# Katz School of Science and Health: Mathematical Sciences Dissertations

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12202/5754

## Browse

### Recent Submissions

Item Open Access Applications of dynamical systems in dissipative mechanics and in topological data analysis(Yeshiva University, 2024-05) Akingbade, Samuel Wale; Gidea, Marian; Chen, Wenxiong; Tere M-Seara, Tere; Marini, Antonella•This dissertation is devoted to two distinct research endeavors on the applications of Dynamical Systems: one focuses on Dissipative Mechanics and the other on Topological Data Analysis. •The first research investigates a simple model of a mechanical system, consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation and also subject to a small dissipative perturbation. The dissipative perturbation transforms the system into a non-Hamiltonian one, altering the symplectic structure to a conformally symplectic one. •We demonstrate that this system exhibits Arnold diffusion, that is, there exist pseudo-orbits along which the energy of the rotator subsystem grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. Our finding's physical significance is that, despite the dissipation effects, it is possible to overall gain a significant amount of energy over time. •In the second research, we propose a heuristic argument for why Topological Data Analysis (TDA) is able to detect early warning signals of critical transitions in financial times series. We focus on the time period of a bubble before the tipping point. The ansatz is that the time series during such a period follows the Log-Periodic Power Law Singularity (LPPLS) model. The application of dynamical systems theory, particularly through Takens' embedding theorem and its generalization, play a role in the analysis using delay coordinate embedding. •The outcome of our work is that whenever the LPPLS model fits the data, the TDA method produces early warning signals of critical transitions. As an illustration, we apply both the LPPLS model and TDA to Bitcoin, which has undergone numerous phases of extreme price growth and massive crashes, providing a rich dataset for exploring financial bubbles. •Our experiments demonstrate a strong agreement between the TDA method and the LPPLS model. Moreover, even in cases where the LPPLS model fits poorly with some of the data, the TDA method still exhibit a relatively strong signal before the tipping point. This research marks the first justification of the TDA method in terms of a deterministic model for the expected dynamics of financial bubbles. •In this dissertation, Chapters 1 through 4 are derived from the first research, while Chapters 5 through 8 are based on the second research. Some of the results presented have been published in [1] and [2].Item Open Access Topics in fractional Laplacian and dynamical systems(Yeshiva University, 2023-06) Liu, Xingyu; Chen, Wenxiong; Nandori, Peter; Gidea, Marian; Marini, Antonella; Catrina, FlorinAbstract In this thesis, we consider problems involving the $n$-dimensional fractional Laplacians including elliptic equations and parabolic equations. We also consider the problems involving fractional Monge-Amp\'ere operators. The thesis is mostly devoted to presenting our original work on the progress obtained in the development of direct methods that can effectively deal with the above problems. ¶ In the second part of the work, we are interested in the length of a few consecutive long free flights in infinite horizon Lorentz Gas. In dimension D=2, it is well known that a flight of length T>>1 is typically followed by a flight of length $C\sqrt{T}$. Here, we extend this result to any dimension $D$.Item Open Access Gravitational and electrostatic potential fields and dynamics of non-spherical systems(2020-07) Lam, Wai-Ting; Gidea, Marian; Zypman, FredyThis thesis is devoted to several aspects of the n-body problem in the context of two models of interest: the gravitational n-body problem and the electrostatic n-body problem. In the case of gravitational n-body problem, we study central configurations of three oblate bodies, the Hill approximation of the restricted four body problem with three oblate heavy bodies, and we find the equilibrium points of the Hill approximation and determine their linear stability. Also in the case of the gravitational n-body problem, we find equilibrium shapes of an irregular body, when the gravitational potential and the rotational potential balance each other. In particular, we find equilibrium dumbbell shapes. In the context of the electrostatic n-body problem, we use variational methods to find approximate solutions of the Poisson-Boltzmann equation, representing the electrostatic potential produced by charged colloidal particles. This research is motivated by applications to astrodynamics, dynamical astronomy and atomic force microscopy.Item Restricted Qualitative Properties for Positive Solutions of Nonlocal Equations(2020-05-28) Hu, Yunyun; Gidea, Marian; Chen, Wenxiong; Marini, AntonellaThis thesis is devoted to the study of properties for nonnegative solutions to nonlocal problems and integral equations. The main tools we use are the method of moving planes and the method of moving spheres._____________ First, we focus on the nonlocal problems involving fractional p-Laplacian (p 2) in unbounded domains. Without assuming any asymptotic behavior of positive solutions near in nity, we develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of positive solutions.