Digital Commons Network^™

Modified Ramsey Numbers, Meaghan Mahoney

Honors Program Theses and Projects

Ramsey theory is a eld of study named after the mathematician Frank P. Ramsey. In general, problems in Ramsey theory look for structure amid a collection of unstructured objects and are often solved using techniques of Graph Theory. For a typical question in Ramsey theory, we use two colors, say red and blue, to color the edges of a complete graph, and then look for either a complete subgraph of order n whose edges are all red or a complete subgraph of order m whose edges are all blue. The minimum number of vertices needed to guarantee one of these …

Optimization Of The Estimation Of Initial Perturbations In Linear Differential Systems, A Kojametov

Karakalpak Scientific Journal

The article deals with constant coefficients of differential system, for this purpose the function of Lyapunova has been used.

A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, Russell Lee Jahn

All Graduate Theses, Dissertations, and Other Capstone Projects

In this paper we demonstrate how the principles of measure theory can be applied effectively to problems of number theory. Initially, necessary concepts from number theory will be presented. Next, we state standard concepts and results from measure theory to which we will need to refer. We then develop our repertoire of measure theoretic machinery by constructing the needed measures and defining a generalized version of the multiplicative convolution of measures. A suitable integration by parts formula, one that is general enough to handle various combinations of measures, will then be derived. At this juncture we will be ready to …

A Pointwise Estimate For The Fourier Transform And The Number Of Maxima Of A Function, Ryan Berndt

Mathematics Faculty Scholarship

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Fibrewise Rational H-Spaces, Gregory Lupton, Samuel Bruce Smith

Mathematics and Statistics Faculty Publications

We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. (2000), 747-768], which is used there as a crucial step in establishing their main finiteness result.

History Of Fermat's Last Theorem, Amanda Brown

Undergraduate Honors Capstone Projects

Around 1637, Pierre de Fermat made a now-famous mathematical conjecture. However, Fermat's conjecture neither began nor ended with him. Fermat's last theorem, as the conjecture is called, has roots approximately 3600 years old. The proof of the theorem was not realized until 1994, over 350 years after it was proposed by Fermat.

Geršgorin And Beyond•••, Jason Knight Belnap

Undergraduate Honors Capstone Projects

Eigenvalues are useful in various areas of mathematics, such as in testing the critical values of a multi variable function to see if it is a local extrema. One of the more common ways to define eigenvalues is:

Definition (1): Given that A is an n by n matrix, λ is an eigenvalue of A if and only if det(A - λI_n) = 0. Any nonzero vector in Null(A - λI) is called an eigenvector associated with λ.

The Frobenius Theorem, Hiroshi Nagao

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Many theorems in differential geometry which deal with the existence of certain geometrical structures or properties depend upon various existence and uniqueness theorems for differential equations. Because of its wide range of applications one of the most important of these theorems is the Frobenius Theorem for systems of total differential equations. There are four different forms of the Frobenius Theorem. In applications of the theorem one form is often preferable to the others. In this report we -shall prove the Frobenius Theorem, establish the equivalence of these various forms, and discuss a few applications.

Classifications Of Plane Continua, Steven Ray Matthews

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In the course of studying continua in the plane it has been asked if a given continuum has uncountably many disjoint duplications in the plane, and if so, what are the consequences of the existence of such a collection. The object of this paper is to study these problems and to develop some machinery useful in their resolution. In Section I, we review the definition of convergence and homeomorphic convergence of point sets in a metric space S. We then consider the space, π of all continuous functions from a compact metric space P to a separable metric space Q …

Barypact Topological Spaces, Bradley Y. Maughan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Recently, Kimber [3] has discovered a general class of topological spaces, the members of which are termed barypact spaces, that includes the compact topological spaces. This class is distinct from the set of all compact topological spaces, but its members possess many of the useful properties associates with compactness. As a consequence, several standard compactness theorems become special cases of corresponding theorems in a more general setting and the techniques of proof applied to these extensions provide new, and sometimes remarkably simple, proofs of the very theorems they generalize. The purpose of this paper is to extend to this …