Organizational Unit:

School of Mathematics

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https://hdl.handle.net/1853/70754

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Organizational Unit

College of Sciences

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https://finding-aids.library.gatech.edu/agents/corporate_entities/245

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Publication Search Results

Now showing 1 - 10 of 101

On difference graph covers and the local dimension of the Boolean lattice

(Georgia Institute of Technology, 2022-05-03) Hall, Ariana

N/A
Decay of Entropy and Information for multidimensional Kac models

(Georgia Institute of Technology, 2021-12-14) Hauger, Lukas Alexander

We study the approach to equilibrium in relative entropy of systems of gas particles modeled via the Kac master equation in arbitrary dimensions. First, we study the Kac system coupled to a thermostat, and secondly connected to a heat reservoir. The use of the Fisher-information allows simple proofs with weak regularity assumptions. As a result, we obtain exponential decay rates for the entropy and information that are essentially independent of the size of the systems.
Numerical Estimation of Several Topological Quantities of the First Passage Percolation Model

(Georgia Institute of Technology, 2021-04-13) Ma, Yuanzhe

In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and the location of each hole for B(t). We discuss the theoretical background of this work, the algorithm we used to conduct simulations, and include an extensive discussion of our simulation results. Our results support some of the prior conjectures and further introduce several interesting open problems.
Numerical estimates for arm exponents and the acceptance profile in two-dimensional invasion percolation

(Georgia Institute of Technology, 2020-05-05) Li, Jiaheng

The main object of this thesis is to numerically estimate some conjectured arm exponents when there exist a number of open paths and closed dual paths that extend to the boundary of different sizes of boxes centering at the origin in bond invasion percolation on a plane square lattice by Monte-Carlo simulations. The results turn out to be supportive of the conjectured value in some case. The numerical estimate for the acceptance profile of invasion percolation at the critical point is also obtained, which suggests a neighborhood in which the liminf and limsup of the acceptance profile might fall. An efficient algorithm to simulate invasion percolation and to find disjoint paths on most regular 2-dimensional lattices are discussed.
Non-negative symmetric polynomials and entangled Bosons

(Georgia Institute of Technology, 2018-09-07) Hebbe Madhusudhana, Bharath

The fundamental relation between quantum entanglement and convex algebraic geometry has unveiled a set of powerful tools, imported from the former to the study the latter. The space of separable mixed states is convex and so is the space of the corresponding observable values. Therefore, the problem of determining whether a set of observable values come from an entangled state is tantamount to checking for membership in a convex set, of a point with coordinates given by the set of observable values. Here, we use techniques from convex algebraic geometry to develop powerful criteria for entanglement in a many-body system of Bosonic atoms with a non-zero spin. The experimentally accessible observables are the spin expectation values and <{S_i, S_j}> which, upon truncating at rank two, are 9 independent quantities. Recently, entanglement criteria in terms of 3 of these 9 quantities have been derived. We develop entanglement criteria using all of these 9 quantities. If we consider these numbers as coordinates of a point in a 9 dimensional space, those with a separable parent state lie within a convex region, also known as the moment cone. Therefore, the problem is to characterize this moment cone. Owing to the Bosonic exchange symmetry, this moment cone is the dual of the cone of non-negative symmetric polynomials. Together with a characterization of this cone, we also develop an entanglement criterion that is asymptotically tight, for large atom numbers.
Open book decompositions in high dimensional contact manifolds

(Georgia Institute of Technology, 2016-05-27) Elmas, Gokhan

In this thesis, we study the open book decompositions in high dimensional contact manifolds. We focus on the results about open book decomposition of manifolds and their relationship with contact geometry.
Limit theorems for a one-dimensional system with random switchings

(Georgia Institute of Technology, 2010-11-15) Hurth, Tobias

We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute its unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero and infinity and derive analogues of classical probabilistic results such as the central limit theorem and large deviations principle.
Mahler's conjecture in convex geometry: a summary and further numerical analysis

(Georgia Institute of Technology, 2010-08-09) Hupp, Philipp

In this thesis we study Mahler's conjecture in convex geometry, give a short summary about its history, gather and explain different approaches that have been used to attack the conjecture, deduce formulas to calculate the Mahler volume and perform numerical analysis on it. The conjecture states that the Mahler volume of any symmetric convex body, i.e. the product of the volume of the symmetric convex body and the volume of its dual body, is minimized by the (hyper-)cube. The conjecture was stated and solved in 1938 for the 2-dimensional case by Kurt Mahler. While the maximizer for this problem is known (it is the ball), the conjecture about the minimizer is still open for all dimensions greater than 2. A lot of effort has benn made to solve this conjecture, and many different ways to attack the conjecture, from simple geometric attempts to ones using sophisticated results from functional analysis, have all been tried unsuccesfully. We will present and discuss the most important approaches. Given the support function of the body, we will then introduce several formulas for the volume of the dual and the original body and hence for the Mahler volume. These formulas are tested for their effectiveness and used to perform numerical work on the conjecture. We examine the conjectured minimizers of the Mahler volume by approximating them in different ways. First the spherical harmonic expansion of their support functions is calculated and then the bodies are analyzed with respect to the length of that expansion. Afterwards the cube is further examined by approximating its principal radii of curvature functions, which involve Dirac delta functions.
Polar - legendre duality in convex geometry and geometric flows

(Georgia Institute of Technology, 2008-07-10) White, Edward C., Jr.

This thesis examines the elegant theory of polar and Legendre duality, and its potential use in convex geometry and geometric analysis. It derives a theorem of polar - Legendre duality for all convex bodies, which is captured in a commutative diagram. A geometric flow on a convex body induces a distortion on its polar dual. In general these distortions are not flows defined by local curvature, but in two dimensions they do have similarities to the inverse flows on the original convex bodies. These ideas can be extended to higher dimensions. Polar - Legendre duality can also be used to examine Mahler's Conjecture in convex geometry. The theory presents new insight on the resolved two-dimensional problem, and presents some ideas on new approaches to the still open three dimensional problem.
Finite Field Models of Roth's Theorem in One and Two Dimensions

(Georgia Institute of Technology, 2006-06-05) Hart, Derrick N.

Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.