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How To Keep Up With Mathematics (Abstract), Paul Campbell

Kenneth C. Schraut Memorial Lectures

You are a mathematics major perhaps because you are good at it and it’s useful, but more likely mainly because you enjoy it.

Most undergraduate mathematics is decades or even centuries old, in part because of the hierarchical nature of the subject. Meanwhile, your fellow students in biology and other sciences use textbooks that feature crucial work of the last five or ten years.

Recall, though, what you were told in orientation to college and often since: Much learning in college has to take place outside the classroom, at your own initiative. After your formal education ends, all mathematics learning …

Third Kenneth C. Schraut Lecture (Poster), University Of Dayton. Department Of Mathematics

Kenneth C. Schraut Memorial Lectures

No abstract provided.

On The Existence Of Nontrivial Solutions To Some Elliptic Variational Inequalities, Vy Khoi Le, Klaus Schmitt

Mathematics and Statistics Faculty Research & Creative Works

The paper is concerned with the existence of nontrivial solutions of the obstacle problem: u ε K: ∫Ω ▽u▽ (v - u) dx - λ ∫ Ω u (v - u) dx ≥ ∫ Ω p (x, u) (v - u) dx ∀x ε K, where K = {v ε Ho1(Ω): v ≤ Ψ a.e. on Ω}. By using a generalized mountain pass theorem for inequalities, we prove, subject to some restrictions on the obstacle Ψ, the existence of nontrivial solutions of the above inequality.

Some Extensions Of Loewner's Theory Of Monotone Operator Functions, Daniel Alpay, Vladimir Bolotnikov, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of -monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.

Dimensionality-Reducing Expansion, Boundary Type Quadrature Formulas, And The Boundary Element Method, Tian-Xiao He

Scholarship

This paper discusses the connection between boundary quadrature formulas constructed by using solutions of partial differential equations and boundary element schemes.

Studying The Functional Genomics Of Stress Responses In Loblolly Pine With The Expresso Microarray Experiment Management System, Lenwood S. Heath, Naren Ramakrishnan, Ronald R. Sederoff, Ross W. Whetten, Boris I. Chevone, Craig Struble, Vincent Y. Jouenne, Dawei Chen, Leonel Van Zyl, Ruth Grene

Mathematics, Statistics and Computer Science Faculty Research and Publications

Conception, design, and implementation of cDNA microarray experiments present a variety of bioinformatics challenges for biologists and computational scientists. The multiple stages of data acquisition and analysis have motivated the design of Expresso, a system for microarray experiment management. Salient aspects of Expresso include support for clone replication and randomized placement; automatic gridding, extraction of expression data from each spot, and quality monitoring; flexible methods of combining data from individual spots into information about clones and functional categories; and the use of inductive logic programming for higher-level data analysis and mining. The development of Expresso is occurring in parallel with …

On Unit Sum Numbers Of Rational Groups, Brendan Goldsmith, Christopher Meehan, S. Wallutis

Articles

The unit sum numbers of rational groups are investigated: the importance of the prime 2 being an automorphism of the rational group is discussed and other results are achieved by considering the number and distribution of rational primes which are, or are not, automorphisms of the group. Proof is given of the existence of rational groups with unit sum numbers greater than 2 but of finite value .

K(Π, 1) For Artin Groups Of Finite Type, Colum Watt, Thomas Brady

Articles

This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in [6] in that their universal covers are simplicial complexes. In [4] a complex is constructed whose top-dimensional cells correspond to minimal factorizations of a Coxeter element as a product of reflections in a finite Coxeter group. Asphericity is established in low dimensions using a metric of non-positive curvature. Since the nonpositive curvature condition is …

On The Evolution Of Simple Material Structures, Marek Elźanowski, Ernst Binz

Mathematics and Statistics Faculty Publications and Presentations

The evolution of a distribution of material inhomogeneities is investigated by analyzing the evolution of the corresponding material connections. Some general geometric relations governing such evolutions are derived. These relations are then analyzed by looking at the restrictions imposed by the material symmetry group.

Asymmetric Two-Colourings Of Graphs In S³, Erica Flapan, David Linnan Li

Pomona Faculty Publications and Research

We prove that for any non-planar graph H, we can choose a two-colouring G of H such that G is intrinsically chiral, and if H is 3-connected and is not K_3,3 or K₅, then G is intrinsically asymmetric. No such asymmetric two-colouring is possible for K_3,3 or K₅.

An Effective Version Of Belyi's Theorem, Lily S. Khadjavi

Mathematics, Statistics and Data Science Faculty Works

We compute bounds on covering maps that arise in Belyi's Theorem. In particular, we construct a library of height properties and then apply it to algorithms that produce Belyi maps. Such maps are used to give coverings from algebraic curves to the projective line ramified over at most three points. The computations here give upper bounds on the degree and coefficients of polynomials and rational functions over the rationals that send a given set of algebraic numbers to the set {0,1,∞} with the additional property that the only critical values are also contained in {0,1,∞}.

