The purpose of this thesis is to study those non-flat space-times in General Relativity admitting high dimensional Lie groups of motions, homotheties, conformals and affines, and to prove a theorem on the relationship between the first three of these. The basic theories and notations of differential geometry are set up first, and a useful theorem on first-order partial differential equations is proved. The concepts of General Relativity are introduced, space-times are defined and a brief account of the well-known Petrov and Segre classifications is given. The interplay between these classifications and the isotropy structure of the various Lie groups is discussed as is the so-called 'Schmidt method'. Generalised p.p. waves are studied, with a special study of the subclass of generalised plane waves undertaken, many different characterisations of these latter are found and their admitted symmetries are completely described. Motions, homotheties and affines are considered. A survey of symmetries in Minkowski space, and a summary of known results on space-times with high dimensional groups of motions is given. The problem of r-dimensional groups of homotheties is studied. The r 6 cases are completely resolved, and examples in the r = 5 cases are given. All examples of non-flat space-times admitting the maximal group of affines are displayed, correcting an error in the literature. The thesis ends with a proof of the Bilyalov-Defrise-Carter theorem, which states that for any non conformally flat space-time there is a conformally related metric for which the original group of conformals is a group of homotheties (motions if not conformal to generalised plane waves). The proof given does not use Bilyalov's analyticity assumption, and is more geometric than Defrise-Carter. The maximum size of the conformal group for a given Petrov type is found. An appendix gives a brief account of some REDUCE routines used to check some algebraic manipulations.

Master of Science - Science / Relationships between symmetries and conservation laws of perturbed partial differential equations are reviewed. Potential symmetries and their applications to perturbed partial differential equations and conservation laws are presented in detail. An example of a perturbed wave equation for an inhomogeneous medium is solved in detail. Proofs of some of the lesser-known theorems are outlined. A wide range of examples is given to further explain these concepts.

<p>In the course of exploring various symmetries of vertex-transitive graphs, we introduce the concept of quasi-normal subgroups in groups. This is done since the symmetries of vertex-transitive graphs are intimately linked to those, fait accompli, of groups. With this, we ask if the concept of strongly simple groups has a place for consideration. We have shown that for n &gt / 5, An, the alternating group on n odd elements, is not strongly simple.</p>

<p>In the course of exploring various symmetries of vertex-transitive graphs, we introduce the concept of quasi-normal subgroups in groups. This is done since the symmetries of vertex-transitive graphs are intimately linked to those, fait accompli, of groups. With this, we ask if the concept of strongly simple groups has a place for consideration. We have shown that for n &gt / 5, An, the alternating group on n odd elements, is not strongly simple.</p>

Thesis advisor: Elisenda Grigsby / The space of parameter vectors for a feedforward ReLU neural networks with any fixed architecture is a high dimensional Euclidean space being used to represent the associated class of functions. However, there exist well-known global symmetries and extra poorly-understood hidden symmetries which do not change the neural network function computed by network with different parameter settings. This makes the true dimension of the space of function to be less than the number of parameters. In this thesis, we are interested in the structure of hidden symmetries for neural networks with various parameter settings, and particular neural networks with architecture \((1,n,1)\). For this class of architectures, we fully categorize the insufficiency of local functional dimension coming from activation patterns and give a complete list of combinatorial criteria guaranteeing a parameter setting admits no hidden symmetries coming from slopes of piecewise linear functions in the parameter space. Furthermore, we compute the probability that these hidden symmetries arise, which is rather small compared to the difference between functional dimension and number of parameters. This suggests the existence of other hidden symmetries. We investigate two mechanisms to explain this phenomenon better. Moreover, we motivate and define the notion of \(\varepsilon\)-effective activation regions and \(\varepsilon\)-effective functional dimension. We also experimentally estimate the difference between \(\varepsilon\)-effective functional dimension and true functional dimension for various parameter settings and different \(\varepsilon\). / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Nucleation is the process by which a metastable phase decays into a stable phase. It is widely observed in nature, and is responsible for many phenomena such as the formation clouds and domains in crystalline solids. The classical theory of nucleation predicts that the objects that initiate the decay from the metastable to the stable phase are compact droplets whose interior has the structure of the stable phase. For quenches deep into the metastable phase, however, the droplets may be ramified, with a structure very different from the stable phase. This difference has profound implications for material properties, especially because predicting the onset of structure early enough is useful for manipulating and controlling nucleation processes. I used molecular dynamics to simulate nucleation in Lennard-Jonesium, a model system for liquid-solid transformations. The system is quenched from a high temperature, where the liquid is stable, to a temperature where the liquid is metastable, and is allowed to nucleate via fluctuation-driven clusters referred to as critical droplets. I determined the occurrence of critical droplets by the intervention method, but found a non-monotonic variation in droplet survival rates near the saddle point. I determined the structure of the critical droplet and found evidence for a core consisting of mostly solid-like particles with hcp symmetry and a previously unknown planar structure around it. Using perturbative techniques, I showed that the planar particles have a significant influence on the nucleation and growth of critical droplets. I also introduced a novel method of learning symmetries to predict the structure and appearance of precursors to the critical nucleus. My results give added evidence for the presence of spinodal nucleation at deep quenches.

