Digital Commons Network^™

Nonstandard Analysis Based Calculus, Kathleen Renae Gibson

Theses Digitization Project

In the first part of the project the elementary development of an extended number system called Hyperreals is discussed. The second half of this project develops the basics of Nonstandard Analysis, including the theory of ultrafilters, and the formal construction of the Hyperreals.

The Splitting Theorem For Orbifolds, Joseph Borzellino, Shun-Hui Zhu

Mathematics

In this paper we wish to examine a generalization of the splitting theorem of Cheeger–Gromoll [CG] to Riemannian orbifolds. Roughly speaking, a Riemannian orbifold is a metric space locally modelled on quotients of Rie- mannian manifolds by finite groups of isometries. The term orbifold was coined by W. Thurston [T] sometime around the year 1976–77. The term is meant to suggest the orbit space of a group action on a manifold. A similar concept was introduced by I. Satake in 1956, where he used the term V–manifold (See [S1]). The “V” was meant to suggest a cone–like singularity. Since then, …

Fault-Tolerant Circuit-Switching Networks, Nicholas Pippenger, Geng Lin

All HMC Faculty Publications and Research

The authors consider fault-tolerant circuit-switching networks under a random switch failure model. Three circuit-switching networks of theoretical importance—nonblocking networks, rearrangeable networks, and superconcentrators—are studied. The authors prove lower bounds for the size (the number of switches) and depth (the largest number of switches on a communication path) of such fault-tolerant networks and explicitly construct such networks with optimal size Θ( n (log n)² ) and depth Θ( log n ).

Advection Of A Passive Scalar By A Vortex Couple In The Small-Diffusion Limit, Joseph F. Lingevitch, Andrew J. Bernoff

All HMC Faculty Publications and Research

We study the advection of a passive scalar by a vortex couple in the small-diffusion (i.e. large Péclet number, Pe) limit. The presence of weak diffusion enhances mixing within the couple and allows the gradual escape of the scalar from the couple into the surrounding flow. An averaging technique is applied to obtain an averaged diffusion equation for the concentration inside the dipole which agrees with earlier results of Rhines & Young for large times. At the outer edge of the dipole, a diffusive boundary layer of width O(Pe^−½) forms; asymptotic matching to the interior …

A Mathematical Analysis Of A Small Corporation: This Stuff Really Works?, Wade Olson

Honors Theses, 1963-2015

This thesis is a mathematical analysis of Foam Enterprises, a small corporation in the Twin Cities. Through interaction with the,. I heard of several problems occurring at that time which were being dealt with in a mathematical fashion. The first involves the profitability of two main clients of Foam Enterprises. The second deals with the production capacity of the two current plants and the distribution of products to customers. Each problem was then described in detail. The general mathematical approaches were stated for each problem and important circumstances which could affect the procedure were given. In turn, the results of …

Triangles, Triangles And, Yes, More Triangles: Explorations In Euclidean Ramsey Theory, Kathleen Wilson

Honors Theses, 1963-2015

Several important general theorems of Euclidean Ramsey Theory are presented with an emphasis on trying to prove or disprove the 1973 conjecture of Erdös et al. that for all triangles, except for equilateral triangles, it is possible to find a monochromatic coloring of the vertices in any two colorings of the plane. Further investigation included looking at triangles in greater dimensions.

Irreducible Polynomials Over A Finite Field Zp, Brian Holbrook

Honors Theses, 1963-2015

The topic of my thesis was counting irreducible polynomials. I began with some preliminary material including relevant theorems discovered in various math textbooks and data generated by a computer program that I wrote. I then used this data to calculate formulas for the number of polynomials which had zeroes in a field Zp of sufficiently low degree. Next, I explained why it would not be practical to use the formulas for lower degrees to synthesize formulas for high degrees. However, I was able to generate a new formula using a different counting technique: the inclusion-exclusion principle. Finally, I discussed the …

Nothing In Moderation, Everything In Excess: A New Weighted Statistic On Permutations, Ann Marie Paulukonis

Honors Theses, 1963-2015

The major index is a well-known statistic on permutations which is computed by summing the positions of descents in a permutation. Instead of considering descents, this paper investigates what happens when excedances are weighted by position in a permutation. Several theorems are presented concerning various symmetries of the resulting distribution.