Ideal Theory In Prüfer Domains - An Unconventional Approach, Edward Mosteig

Mathematics, Statistics and Data Science Faculty Works

In Prüfer domains of finite character, ideals are represented as finite intersections of special ideals which are proper generalizations of the classical primary ideals. We show that representations of ideals as shortest intersections of primal or quasi-primary ideals exist and are unique. Moreover, every non-zero ideal is the product of uniquely determined pairwise comaximal quasi-primary ideals. Semigroups of primal and quasi-primary ideals with fixed associated primes are also investigated in arbitrary Prüfer domains. Their structures can be described in terms of the value groups of localizations.

Valuations And Filtrations, Edward Mosteig

Mathematics, Statistics and Data Science Faculty Works

The classical theory of Gröbner bases, as developed by Bruno Buchberger, can be expanded to utilize objects more general than term orders. Each term order on the polynomial ring k[x] produces a filtration of k[x] and a valuation ring of the rational function field k(x). The algorithms developed by Buchberger can be performed by using directly the induced valuation or filtration in place of the term order. There are many valuations and filtrations that are suitable for this general computational framework that are not derived from term orders, even after a change of variables. Here we study how to translate …

Intrinsic Knotting And Linking Of Complete Graphs, Erica Flapan

Pomona Faculty Publications and Research

We show that for every m∈N, there exists an n∈N such that every embedding of the complete graph K_n in R³ contains a link of two components whose linking number is at least m. Furthermore, there exists an r∈N such that every embedding of K_r in R³ contains a knot Q with |a₂(Q)| ≥ m, where a₂(Q) denotes the second coefficient of the Conway polynomial of Q.

A Sufficient Condition For The Uniqueness Of Positive Steady State To A Reaction Diffusion System, Joon Hyuk Kang, Yun Oh

Faculty Publications

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system { Δu + u(g1(u) - g 2(v)) = 0 in R+ × Ω, Δν + v(h 1(u) - h2(v)) = 0 u|∂Ω = v| ∂Ω = 0. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

Flow Patterns In A Two-Roll Mill, Christopher Hills

Articles

The two-dimensional flow of a Newtonian fluid in a rectangular box that contains two disjoint, independently-rotating, circular boundaries is studied. The flow field for this two-roll mill is determined numerically using a finite-difference scheme over a Cartesian grid with variable horizontal and vertical spacing to accommodate satisfactorily the circular boundaries. To make the streamfunction numerically determinate we insist that the pressure field is everywhere single-valued. The physical character, streamline topology and transitions of the flow are discussed for a range of geometries, rotation rates and Reynolds numbers in the underlying seven-parameter space. An account of a preliminary experimental study of …

Some Irrational Generalised Moonshine From Orbifolds, Rossen Ivanov, Michael Tuite

Articles

We verify the Generalised Moonshine conjectures for some irrational modular functions for theMonster centralisers related to the Harada-Norton, Held, M12 and L3(3) simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case.

Rational Generalized Moonshine From Abelian Orbifoldings Of The Moonshine Module, Rossen Ivanov, Michael Tuite

Articles

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order p = 2, 3, 5, 7 and the other of order pk for k = 1 or k prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational …

Proceedings Of The First International Conference On Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability And Statistics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positive integers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite numbers 1+ = 1+&, where “1” is its standard part and “&” its non-standard part, …

Randomness And Optimal Estimation In Data Sampling, Florentin Smarandache, Mohammad Khosnevisan, Housila P. Singh, S Saxena, Sarjinder Singh

Branch Mathematics and Statistics Faculty and Staff Publications

The purpose of this book is to postulate some theories and test them numerically. Estimation is often a difficult task and it has wide application in social sciences and financial market. In order to obtain the optimum efficiency for some classes of estimators, we have devoted this book into three specialized sections: Part 1. In this section we have studied a class of shrinkage estimators for shape parameter beta in failure censored samples from two-parameter Weibull distribution when some 'apriori' or guessed interval containing the parameter beta is available in addition to sample information and analyses their properties. Some estimators …

Like A Bridge Over Colored Water: A Mathematical Review Of The Rainbow Bridge: Rainbows In Art, Myth, And Science, John A. Adam

Mathematics & Statistics Faculty Publications

Commenting on a recent book, the author discusses various views of the rainbow: its role in culture, its scientific description, and its mathematical theory.

On Torsion And Mixed Minimal Abelian Groups, Brendan Goldsmith, S. O. Hogain

Articles

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. We obtain a complete characterisation of such groups in the torsion case; in the case of mixed groups of rank 1 we obtain a characterisation for some large classes of such groups.