Physical Sciences and Mathematics Commons^™

Hyperbolic Billiard Paths, Rebecca Lehman, Chad White

Mathematical Sciences Technical Reports (MSTR)

A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7)-tiling.

Pigeon-Holing Monodromy Groups, Niles G. Johnson

Mathematical Sciences Technical Reports (MSTR)

A simple tiling on a sphere can be used to construct a tiling on a d-fold branched cover of the sphere. By lifting a so-called equatorial tiling on the sphere, the lifted tiling is locally kaleidoscopic, yielding an attractive tiling on the surface. This construction is via a correspondence between loops around vertices on the sphere and paths across tiles on the cover. The branched cover and lifted tiling give rise to an associated monodromy group in the symmetric group on d symbols. This monodromy group provides a beautiful connection between the cover and its base space. Our investigation …

Invariant Sets And Inverse Limits, William Thomas Ingram

Mathematics and Statistics Faculty Research & Creative Works

In this paper we investigate the nature of inverse limits from the point of view of invariant sets. We then introduce a special class of examples of inverse limits on [0,1] using Markov bonding maps determined by members of the group of permutations on n elements. © 2002 Elsevier Science B.V. All rights reserved.

Sign-Changing And Multiple Solutions For The P-Laplacian, Siegfried Carl, Kanishka Perera

Mathematics and System Engineering Faculty Publications

We obtain a positive solution, a negative solution, and a sign-changing solution for a class of p-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.

Normal Forms For Two-Inputs Nonlinear Control Systems, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback group action on two-inputs non-linear control systems. We follow an approach proposed by Kang and Krener which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes that obtained in the single-input case. We also give homogeneous m-invariants of the action of the group of homogeneous transformations on the homogeneous systems of the same degree. We illustrate our results by analyzing the normal form and invariants of homogeneous systems of degree two.

On Choosing And Bounding Probability Metrics, Alison L. Gibbs, Francis E. Su

All HMC Faculty Publications and Research

When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.

The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong

Mathematical Sciences Technical Reports (MSTR)

There is a very beautiful correspondence between branched covers of the Riemann sphere P¹ and subgroups of the fundamental group π₁(P¹ − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non- Galois covers S/P¹. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of …

Poincaré Types Solutions Of Systems Of Difference Equations, Raghib Abu-Saris, Saber Elaydi, Sophia Jang

Mathematics Faculty Research

No abstract provided.

Two Quick Combinatorial Proofs, Arthur T. Benjamin, Michael E. Orrison

All HMC Faculty Publications and Research

Presentation of two simple combinatorial proofs.

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

Long-Step Homogeneous Interior-Point Method For P*-Nonlinear Complementarity Problem, Goran Lesaja

Department of Mathematical Sciences Faculty Publications

A P*-Nonlinear Complementarity Problem as a generalization of the P*Linear Complementarity Problem is considered. We show that the long-step version of the homogeneous self-dual interior-point algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set.

On Simplicial Commutative Algebras With Noetherian Homotopy, James M. Turner

University Faculty Publications and Creative Works

In this paper, we introduce a strategy for studying simplicial commutative algebras over general commutative rings R. Given such a simplicial algebra A, this strategy involves replacing A with a connected simplicial commutative k(℘)-algebra A(℘), for each ℘ ε Spec(π0A), which we call the connected component of A at ℘. These components retain most of the André-Quillen homology of A when the coefficients are k(℘)-modules (k(℘)=residue field of ℘ in π0A). Thus, these components should carry quite a bit of the homotopy theoretic information for A. Our aim will be to apply this strategy to those simplicial algebras which possess …

Intersection Multiplicities Over Gorenstein Rings, Claudia M. Miller, Anurag K. Singh

Mathematics - All Scholarship

We construct a complex of free-modules over a Gorenstein ring R of dimension five, for which the Euler characteristic and Dutta multiplicity are different. This complex is the resolution of an R-module of finite length and finite projective dimension. As a consequence, the ring R has a nonzero Todd class tau_3(R) and a bounded free complex whose local Chern character does not vanish on this class.
In the course of our work, we construct a module N of finite length and finite projective dimension over the hypersurface A=K[u,v,w,x,y,z]/(ux+vy+wz), such that the Serre intersection multiplicity of the modules N and A/(u,v,w)A …

The Fine Structure Of The Kasparov Groups Ii: Topologizing The Uct, Claude Schochet

Mathematics Faculty Research Publications

The Kasparov Groups KK∗(A,B) have a natural structure as pseudopolonais groups. In this paper we analyze how this topology interacts with the terms of the Universal Coefficient Theorem (UCT) and the splitting sof the UCT constructed by J. Rosenberg and the author, as well as its canonical three term decomposition which exists under bootstrap hypotheses. We show that the various topologies on [cursive]Ext^{1}_{ℤ}(K∗(A),K∗(B)) and other related groups mostly coincide. Then we focus attention on the Milnor sequence and the fine structure subgroup of KK∗(A,B). …

Stochastic 2-D Navier-Stokes Equation, J. L. Menaldi, S. S. Sritharan

Mathematics Faculty Research Publications

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this signi cantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution.

