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enumeration of lattice paths and walks
Title
The enumeration of lattice paths and walks.
Creator
Gao, Shanzhen., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
A well-known long standing problem in combinatorics and statistical mechanics is to find the generating function for self-avoiding walks (SAW) on a two-dimensional lattice, enumerated by perimeter. A SAW is a sequence of moves on a square lattice which does not visit the same point more than once. It has been considered by more than one hundred researchers in the pass one hundred years, including George Polya, Tony Guttmann, Laszlo Lovasz, Donald Knuth, Richard Stanley, Doron Zeilberger,...
Show moreA well-known long standing problem in combinatorics and statistical mechanics is to find the generating function for self-avoiding walks (SAW) on a two-dimensional lattice, enumerated by perimeter. A SAW is a sequence of moves on a square lattice which does not visit the same point more than once. It has been considered by more than one hundred researchers in the pass one hundred years, including George Polya, Tony Guttmann, Laszlo Lovasz, Donald Knuth, Richard Stanley, Doron Zeilberger, Mireille Bousquet-Mlou, Thomas Prellberg, Neal Madras, Gordon Slade, Agnes Dit- tel, E.J. Janse van Rensburg, Harry Kesten, Stuart G. Whittington, Lincoln Chayes, Iwan Jensen, Arthur T. Benjamin, and many others. More than three hundred papers and a few volumes of books were published in this area. A SAW is interesting for simulations because its properties cannot be calculated analytically. Calculating the number of self-avoiding walks is a common computational problem. A recently proposed model called prudent self-avoiding walks (PSAW) was first introduced to the mathematics community in an unpublished manuscript of Pra, who called them exterior walks. A prudent walk is a connected path on square lattice such that, at each step, the extension of that step along its current trajectory will never intersect any previously occupied vertex. A lattice path composed of connected horizontal and vertical line segments, each passing between adjacent lattice points. We will discuss some enumerative problems in self-avoiding walks, lattice paths and walks with several step vectors. Many open problems are posted.
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Date Issued
2011
PURL
http://purl.flvc.org/FAU/3183129
Subject Headings
Combinatorial analysis, Approximation theory, Mathematical statistics, Limit theorems (Probabilty theory)
Format
Document (PDF)
Geometric Representation of Continued Fractions
Title
Geometric Representation of Continued Fractions.
Creator
Escuder, Ana, Peitgen, Heinz-Otto, Florida Atlantic University
Abstract/Description
The theory of continued fractions goes possibly as far back as 300 B.C. to Euclid. Some consider continued fractions as part of the "lost mathematics". I came across continued fractions for the first time while taking a graduate math class and I was amazed as how beautiful the representation of some numbers were using them and also how easy it was to understand the theory by making pictures. As an educator, I want to share this knowledge with my students and make it something that they could...
Show moreThe theory of continued fractions goes possibly as far back as 300 B.C. to Euclid. Some consider continued fractions as part of the "lost mathematics". I came across continued fractions for the first time while taking a graduate math class and I was amazed as how beautiful the representation of some numbers were using them and also how easy it was to understand the theory by making pictures. As an educator, I want to share this knowledge with my students and make it something that they could discover, manipulate, and appreciate. This paper is divided m three sections. The first deals with the basic theory and definitions of continued fractions. The second shows how to use technology, especially Dynamic Geometry Software to make the basic theory accessible to students in middle and high school. The third section gives some reflections of my experience working with students on continued fractions.
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Date Issued
2007
PURL
http://purl.flvc.org/fau/fd/FA00000751
Subject Headings
Combinatorial enumeration problems, Continued fractions, Metric spaces, Approximation theory
Format
Document (PDF)
Sparse and low rank constraints on optical flow and trajectories
Title
Sparse and low rank constraints on optical flow and trajectories.
Creator
Gibson, Joel, Marques, Oge, Florida Atlantic University, College of Engineering and Computer Science, Department of Computer and Electrical Engineering and Computer Science
Abstract/Description
In this dissertation we apply sparse constraints to improve optical flow and trajectories. We apply sparsity in two ways. First, with 2-frame optical flow, we enforce a sparse representation of flow patches using a learned overcomplete dictionary. Second, we apply a low rank constraint to trajectories via robust coupling. We begin with a review of optical flow fundamentals. We discuss the commonly used flow estimation strategies and the advantages and shortcomings of each. We introduce the...
Show moreIn this dissertation we apply sparse constraints to improve optical flow and trajectories. We apply sparsity in two ways. First, with 2-frame optical flow, we enforce a sparse representation of flow patches using a learned overcomplete dictionary. Second, we apply a low rank constraint to trajectories via robust coupling. We begin with a review of optical flow fundamentals. We discuss the commonly used flow estimation strategies and the advantages and shortcomings of each. We introduce the concepts associated with sparsity including dictionaries and low rank matrices.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004286, http://purl.flvc.org/fau/fd/FA00004286
Subject Headings
Approximation theory -- Mathematical models, Computer vision, Image processing -- Digital techniques, Information visualization
Format
Document (PDF)
Maximally Prüfer rings
Title
Maximally Prüfer rings.
Creator
Sharma, Madhav, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong...
Show moreIn this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004465, http://purl.flvc.org/fau/fd/FA00004465
Subject Headings
Approximation theory, Commutative algebra, Commutative rings, Geometry, Algebraic, Ideals (Algebra), Mathematical analysis, Prüfer rings, Rings (Algebra), Rings of integers
Format
Document (PDF)