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Some applications of algebraic methods in combinatorial geometry

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Some applications of algebraic methods in combinatorial geometry
Name (type = personal)
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Basit
NamePart (type = given)
Abdul
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1987-
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Abdul Basit
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author
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Steiger
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William
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William Steiger
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Advisory Committee
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chair
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Kahn
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Jeff
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Jeff Kahn
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Advisory Committee
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internal member
Name (type = personal)
NamePart (type = family)
Saraf
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Shubhangi
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Shubhangi Saraf
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Advisory Committee
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internal member
Name (type = personal)
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Aronov
NamePart (type = given)
Boris
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Boris Aronov
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Advisory Committee
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outside member
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Rutgers University
Role
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degree grantor
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Graduate School - New Brunswick
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school
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Text
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theses
OriginInfo
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2017
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2017-05
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2017
Place
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xx
Language
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eng
Subject (authority = RUETD)
Topic
Computer Science
Subject (authority = ETD-LCSH)
Topic
Combinatorial geometry
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Title
Rutgers University Electronic Theses and Dissertations
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ETD_7964
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electronic resource
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application/pdf
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Extent
1 online resource (vii, 77 p.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Subject (authority = ETD-LCSH)
Topic
Combinatorial analysis
Abstract (type = abstract)
This dissertation explores problems in combinatorial geometry relating to incidences and to applications of incidence problems in other areas of combinatorics. In recent years, various tools from algebra have been applied to make significant progress on longstanding combinatorial geometric problems. These breakthroughs have stimulated work in developing additional algebraic tools and applying them to other problems. We study two different flavors of incidence problems using algebraic techniques.
• Given a set of points P and a set of objects V, an incidence is defined to be a point-object pair (p, v) ∈ P × V such that p ∈ v, i.e., the point is contained in the object. In Chapter 2, we introduce a new notion of degeneracy and give bounds on the maximum number of point-plane and point-sphere incidences in R 3 that are non-degenerate under this notion.
• Given a set of points P, an ordinary line is defined to be a line incident to exactly two points. In 1893, Sylvester posed the following question: “Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line.” In other words, Sylvester asked if every finite point set in R 2 , not all on a ii line, determines an ordinary line. The question was resolved in the affirmative by Gallai in 1944 and many others subsequently. For point sets in complex space, Kelly’s theorem states that point sets in C 3 , not all on a plane, must determine an ordinary line. In Chapter 3, we give bounds on the minimum number of ordinary lines determined by sets of points in C 3 .
Lastly, we give some applications of incidence bounds to other combinatorial problems. We study the k-most-frequent distances problem in R 3 . This generalizes the unit distance problem of Erdos, which asks for the maximum number of times a distance can be realized among the n 2 pairs of n given points. In the k-most-frequent distances problem, we give a bound on the number of times a set of k distances can be realized by a set of n points in R 3 . Next, we consider the sum product conjecture, first stated by Erdos and Szemeredi in 1983. Informally, the conjecture states that a finite subset of the reals can not have both additive and multiplicative structure at the same time. We give new bounds for a more general version of this problem which considers subsets of complex numbers.
Note (type = statement of responsibility)
by Abdul Basit
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3QJ7M8T
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
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Basit
GivenName
Abdul
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Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2017-04-08 21:11:35
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Name
Abdul Basit
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
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Open
Reason
Permission or license
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2017-04-08T21:03:18
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