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School of Mathematics

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College of Sciences

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

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Now showing 1 - 10 of 14

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.
Automate Reasoning: Computer Assisted Proofs in Set Theory Using Godel's Algorithm for Class Formation

(Georgia Institute of Technology, 2004-08-17) Goble, Tiffany Danielle

Automated reasoning, and in particular automated theorem proving, has become a very important research field within the world of mathematics. Besides being used to verify proofs of theorems, it has also been used to discover proofs of theorems which were previously open problems. In this thesis, an automated reasoning assistant based on Godel's class theory is used to deduce several theorems.
Minimal Surfaces in three-sphere with special spherical symmetry

(Georgia Institute of Technology, 2004-07-14) Hynd, Ryan Charles

We introduce the notion of special spherical symmetry and classify the complete regular minimal surfaces in the three sphere having this symmetry. We also show that the Clifford torus is the unique embedded minimal torus in three sphere possessing special spherical symmetry.
On the Structure of Counterexamples to the Coloring Conjecture of Hajós

(Georgia Institute of Technology, 2004-05-20) Zickfeld, Florian

Hajós conjectured that, for any positive integer k, every graph containing no K_(k+1)-subdivision is k-colorable. This is true when k is at most three, and false when k exceeds six. Hajós' conjecture remains open for k=4,5. We will first present some known results on Hajós' conjecture. Then we derive a result on the structure of 2-connected graphs with no cycle through three specified vertices. This result will then be used for the proof of the main result of this thesis. We show that any possible counterexample to Hajós' conjecture for k=4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajós' conjecture for k=4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K_5-subdivision.
Density conditions on Gabor frames

(Georgia Institute of Technology, 2003-12-01) Leach, Sandie Patricia