A Qualitative Analysis Of Differential Equations Of Population Dynamics, Kevin Brenner

Honors Theses, 1963-2015

An exploration of systems of equations modeling closed hypothetical ecosystems. Models are formed from biological assumptions about population. Models are constructed by combining or modifying simpler systems. The simple models used in constructing systems are the logistic, competition, and Lotka-Volterra models. Eigenvalues and eigenvectors are used to prove some results about the equilibria of the systems. Geometric perspectives of the systems are used to develop the intuitive approach of the work. Examples of graphed systems are included. Partial results are proven. Conjectures are formed. Difficulties in the work are described and suggestions for further work are given.

Regularity Of The Free-Boundary In Singular Stochastic Control, S. A. Williams, P. L. Chow, J. L. Menaldi

Mathematics Faculty Research Publications

No abstract provided.

Characterization Of The Local Lipschitz Constant, M. W. Bartelt, J. J. Swetits

Mathematics & Statistics Faculty Publications

A characterization, using polynomials introduced by A. V. Kolushov, is given for the local Lipschitz constant for the best approximation operator in Chebyshev approximation from a Haar set. The characterization is then used to study the existence of uniform local Lipschitz constants.

Finite Amplitude Convection Between Stress-Free Boundaries; Ginzburg-Landau Equations And Modulation Theory, Andrew J. Bernoff

All HMC Faculty Publications and Research

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg-Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.

Multiage Classrooms: A New Way To Learn Math, Martha Taylor Dever, Randy Zila, Noel N. Mansano

Teacher Education and Leadership Faculty Publications

No abstract provided.

Random Fixed Point Theorems For Nonexpansive And Contractive-Type Random Operators On Banach Spaces, Ismat Beg, Naseer Shahzad

Mathematics and System Engineering Faculty Publications

The existence of random fixed points for nonexpansive and pseudocontractive random multivalued operators defined on unbounded subsets of a Banach space is proved. A random coincidence point theorem for a pair of compatible random multivalued operators is established.

Linear Codes And Error-Correction, Karen Brown

Presidential Scholars Theses (1990 – 2006)

The process of encoding information for transmission from one source to another is a vital process in many areas of science and technology. Whenever coded information is sent, there arises a certain possibility that an error will occur, either during transmission or in decoding. Therefore, it is imperative to develop methods to detect and correct errors in a code. The study of coding theory is a "new" area of mathematics which is relatively undeveloped.

This paper focuses on the properties of linear codes -and their corresponding methods of error-correction. To simplify the issue, only binary block codes are studied; hence …

Some Conjectures Concerning Triangular Numbers, Bruce Brandt

Journal of the Minnesota Academy of Science

Strong empirical evidence supports conjectures that certain number patterns always hold. These patterns concern the function cr, defined by the equation cr(n) = n - m2, m2 being the nearest square to n, on the domain of the triangular numbers. Triangular squares or triangular numbers of the form m2+m are also mentioned in most of the conjectures. One of the conjectures, for example, is that the sum of cr over the triangular numbers up to a triangular square is 0. Some of these patterns can be described by strings of symbols, such as "S" and "L," formed by first writing …

Supplement To "Some Conjectures Concerning Triangular Numbers", Bruce Brandt

Journal of the Minnesota Academy of Science

In a previous paper (1), I stated many conjectures about triangular numbers. Since submitting that paper I have discovered many more results, including generalizations, which are presented here.

Group Cohomology, Modular Theory And Space-Time Symmetries, R. Brunetti, D. Guido, R. Longo

Physics - All Scholarship

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincar'e group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of …

An Invariant Subspace Problem For P = 1 Bergman Spaces On Slit Domains, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces A¹(G) and A¹(G/K), where G is a bounded region in the complex plane and K is a compact subset of a simple arc of class C¹.

