Mathematical Sciences Dissertations https://hdl.handle.net/20.500.12202/5754 2020-08-04T17:43:13Z 2020-08-04T17:43:13Z Qualitative Properties for Positive Solutions of Nonlocal Equations. Hu, Yunyun https://hdl.handle.net/20.500.12202/5805 2020-07-17T14:27:38Z 2020-05-28T00:00:00Z Qualitative Properties for Positive Solutions of Nonlocal Equations. Hu, Yunyun This 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. Doctoral Dissertation, Ph.D. --- Opt-Out. For access, please contact: yair@yu.edu 2020-05-28T00:00:00Z Classical mechanics over fields of characteristic p greater than 0 Merewether, James William https://hdl.handle.net/20.500.12202/3200 2020-06-29T19:04:17Z 1986-01-01T00:00:00Z Classical mechanics over fields of characteristic p greater than 0 Merewether, James William A 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. 1986-01-01T00:00:00Z THE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS GELMAN, ALEXANDER https://hdl.handle.net/20.500.12202/2983 2020-06-26T17:10:18Z 1984-01-01T00:00:00Z THE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS GELMAN, ALEXANDER This 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. 1984-01-01T00:00:00Z THE SPECTRA OF THE SCHROEDINGER OPERATOR BRENNER, TERENCE https://hdl.handle.net/20.500.12202/2976 2020-06-26T17:10:19Z 1984-01-01T00:00:00Z THE SPECTRA OF THE SCHROEDINGER OPERATOR BRENNER, TERENCE We 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). 1984-01-01T00:00:00Z