Mathematics Commons^™

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Erica Flapan, Blake Mellor, Ramin Naimi

Pomona Faculty Publications and Research

We determine for which m the complete graph K_m has an embedding in S³ whose topological symmetry group is isomorphic to one of the polyhedral groupsA₄, A₅ or S₄.

Connecting Mathematics Students To Philosophy And Faith, Nathan Moyer

ACMS Journal 2010-2011

This paper describes an upper-division class project that helps students wrestle with the deep philosophical questions of mathematics in a relevant way. Through article readings and group discussions, various historical views of mathematical philosophy are examined and critiqued. An emphasis is placed upon students making meaningful connections between their own views of mathematics and their personal Christian faith or worldview.

Structure Of Wave Wperators In R^3, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

We prove a structure formula for the wave operators in R^3 and their adjoints for a scaling-invariant class of scalar potentials V, under the assumption that zero is neither an eigenvalue, nor a resonance for -\Delta+V. The formula implies the boundedness of wave operators on L^p spaces, 1 \leq p \leq \infty, on weighted L^p spaces, and on Sobolev spaces, as well as multilinear estimates for e^{itH} P_c. When V decreases rapidly at infinity, we obtain an asymptotic expansion of the wave operators. The first term of the expansion is of order < y >^{-4}, commutes with the Laplacian, and exists when …

Genus 0, 1, 2 Actions Of Some Almost Simple Groups Of Lie Rank 2, Xianfen Kong

Wayne State University Dissertations

Please see the paper.

Thanks.

Numerical Methods For Problems Arising In Risk Management And Insurance, Zhuo Jin

Wayne State University Dissertations

In this dissertation we investigate numerical methods for problems annuity purchasing and dividend optimization arising in risk management and insurance. We consider the models with Markov regime-switching process. The regime-switching model contains both continuous and discrete components in their evolution and is referred to as a hybrid system. The discrete events are used to model the random factors that cannot formulated by differential equations. The switching process between regimes is modulated as a finite state Markov chain.

As is widely recognized, this regime-switching model appears to be more versatile and more realistic. However, because of the regime switching and the …

New Variational Principles With Applications To Optimization Theory And Algorithms, Hung Minh Phan

Wayne State University Dissertations

In this dissertation we investigate some applications of variational analysis in optimization theory and algorithms. In the first part we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, under the name of tangential extremal principles and rated extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. These developments are in the core geometric theory of variational analysis. Our study includes the basic theory and applications to problems of semi-infinite programming …

Variational Analysis And Optimal Control Of The Sweeping Process, Hoang Dinh Nguyen

Wayne State University Dissertations

We formulate and study an optimal control problem for the sweeping(Moreau) process, where control functions enter the moving sweeping

set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and …

Moduli Spaces And Cw Structures Arising From Morse Theory, Lizhen Qin

Wayne State University Dissertations

In this dissertation, we study the moduli spaces and CW Structures arising from Morse theory.

Suppose M is a smooth manifold and f is a Morse function on it. We consider the negative gradient flow of f. Suppose the flow satisfies transversality. This naturally defines the moduli spaces of flow lines and gives a stratication of M by its unstable manifolds. The gluing of broken flow lines can also be constructed.

We prove that, under certain assumptions, these moduli spaces can be compactified and the compactified spaces are smooth manifolds with corners. Moreover, these compactified manifolds satisfy certain orientation formulas. …

Analyzing The Galois Groups Of Fifth-Degree And Fourth-Degree Polynomials, Jesse Berglund

Undergraduate Review

It is known that the general equations of fourth-degree or lower are solvable by formula and general equations of fifth-degree or higher are not. To get an understanding of the differences between these two types of equations, Galois theory and Field theory will be applied. The Galois groups of field extensions will be analyzed, and give the solution to the query “What is the difference between unsolvable fifth-degree equations and fourth-degree equations?”

A Noneuclidean Universe, Frank A. Farris

Mathematics and Computer Science

Let us construct a hypothetical universe, if for no other reason than to challenge our preconceptions about space. We call it a noneuclidean universe because it contradicts some of the notions central to euclidean geometry, where, for instance, the angle measures in a triangle add up to 180 degrees. There are many noneuclidean universes; ours is of a type called hyperbolic. This hypothetical universe has been constructed before. It is often called the Poincaré Upper Halfplane, in honor of French mathematician Henri Poincaré (1854-1912). I admire Poincaré because he is said to have been the last person in the world …

Higher Dimensional Lattice Chains And Delannoy Numbers, John S. Caughman, Charles L. Dunn, Nancy Ann Neudauer, Colin L. Starr

Faculty Publications

Fix nonnegative integers n₁ , . . ., n_d, and let L denote the lattice of points (a₁ , . . ., a_d) ∈ ℤ^d that satisfy 0 ≤ a_i ≤ n_i for 1 ≤ i ≤ d. Let L be partially ordered by the usual dominance ordering. In this paper we use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in L. Setting n_i = n (for all i) in these expressions yields a new …

Clique-Relaxed Graph Coloring, Charles Dunn, Jennifer Firkins Nordstrom, Cassandra Naymie, Erin Pitney, William Sehorn, Charlie Suer

Faculty Publications

We define a generalization of the chromatic number of a graph G called the k-clique-relaxed chromatic number, denoted χ^(k)(G). We prove bounds on χ^(k)(G) for all graphs G, including corollaries for outerplanar and planar graphs. We also define the k-clique-relaxed game chromatic number, χ_g^(k)(G), of a graph G. We prove χ_g⁽²⁾(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χ_g⁽²⁾(H) ≥ 3. Finally, we prove that if H is a member …