Analytic Besov Spaces And Invariant Subspaces Of Bergman Spaces, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we examine the invariant subspaces (under the operator f -->z f) M of the Bergman space ^p_a (G\T) (where 1 < p < 2, G is a bounded region in C containing D, T is the unit circle, and D is the unit disk) which contain the characteristic functions x_D and x_G, i.e. the constant functions on the components of G\T. We will show that such M are in one-to-one correspondence with the invariant subspaces of the analytic Besov space AB_q (q is the conjugate index to p) and …

Hyperinvariant Subspaces Of The Harmonic Dirichlet Space, William T. Ross, Stefan Richter, Carl Sundberg

Department of Math & Statistics Faculty Publications

No abstract provided.

Prediction Of The Stochastic Behavior Of Nonlinear Systems By Deterministic Models As A Classical Time-Passage Probabilistic Problem, L. M. Ivanov, A. D. Kirwan Jr., O. V. Melnichenko

CCPO Publications

Assuming that the behaviour of a nonlinear stochastic system can be described by a Markovian diffusion approximation and that the evolution equations can be reduced to a system of ordinary differential equations, a method for the calculation of prediction time is developed. In this approach, the prediction time depends upon the accuracy of prediction, the intensity of turbulence, the accuracy of the initial conditions, the physics contained in the mathematical model, the measurement errors, and the number of prediction variables. A numerical application to zonal channel flow illustrates the theory. Some possible generalizations of the theory are also discussed.

Density Of The Numerators Or Denominators Of A Continued Fraction, Seyed J. Vafabakhsh

UNF Graduate Theses and Dissertations

Let A = {a_n}^∞_{n = 1} be a sequence of positive integers. There are two related sequences P_n and Q_n obtained from A by taking partial convergents out of the number [0; a₁, a₂, ..., a_n, ...], where P_n and Q_n are the numerators and denominators of the finite continued fraction [0; a₁, a₂, ...,a_n].

Let P(n) be the largest positive integer k , such that P_k ≤ n. The sequence Q(n …

A Study Of The Two Major Causes Of Neonatal Deaths: Perinatal Conditions And Congenital Anomalies, Felipe Lorenzo-Luaces

UNF Graduate Theses and Dissertations

Infant mortality is a public health concern in the United states. We concentrate on neonatal mortality for its high accountability of infant mortality. In this paper we study the neonatal mortality of Florida's 1989 live birth cohort.

The data has been analyzed for two major causes of deaths: perinatal conditions and congenital anomalies. We use the KAPLAN-MEIER method to estimate the survival probabilities. For each cause, data were fit to the Weibull models and Extreme Value models to estimate the parameters of the survival curves. The results indicate that primary factors for each cause of neonatal deaths are very low …

Operator Equations And Invariant Subspaces, Valentin Matache

Mathematics Faculty Publications

Banach space operators acting on some fixed space X are considered. If two such operators A and B verify the condition A² = B² and if A has non-trivial hyperinvariant subspaces, then B has nontrivial invariant subspaces. If A and B commute and satisfy a special type of functional equation, and if A is not a scalar multiple of the identity, the author proves that if A has nontrivial invariant subspaces, then so does B.

On Endomorphisms And Automorphisms Of Some Pure Subgroups Of The Baer-Specker Group, A. L.S. Corner, Brendan Goldsmith

Articles

The endomoprhism algebras of certain pure subgroups of the Baer-Specker group are investigated to answer a question of Irwin. Automorphism groups of these Abelian groups are investigated and a realization theme is established.

Asymptotic Behavior Of Solutions Of Model Problems For A Coupled System, S. Shao

Mathematics and Statistics Faculty Publications

We study the asymptotic behavior of solutions of model problems for a coupled system. By consistent use of a priori estimates and asymptotic analysis, we present here an more efficient approach which provides precise descriptions of the asymptotic behabior of solutions of this system. Our results amplify and extend earlier results of Dorr, Patter, and Shampine and their treatment of this system. Meanwhile, the stability of the steady-state solutions of the corresponding time-dependent system is discussed.

Boundary And Interior Layer Behavior In A Time-Dependent Singularly Perturbed System, S. Shao

Mathematics and Statistics Faculty Publications

No abstract provided.

First Excess Levels Of Vector Processes, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic …

Invariant Subspaces Of Bergman Spaces On Slit Domains, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces A^p(G) and A^p(G\K), where 1 < p < ∞, G is a bounded region in C, and K is a closed subset of a simple, compact, C¹ arc.