DSpace Collection:
https://hdl.handle.net/20.500.12202/5754
2024-03-28T09:14:11ZTopics in fractional Laplacian and dynamical systems
https://hdl.handle.net/20.500.12202/9240
Title: Topics in fractional Laplacian and dynamical systems
Authors: Liu, Xingyu
Abstract: Abstract
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.
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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 $D$.
Description: Doctoral dissertation, PhD / Open Access2023-06-01T00:00:00ZGravitational and electrostatic potential fields and dynamics of non-spherical systems
https://hdl.handle.net/20.500.12202/6621
Title: Gravitational and electrostatic potential fields and dynamics of non-spherical systems
Authors: Lam, Wai-Ting
Abstract: This 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.
Description: Doctoral Dissertation, Ph.D., Katz School of Science and Health, Open-Access2020-07-01T00:00:00ZQualitative Properties for Positive Solutions of Nonlocal Equations
https://hdl.handle.net/20.500.12202/5805
Title: Qualitative Properties for Positive Solutions of Nonlocal Equations
Authors: Hu, Yunyun
Abstract: 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.
Description: Doctoral Dissertation, Ph.D., Department of Mathematical Science. --- Opt-Out. For access, please contact: yair@yu.edu2020-05-28T00:00:00ZClassical mechanics over fields of characteristic p greater than 0
https://hdl.handle.net/20.500.12202/3200
Title: Classical mechanics over fields of characteristic p greater than 0
Authors: Merewether, James William
Abstract: 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