The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations.In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation.After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups.The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.

An examination of some properties that interrelate the computational complexities of evaluating multivariate polynomial functions is presented. The kind of relationship between polynomial functions that is studied takes the form of linear transformations of the arguments and results of a polynomial function that transform it into another such function. Such transformations are a generalisation of projection (a form of reduction in algebraic complexity first introduced by Valiant, whereby variables and constants are substituted for the arguments of a polynomial function in order to transform it into another polynomial function). In particular, two restricted forms of this generalised projection are considered: firstly, those that relate a polynomial function to itself, and secondly, those that are invertable. Call these symmetries and similarities, respectively. The structure of the set of symmetries of a polynomial function is explored, and the computationally useful members of the set identified; a technique for finding all such symmetries is presented. It is shown that polynomials related by similarity have "isomorphic" sets of symmetries, and this condition may be used as a criterion for similarity. Similarity of polynomial functions is shown to be an equivalence relation, and "similar polynomials" can be seen to possess closely comparable complexities. A fast probabilistic algorithm for finding the symmetries of a polynomial function is given. The symmetries of the determinant and of the permanent (which differs from the determinant only in that all of its monomials have coefficients of +1), and those of some other polynomials, are explicitly found using the above theory. Fast algorithms using linear algebra for evaluating the determinant are known, whereas evaluating the permanent is known to be a #p-complete problem, and is apparently intractable; the reasons for this are exposed. As an easy corollary it is shown that the permanent is not preserved by any bilinear product of matrices, in con'trast to the determinant which is preserved by matrix multiplication. The result of Marcus and Minc, that the determinant cannot be transformed into the permanent by substitution of linear combinations of variables for its arguments (i.e. the permanent and determinant are not similar), also follows as an easy corollary. The relationship between symmetries and ease of evaluation is discussed.

In recent decades, the accessibility of nuclear physics has been greatly improved due to the advent of modern supercomputers, as well as theoretical developments in effective field theory and ab initio (first--principles) nuclear approaches. As a result, in modern nuclear theory it is possible to perform realistic quantum many--body calculations of nuclear systems, beginning solely from underlying Standard Model symmetries. A fundamental object of interest in nuclear structure are the nuclear densities, which may be abundantly used in calculation of other nuclear observables. Utilizing the ab initio no--core shell model, a rigorous theoretical approach for calculations involving light--nuclei, we study the coordinate space densities of various nuclear systems and discuss the importance of nonlocality and translation invariance in the densities. In particular, this property is investigated at length in the context of scattering theory, in which optical potentials are constructed from the ab initio no--core shell model densities. We explore the impacts of nonlocality and translation invariance in proton and antiproton scattering, and in the latter we review the first fully microscopic optical potential for antiproton--nucleus scattering. In addition, while the full problem is intractable at present, we assess the potential impact of many--nucleon dynamics on scattering observables. We additionally present an analytic computation of the nuclear kinetic density distribution, derived from the nonlocal nuclear densities. While the nuclear problem has become increasingly tractable, the computational barrier is still ever present, with nuclear calculations pushing the frontier of modern supercomputing. Many approaches have been developed to quell the computational demand, e.g. the similarity renormalization group approach. We introduce and discuss another approach, namely the natural orbitals unitary transformation, which has been shown increase the convergence rate of quantum many--body calculations. Lastly, in the past three years there has revitalized interest in reevaluation of particular Standard Model symmetries. Notably, the Cabibbo--Kobayashi--Maskawa quark mixing matrix has been established as a high--precision test of the Standard Model, capable of pointing to novel physics. Recent theoretical advances in corrections needed to evaluate unitarity of the Cabibbo--Kobayashi--Maskawa matrix have indicated a statistical discrepancy with the Standard Model expectation. In light of this development, using the ab initio no--core shell model with continuum, we pursue a high--precision calculation of the isospin symmetry breaking correction, $\delta_C$. This correction is one of two nuclear structure dependent corrections needed to shed light on this discrepancy, and potentially identify physics beyond the Standard Model. / Graduate

Orientador: Yuri Dimitrov Bozhkov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T01:04:06Z (GMT). No. of bitstreams: 1 SilvaJunior_ValterAparecido_M.pdf: 1356319 bytes, checksum: e64224c844e48a46487c6d387ec0f3e7 (MD5) Previous issue date: 2015 / Resumo: Nesta dissertação, obteremos as soluções exatas do Problema de Liouville-Gelfand (em uma e em duas dimensões) via grupos de Lie de simetrias / Abstract: In this dissertation, we shall obtain the exact solutions of the Liouville-Gelfand Problem (in one and in two dimensions) via Lie groups of symmetries / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada