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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Now showing 1 - 10 of 10

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
Model and analysis of the geometric characteristics of primary carpet backing

(Georgia Institute of Technology, 2003-12-01) Ford, Allison Elaine
The maximum clique problem - on finding an upper bound with application to protein structure alignment

(Georgia Institute of Technology, 2003-08) Baamann, Katharina
Conditional uniform convexity in Orlicz spaces and minimization problems

(Georgia Institute of Technology, 2002-08) Doto, James William
Bounding entropy and finding symbolic dynamics via the spectrum of the Conley index

(Georgia Institute of Technology, 2000-08) Baker, Anthony W.