Mathematics Commons^™

Equivariant Structure Constants For Hamiltonian-T-Spaces, Ho Hon Leung

Turkish Journal of Mathematics

If there exists a set of canonical classes on a compact Hamiltonian-T-space in the sense of R Goldin and S Tolman, we derive some formulas for certain equivariant structure constants in terms of other equivariant structure constants and the values of canonical classes restricted to some fixed points. These formulas can be regarded as a generalization of Tymoczko's results.

Characteristic Classes On Grassmannians, Jin Shi, Jianwei Zhou

Turkish Journal of Mathematics

In this paper, we study the geometry and topology on the oriented Grassmann manifolds. In particular, we use characteristic classes and the Poincaré duality to study the homology groups of Grassmann manifolds. We show that for k=2 or n \leq 8, the cohomology groups H^*(G(k,n), R) are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincaré duality: H^q(G(k,n), R) \to H_{k(n-k)-q}(G(k,n), R) can be expressed explicitly.

W.R.I.T.E: Writing’S Role In Thoughtful Endeavors, Catherine M. Miller

Faculty Work

“Mathematicians actually spend a great deal of time writing. If a mathematician wants to contribute to the greater body of mathematical knowledge, she must be able communicate her ideas in a way which is comprehensible to others.” Lee, 2014, pg. 1

Why should our students be different from the mathematicians Lee (2014) refers to in the quote above? Learning mathematics requires a confluence of spoken, written and mathematically abstract languages. After all, who thinks entirely in mathematical symbols? No one, except a few mathematical thinkers who are live in the abstract worlds they construct. Thinking about mathematics in the language …

The Geometry Of The Projective Joint Spectrum And The Commutativity Of Self-Adjoint Operators, Isaak Chagouel

Legacy Theses & Dissertations (2009 - 2024)

The theory of single operators is by now a very mature subject, with the notion of spectrum playing a key role in the theory. However, multivariate operator theory is only in its very early stages of development. There is not even wide agreement about how "the joint spectrum" of an n-tuple A = (A₁, · · · , A_n) of bounded linear operators on the same Hilbert space H should be defined.

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

Scholarship

style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;"> prime factors for every style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;"> between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">with the property that …

On An Extension Of Riordan Array And Its Application In The Construction Of Convolution-Type And Abel-Type Identities, Tian-Xiao He, Leetsch Hsu, Xing Ron Ma

Scholarship

Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {pn(t)}{pn(t)}, where pn(t)pn(t) is of degree nn, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {pn(t)}{pn(t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed.

Hyperbolic Expressions Of Polynomial Sequences And Parametric Number Sequences Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter Shiue, Tsui-Wei Weng

Scholarship

A sequence of polynomial {an(x)} is called a function sequence of order 2 if it satisfies the linear recurrence relation of order 2: an(x) = p(x)an-1(x) + q(x)an-2(x) with initial conditions a0(x) and a1(x). In this paper we derive a parametric form of an(x) in terms of eθ with q(x) = B constant, inspired by Askey's and Ismail's works shown in [2] [6], and [18], respectively. With this method, we give the hyperbolic expressions of Chebyshev polynomials and Gegenbauer-Humbert Polynomials. The applications of the method to construct corresponding hyperbolic form of several well-known identities are also discussed in this paper.

Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn

Mathematics, Statistics and Data Science Faculty Works

We characterize which automorphisms of an arbitrary complete bipartite graph K_n,m can be induced by a homeomorphism of some embedding of the graph in S³.

The Classification Of V -Transverse Knots And Loose Legendrians, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

We classify knots in a 3-manifold M that are transverse to a nowhere zero vector field V up to the corresponding isotopy relation. When V is the coorienting vector field of a contact structure, these knots are the same as pseudo-Legendrian knots, which were introduced by Benedetti and Petronio. We show that two loose Legendrian knots with the same overtwisted disk in their complement are Legendrian isotopic if and only if they are pseudoLegendrian isotopic.

V -transverse knots are naturally framed. We show that each framed isotopy class corresponds to infinitely many V -transverse isotopy classes whose elements are pairwise …

L∞-Extremal Mappings In Amle And Teichmüller Theory, Luca Capogna

Mathematics Sciences: Faculty Publications

These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L ∞ norm of an energy density rather than the more familiar L p norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.

