. / Bibliography: p. 173-177. / x, 177 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / The classical dynamical structure of cosomological models in which the matter content of the universe consists of a scalar field with arbitrary non-negative potential is analyzed in full. (abstract) / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1996?

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.