Reverse Mathematics And Uniformity In Proofs Without Excluded Middle, Jeffry L. Hirst, Carl Mummert

Mathematics Faculty Research

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a \Pi^1_2 sentence of a certain form is provable using E-HA^ω along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles …

Geometrical Objects Associated To A Substructure, Fatma Özdemi̇r, Mircea Craşmareanu

Turkish Journal of Mathematics

Several geometric objects, namely global tensor fields of (1,1)-type, linear connections and Riemannian metrics, associated to a given substructure on a splitting of tangent bundle, are studied. From the point of view of lifting to entire manifold, two types of polynomial substructures are distinguished according to the vanishing of not of the sum of the coefficients. Conditions of parallelism for the extended structure with respect to some remarkable linear connections are given in two forms, firstly in a global description and secondly using the decomposition in distributions. A generalization of both Hermitian and anti-Hermitian geometry is proposed.

Existence Theory For Positive Solutions Of P-Laplacian Multi-Point Bvps On Time Scales, You-Hui Su

Turkish Journal of Mathematics

This paper is concerned with the one-dimensional p-Laplacian multi-point boundary value problem on time scales T: (\varphi_p(u^{\Delta}))^{\nabla} + h(t)f(u) = 0, t \in [0,T]_T, subject to multi-point boundary conditions u(0) - B_0(\sum_{i=1}^{m-2}a_i u^{\Delta}(\xi_i)) = 0, u^{\Delta}(T) = 0, or u^{\Delta}(0) = 0, u(T) + B_1(\sum_{i=1}^{m-2}b_iu^{\Delta}(\xi'_i)) = 0, where \varphi_p(u) is p-Laplacian operator, i.e., \varphi_p(u = u ^{p-2}u, p>1, \xi_i,\xi'_i\in [0,T]_T, m \geq 3 and satisfy 0 \leq \xi_1 < \xi_2 < ... < \xi_{m-2} < \rho(T), \sigma(0) < \xi'_1 < \xi'_2 < ... < \xi'_{m-2} \leq T, a_i, b_i\in [0,\infty) (i=1,2,..., m-2). Some new sufficient conditions are obtained for the existence of at least one positive solution by using Krasnosel'skii's fixed-point theorem and new sufficient conditions are obtained for the existence of twin, triple or arbitrary odd positive solutions by using generalized Avery and Henderson fixed-point theorem and Avery-Peterson fixed-point theorem. Our results include and extend some known results. As applications, two examples are given to illustrate the main results and their differences. These results are new even for the special cases of continuous and discrete equations, as well as in the general time scale setting.

Approximation By Complex Potentials Generated By The Gamma Function, Sorin G. Gal

Turkish Journal of Mathematics

In this paper we find the exact orders of approximation of analytic functions by the complex versions of several potentials (including the Flett potential) generated by the Gamma function and by some singular integrals.

Cyclic Codes Over {Z_2+Uz_2+U^2z_2+\Ldots+U^{K-1}Z_2}, Mohammed Al Ashker, Mohammed Hamoudeh

Turkish Journal of Mathematics

In this paper, we study the structure of cyclic codes of an arbitrary length n over the ring Z_2+uZ_2+u^2Z_2+\ldots+u^{k-1}Z_2, where u^k=0. Also we study the rank for these codes, and we find their minimal spanning sets. This study is a generalization and extension of the work in reference [1].

Models For Discrete Quantum Gravity, S. Gudder

Mathematics Preprint Series

We first discuss a framework for discrete quantum processes (DQP). It is shown that the set of q-probability operators is convex and its set of extreme elements is found. The property of consistency for a DQP is studied and the quadratic algebra of suitable sets is introduced. A classical sequential growth process is “quantized” to obtain a model for discrete quantum gravity called a quantum sequential growth process (QSGP). Two methods for constructing concrete examples of QSGP are provided.

Vascular Countercurrent Network For 3d Triple-Layered Skin Structure With Radiation Heating, Xiaoqi Zeng

Doctoral Dissertations

Heat transfer in living tissue has become more and more attention for researchers, because high thermal radiation produced by intense fire, such as wild fires, chemical fires, accidents, warfare, terrorism, etc, is often encountered in human's daily life. Living tissue is a heterogeneous organ consisting of cellular tissue and blood vessels, and heat transfer in cellular tissue and blood vessel is quite different, because the blood vessels provide channels for fast heat transfer. The metabolic heat generation, heat conduction and blood perfusion in soft tissue, convection and perfusion of the arterial-venous blood through the capillary, and interaction with the environment …

Spectral Collocation Method For Compact Integral Operators, Can Huang

Wayne State University Dissertations

We propose and analyze a spectral collocation method for integral

equations with compact kernels, e.g. piecewise smooth kernels and

weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \;

0<\mu<1. $ We prove that 1) for integral equations, the convergence

rate depends on the smoothness of true solutions $y(t)$. If $y(t)$

satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq

ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$

satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k $,

we obtain supergeometric rate of convergence for both Volterra

equations and Fredholm equations and related integro differential

equations; 2) for eigenvalue problems, the convergence rate depends

on the smoothness of eigenfunctions. The same convergence rate for

the largest modulus eigenvalue approximation …