__________ Second, we study the symmetry of positive solutions for nonlinear equations involving fractional Laplacian. In bounded domain, we prove that all positive solutions of fractional equations with Hardy Leray potential are radically symmetric about the origin. Then we consider a nonlocal problem in unbounded cylinders. By using the method of moving planes, we establish the symmetry and monotonicity of positive solutions. Furthermore, we obtain the nonexistence of nonnegative solutions for nonlocal problems in the whole space Rn.___________ Third, we establish a strong maximum principle and a Hopf type lemma for antisymmetric solutions of fractional parabolic equations in unbounded domains. These will become most commonly used basic techniques in the study of monotonicity and symmetry of solutions.________ Finally, we consider general integral systems on a half space and integral equations in bounded domains. Under natural integrability conditions, we obtain a classi cation of positive solutions for an integral system on half space by using a slight variant of the method of moving spheres. Here we removed the global integrability hypothesis on positive solutions by introducing some new ideas. In addition, we study the symmetry and monotonicity of positive solutions to over-determined problems and partially overdetermined problems. The main technique we use is the method of moving planes in an integral form.Item Metadata only Classical mechanics over fields of characteristic p greater than 0(ProQuest Dissertations & Theses, 1986) Merewether, James WilliamA formulation of the algebraic structures of a classical mechanics over fields of characteristic p {dollar}>{dollar} 0 is presented. It is shown how a definition of an abstract mechanics, which is a composition class of two-product algebras, allows for realizations over fields of characteristic p {dollar}>{dollar} 0. A broad class of such realizations of simple two-product algebras is derived and their structure examined. They are shown to be simple nodal noncommutative Jordan algebras, defined by a non-degenerate skew-symmetric bilinear form, over the field of intergers modulo p (p prime). Equations for determination of the automorphism group of the algebras are deduced. The affine restriction of the canonical group is given explicitly. Finally, a characteristic p {dollar}>{dollar} 0 analog of the Galilei group is given and its canonical realizations are produced.Item Metadata only THE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS(ProQuest Dissertations & Theses, 1984) GELMAN, ALEXANDERThis dissertation considers three problems associated with the Klein-Gordon Equation: (a) The conditions for the operator to be self-adjoint; (b) The existence of the wave operator; and (c) The completeness of the wave operator. These problems are considered for the operator with general and oscillating potentials.;For problem (a) the work is based on the theory of forms extensions originated by K. Friederichs; and for problems (b) and (c), the abstract theory of scattering which originated in the work of Kato and Birman. The particular result which we use for problems (a) and (b) is the recent theorem proven by M. Schechter, in which he was able to relax requirements on J (no requirement for the bijectivity of J, and no reference to R(z), for example).;Application of the methods described above to the Klein-Gordon operator allowed us to solve the three problems above for an unbounded operator J and also for the oscillating potential.Item Metadata only THE SPECTRA OF THE SCHROEDINGER OPERATOR(ProQuest Dissertations & Theses, 1984) BRENNER, TERENCEWe look at the Schrodinger operator H=-(DELTA)+q(x) where (DELTA) is the Laplacian and q(x)(epsilon)R('n). We give sufficient conditions for the spectrum of H to contain the interval of the form {lcub}a,(INFIN)) and sufficient conditions for the essential spectrum of H to contain the interval of the form {lcub}b,(INFIN)). Our estimates for the lower bounds of a and b are positive numbers. We allow q(x) to be negative in some region. Our results are in R('2) and in R('n).Item Metadata only THE INVARIANCE PRINCIPLE AND ASYMPTOTIC COMPLETENESS FOR A QUANTUM MECHANICAL SYSTEM(ProQuest Dissertations & Theses, 1982) HERZBERG, MARTINWe study operators H and H(,0) acting on the Hilbert Space L('2)(R). The free or unperturbed operator H(,0) is the multiplication (also known as the position) operator x. We consider H = H(,0) + A, where the perturbation A is an integral operator which may be unbounded. H(,0) is known to be self-adjoint. We give conditions on the perturbation A for H to be self-adjoint as well.;The primary objective of this dissertation is to prove the main conclusions of scattering theory for operators of the type just described. Both time-dependent and stationary methods are utilized. We use a factored perturbation technique to prove existence and completeness of the wave operators W (+OR-) (H,H(,0)). We also obtain conditions for the existence of the wave operators for a generalized integral perturbation where A is not factorable. This is accomplished by first finding operators L(,0) and B in momentum (Fourier-transform) space which are equivalent to our operators H and H(,0) in configuration space. Then, by showing that the two pairs of Hamiltonians are equivalent, we have in effect proved that all theorems proved for H and H(,0) are true for L (= L(,0) + B) and