An Extension Of The Fundamental Theorem Of Linear Programming, A Brown, A Gedlaman, Allen G. Holder, S Martinez

Mathematics Faculty Research

In 1947 George Dantzig developed the Simplex Algorithm for linear programming, and in doing so became known as The Father of Linear Programming. The invention of the Simplex Algorithm has been called "one of the most important discoveries of the 20th century," and linear programming techniques have proven useful in numerous fields of study. As such, topics in linear optimization are taught in a variety of disciplines. The finite convergence of the simplex algorithm hinges on a result stating that every linear program with an optimal solution has a basic optimal solution; a result known as the Fundamental Theorem of …

On The Number Of Factorizations Of An Element In An Atomic Monoid, Scott T. Chapman, Juan Ignacio García-García, Pedro A. García Sánchez, José Carlos Rosales

Mathematics Faculty Research

Let S be a reduced commutative cancellative atomic monoid. If s is a nonzero element of S, then we explore problems related to the computation of η(s), which represents the number of distinct irreducible factorizations of s∈S. In particular, if S is a saturated submonoid of N^d, then we provide an algorithm for computing the positive integer r(s) for which

0 < limn→∞η(sⁿ)n^r(s)-1∞.

We further show that r(s) is constant on the Archimedean components of S. We apply the algorithm to show how to …

Teaching Mathematics In The Seventeenth And Twenty-First Centuries, Dennis C. Smolarski

Mathematics and Computer Science

In the late 1960s, many people saw a fictional vision of the beginning of the twenty-first century via the movie, 2001: A Space Odyssey. Early in the movie, a lunar expedition uncovers a large, black monolith in the crater Clavius. Although the movie was fictional, and computers have not yet reached HAL's ability to speak and read lips, the lunar crater Clavius does exist and is named after a sixteenth century scholar who was instrumental in introducing mathematics into the university curriculum.

Christopher Clavius (1538-1612) is often associated with the astronomical and mathematical justification for shifting from the Julian to …

Advanced Portfolio Theory: Why Understanding The Math Matters, Tom Arnold

Finance Faculty Publications

The goal of this paper is to motivate the use of efficient set mathematics for portfolio analysis [as seen in Roll, 1977] in the classroom. Many treatments stop at the two asset portfolio case (avoiding the use of matrix algebra) and an alarming number of treatments rely on illustration and templates to provide a heuristic sense of the material without really teaching how efficient portfolios are generated. This is problematic considering that the benefits of understanding efficient set mathematics go beyond portfolio analysis and into such topics as regression analysis (as demonstrated here).

Bidding For Envy-Freeness: A Procedural Approach To N-Player Fair-Division Problems, Claus-Jochen Haake, Matthias G. Raith, Francis E. Su

All HMC Faculty Publications and Research

We develop a procedure for implementing an efficient and envy-free allocation of m objects among n individuals with the possibility of monetary side-payments, assuming that players have quasi–linear utility functions. The procedure eliminates envy by compensating envious players. It is fully descriptive and says explicitly which compensations should be made, and in what order. Moreover, it is simple enough to be carried out without computer support. We formally characterize the properties of the procedure, show how it establishes envy-freeness with minimal resources, and demonstrate its application to a wide class of fair-division problems.

A Polytopal Generalization Of Sperner's Lemma, Jesus A. De Loera, Elisha Peterson '00, Francis E. Su

All HMC Faculty Publications and Research

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar.32 (1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices v₁, v₂,…,v_n. Label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if v_j is on F. Then there are at least n−d full dimensional simplices of T, each labelled with d+1 different labels. We …

Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of locating and describing an internal point defect in a one­ dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.

An Extension Of Elton's L(1)(N) Theorem To Complex Banach Spaces, S J. Dilworth, Joseph P. Patterson

Faculty Publications

No abstract provided.

A Posteriori Error Estimates Based On Polynomial Preserving Recovery, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

Superconvergence of order O(h^1+rho), for some rho is greater than 0, is established for gradients recovered using Polynomial Preserving Recovery technique when the mesh is mildly structured. Consequently this technique can be used in building a posteriori error estimator that is asymptotically exact.

Using Composition Techniques To Improve Classroom Instruction And Students’ Understanding Of Proof, Christopher D. Goff

College of the Pacific Faculty Articles

This paper describes an effort to incorporate standard composition exercises into a sophomore-level discrete mathematics class. It provides an example of how peer review can be integrated with a mathematical curriculum through the writing of proofs.

Writing Mathematics-A Nut And A Bolt Of Style, Frank A. Farris

Mathematics and Computer Science

As editor of Mathematics Magazine, I see a lot of manuscripts. Some of them are written with a charming sense of style, but many of them leave me thinking that the author's only concern was to set out the mathematics clearly. This is a fine place to start, but the tradition of the Magazine is to offer things that people will enjoy reading, and this requires more than clarity. Let me explain an important step authors can take in order to make their work more attractive.

Matroid Duality From Topological Duality In Surfaces Of Nonnegative Euler Characteristic, Dan Slilaty

Mathematics and Statistics Faculty Publications

Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.

The Liouville-Bratu-Gelfand Problem For Radial Operators, Jon T. Jacobsen, Klaus Schmitt

All HMC Faculty Publications and Research

We determine precise existence and multiplicity results for radial solutions of the Liouville–Bratu–Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators.

Analysis Of The N-Card Version Of The Game Le Her, Arthur T. Benjamin, Alan J. Goldman

All HMC Faculty Publications and Research

We present a complete solution to a card game with historical origins. Our analysis exploits the convexity properties in the payoff matrix, allowing this discrete game to be resolved by continuous methods.

Infinitely Many Nonradial Solutions To A Superlinear Dirichlet Problem, Hugo Aduén, Alfonso Castro

All HMC Faculty Publications and Research

In this article we provide sufficient conditions for a superlinear Dirichlet problem to have infinitely many nonradial solutions. Our hypotheses do not require the nonlinearity to be an odd function. For the sake of simplicity in the calculations we carry out details of proofs in a ball. However, the proofs go through for any annulus.