Open Access. Powered by Scholars. Published by Universities.®
- Discipline
- Institution
Articles 1 - 6 of 6
Full-Text Articles in Entire DC Network
The P-Adic Numbers And Conic Sections, Abdelhadi Zaoui
The P-Adic Numbers And Conic Sections, Abdelhadi Zaoui
Electronic Theses and Dissertations
This thesis introduces the p-adic metric on the rational numbers. We then present the basic properties of this metric. Using this metric, we explore conic sections, viewed as equidistant sets. Lastly, we move on the sequences and series, and from there, we define p-adic expansions and the analytic completion of Q with respect to the p-adic metric, which leads to exploring some arithmetic properties of Qp.
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
Electronic Theses and Dissertations
While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …
Vector Partitions, Jennifer French
Vector Partitions, Jennifer French
Electronic Theses and Dissertations
Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose …
A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot
A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot
Electronic Theses and Dissertations
Nearly a century ago, the mathematicians Hardy and Ramanujan established their celebrated circle method to give a remarkable asymptotic expression for the unrestricted partition function. Following later improvements by Rademacher, the method was utilized by Niven, Lehner, Iseki, and others to develop rapidly convergent series representations of various restricted partition functions. Following in this tradition, we use the circle method to develop formulas for counting the restricted classes of partitions that arise in the Gollnitz-Gordon identities. We then show that our results are strongly supported by numerical tests. As a side note, we also derive and compare the asymptotic behavior …
Avoiding, Pretending, And Querying : Three Combinatorial Problems., Adam S. Jobson
Avoiding, Pretending, And Querying : Three Combinatorial Problems., Adam S. Jobson
Electronic Theses and Dissertations
A k-term quasi-progression of diameter d is a sequence {Xl,... ,xk} for which there exists a positive integer l such that l < Xi-Xi-1 < l+d, for all i = 2, ... ,k. Quasi-progressions may be thought of as arithmetic progressions with a certain amount of 'wiggle-room' allowed. Let Q(d, k) be the least positive integer such that every 2-coloring of {1, ... , Q(d, k)} contains a monochromatic k-term quasi-progression of diameter d. We prove a conjecture of Landman for certain values of k and d, provide counterexamples for some other cases, and determine that the conjecture always has the correct order of growth. Let A be the adjacency matrix of a non empty graph. Is there always a nonzero {0, 1}-vector in the row space of A that is not a row of A? Akbari, Cameron, and Khosrovshahi have shown that an affirmative answer to this question would imply bounds on many graph parameters as a function of the rank of the adjacency matrix. We demonstrate the existence of such vectors for certain families of graphs, examine techniques to find and verify the existence of such vectors, and show that if you generalize the problem to allow asymmetry in the matrices then some {0, 1 }-matrices do not have such vectors. In 1981, Andrew Yao asked "Should tables be sorted?". When the table has n cells that are filled with entries taken from a key space of m possibilities, he showed that it is possible to decide whether any member of the key space is present in the table by inspecting (querying) only one cell of the table if and only if m < 2n - 2. We make steps toward extending his result to the case where you are permitted two queries by considering several variations of the problem.
An Introduction To Number Theory Prime Numbers And Their Applications., Crystal Lynn Anderson
An Introduction To Number Theory Prime Numbers And Their Applications., Crystal Lynn Anderson
Electronic Theses and Dissertations
The author has found, during her experience teaching students on the fourth grade level, that some concepts of number theory haven't even been introduced to the students. Some of these concepts include prime and composite numbers and their applications. Through personal research, the author has found that prime numbers are vital to the understanding of the grade level curriculum. Prime numbers are used to aide in determining divisibility, finding greatest common factors, least common multiples, and common denominators. Through experimentation, classroom examples, and homework, the author has introduced students to prime numbers and their applications.