L(,0), and vice versa. We can then examine the operators L and L(,0) and transfer our results to H and H(,0).Item Metadata only ERGODIC PROPERTIES OF A PARTICLE IN CONTACT WITH A HEAT BATH(ProQuest Dissertations & Theses, 1980) RAVISHANKAR, KRISHNAMURTHIWe study the ergodic properties of a system of point particles on the semi-infinite line. The particles evolve under a dynamics where the only interaction is between the first particle and each of the other particles through elastic collision. The first particle is constrained to stay within the spatial region {lcub}0,1{rcub} by a reflective boundary condition. We choose the Grand Canonical Gibbs measure (mu) appropriately normalized as our invariant measure.;The motivation for this work comes from physics. This system can be considered as a model for a finite system (first particle) in equilibrium with a reservoir (the rest of the particles). We show that the system, starting from an initial state (measure) concentrated on any point in its phase-space with the reservoir being in the equilibrium state, evolves under the time evolution described above to the equilibrium state (defined appropriately w.r.t. the reservoir state).;In order to study the ergodic properties of the finite system under the stochastic time evolution defined by the reservoir, we study the stochastic process (Q(w,t,),V(w,t)) of the position and velocity of the first particle. This is a non-Markovian process. By appropriately enlarging the state space (giving more information!) we obtain a Markov process. We prove that this process is an ergodic, Harris process with no cyclic classes. This we do by essentially showing that the transition probability grows an absolutely continuous component. This enables us to invoke the powerful theorems from erodic theory of Harris processes, which yield strong convergence of measures under time evolution. We then use this convergence result on the Markov process to show that.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;in absolute variations norm for all (q,v) (ELEM) {lcub}0,1{rcub} X (//R). This result implies in particular that the whole semi-infinite system is a Bernoulli flow.;Most of the results obtained are valid in the case where the mass of the first particle is either equal or different from the mass of the other particles.Item Metadata only WISHART EXPECTATION OPERATORS AND INVARIANT DIFFERENTIAL OPERATORS(ProQuest Dissertations & Theses, 1980) KUSHNER, HOWARD BURTLet E(,n) denote the expectation operator of the Wishart distribution W(k,n,(SUMM)) and let E(,n)('t) denote the "conditional" expectation operator defined by.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where (VBAR)V(VBAR) denotes the determinant of the k x k positive definite symmetric matrix V, trA is the trace of A, (SUMM) is a k x k positive definite symmetric matrix, C(,n) is a constant depending on n, and dV denotes Lebesgue measure on the space of k x k positive definite matrices, V > 0. We study the common eigenfunctions of the operators E(,n) (n (GREATERTHEQ) n(,0)) and the common eigenfunctions of the operators E(,n)('t) (n (GREATERTHEQ) n(,0)) which are shown to be identical with the common eigenfunctions of certain invariant partial differential operators. The eigenvalues of the expectation operators are determined in terms of the eigenvalues of the differential operators, and vice-versa. Explicit expressions for (lamda)(,n), the eigenvalues of E(,n), somewhat more complete than those obtained previously by Maass, are given. (lamda)(,n)('t), the eigenvalues of E(,n)('t), are characterized via a unique solution to a certain ordinary differential equation. An asymptotic formula for (lamda)(,n)('t), as t (--->) (INFIN), is proved, as well as for functions defined by integrals of the form.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where f(V) is homogeneous function. Irreducible class 1 subspaces, with respect to the congruence transformations of G (k) are constructed from the common eigenfunctions of E(,n). An integral formula for the most general common eigenfunction of E(,n) is proved.Item Metadata only A COVARIANT FRAMEWORK FOR MASS MONOPOLE IN GENERAL RELATIVITY(ProQuest Dissertations & Theses, 1979) SCHLEIFER, NATHANItem Metadata only THE FOURIER ANALYSIS THEORY OF UNSOLVABILITY IN PARTIAL DIFFERENTIAL EQUATIONS(ProQuest Dissertations & Theses, 1979) LEVIN, RONALD STUARTItem Metadata only COMPUTER SIMULATION STUDIES OF BINARY ALLOYS(ProQuest Dissertations & Theses, 1978) PHANI, MOHAN KUMARItem Metadata only DIELECTRIC AND PYROELECTRIC PROPERTIES OF FERROELECTRICS(ProQuest Dissertations & Theses, 1978) TSUO, YUAN-HUAIItem Metadata only PINCHED RELATIVISTIC ELECTRON BEAMS AND COLLECTIVE ACCELERATION OF IONS(ProQuest Dissertations & Theses, 1979) SERLIN, VICTORItem Metadata only STABILITY IN A RANDOM ENVIRONMENT(ProQuest Dissertations & Theses, 1979) SCHWARTZ, MICHAEL STEPHENItem Metadata only A MATHEMATICAL THEORY OF MUSICAL COMBINATION TONES(ProQuest Dissertations & Theses, 1977) VANDE KOPPLE, JULIUS JOHNItem Metadata only RESULTS ON THE STEPPING STONE MODEL(ProQuest Dissertations & Theses, 1977) RUSINEK, ROZAItem Metadata only ELECTROREFLECTANCE IN SEMICONDUCTORS AND ITS APPLICATION AS A METHOD OF MATERIALS CHARACTERIZATION(ProQuest Dissertations & Theses, 1977) OKEKE, CAJETAN EZEANIItem Metadata only FATOU THEOREMS FOR EIGENFUNCTIONS OF THE LAPLACE-BELTRAMI OPERATOR(ProQuest Dissertations & Theses, 1977) LINDEN, ORIN M.