Ramsey Theory Using Matroid Minors, Dixie Smith Horne

Electronic Theses and Dissertations

This thesis considers a Ramsey Theory question for graphs and regular matroids. Specifically, how many elements N are required in a 3-connected graphic or regular matroid to force the existence of certain specified minors in that matroid? This question cannot be answered for an arbitrary collection of specified minors. However, there are results from the literature for which the number N exists for certain collections of minors. We first encode totally unimodular matrix representations of certain matroids. We use the computer program MACEK to investigate this question for certain classes of specified minors.

The Characterization Of Graphs With Small Bicycle Spectrum, Bette Catherine Putnam

Electronic Theses and Dissertations

Matroids designs are defined to be matroids in which the hyperplanes all have the same size. The dual of a matroid design is a matroid with all circuits of the same size, called a dual matroid design. The connected bicircular dual matroid designs have been characterized previously. In addition, these results have been extended to connected bicircular matroids with circuits of two sizes in the case that the associated graph is a subdivision of a 3-connected graph. In this dissertation, we will use a graph theoretic approach to discuss the characterizations of bicircular matroids with circuits of two and three …

Moments Of Products Of L-Functions, Caroline Laroche Turnage-Butterbaugh

Electronic Theses and Dissertations

We first consider questions on the distribution of the primes. Using the recent advancement towards the Prime k-tuple Conjecture by Maynard and Tao, we show how to produce infinitely many strings of consecutive primes satisfying specified congruence conditions. We answer an old question of Erdös and Turán by producing strings of consecutive primes whose successive gaps form an increasing (respectively decreasing) sequence. We also show that such strings exist whose successive gaps follow a certain divisibility pattern. Finally, for any coprime integers a and D ≥ 1, we refine a theorem of D. Shiu and find strings of consecutive primes …

An Introduction To Set Theory And Topology, Ronald C. Freiwald

Books and Monographs

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. …

A Method For Exact Solutions To Integrable Evolution Equations In 2+1 Dimensions, Alicia Machuca

Mathematics Dissertations - Archive

A systematic method is developed to obtain solution formulas for certain explicit solutions to integrable nonlinear partial differential equations in two spatial variables and one time variable. The method utilizes an underlying Marchenko integral equation that arises in the corresponding inverse scattering problem. The method is demonstrated for the Kadomtsev-Petviashvili and the generalized Davey-Stewartson II system. A derivation and analysis of the solution formulas to these two nonlinear partial differential equations are given, and an independent verification of the solution formulas is presented. Such solution formulas are expressed in a compact form in terms of matrix exponentials, by using a …

Estimation Of Variance In Bivariate Normal Distribution After Preliminary Test Of Homogeneity, Juan Manuel Romero Padilla

Mathematics Dissertations - Archive

A preliminary test estimator of variance in the bivariate normal distribution is proposed after Pitman-Morgan test of homogeneity of two variances. We propose one estimator of variance after preliminary test of two tails and another one for one tail test. The biases and mean square errors of both estimators are derived. The relative efficiency (RE) of the preliminary test estimator is studied. Computations and 3D graphs of RE for different parameters are analyzed. In order to get the maximum RE, recommendations of the significance level for the preliminary test are given for various sample sizes by using the max-min criterion.

Proceedings Of The Eighth Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

• GAMTE 2014 Officers
• GAMTE 2014 Conference Committee
• GAMTE 2014 Proceedings Committee
• Purposes and Goals of GAMTE
• Letter from the President

Discrete Fourier Restriction Associated With Schrödinger Equations, Yi Hu, Xiaochun Li

Mathematical Sciences: Faculty Publications

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain’s level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on L 2(d+2) d norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on L 2 (Z).

Superconvergence And A Posteriori Error Estimates Of A Local Discontinuous Galerkin Method For The Fourth-Order Initial-Boundary Value Problems Arising In Beam Theory, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k + 1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^k+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate …

Fast Algorithm For Finding Lattice Subspaces In Rn And Its Implementation, Andrew Martin Pownuk

Open Access Theses & Dissertations

There are known necessary and sufficient conditions for a subspace of Rm to be lattice-ordered. Let Y = {y1,…,ym} and yi are rows of the matrix X. A subspace ⟨X⟩, of linear space generated by the set X of n linearly independent positive vectors is lattice-ordered if and only the set X admits a fundamental set of indices I, which means that the subset YI ⊆ Y of vectors indexed by I is linearly independent, and every vector from Y\YI is a nonnegative linear combination of vectors form YI.

In economics it is possible to prove that the minimum-cost insured …