Physical Sciences and Mathematics Commons^™

The Stabilizing Effect Of Noise On The Dynamics Of A Boolean Network, Christopher S. Goodrich, Mihaela Teodora Matache

Mathematics Faculty Publications

In this paper, we explore both numerically and analytically the robustness of a synchronous Boolean network governed by rule 126 of cellular automata. In particular, we explore whether or not the introduction of noise into the system has any discernable effect on the evolution of the system. This noise is introduced by changing the states of a given number of nodes in the system according to certain rules. New mathematical models are developed for this purpose. We use MATLAB to run the numerical simulations including iterations of the real system and the model, computation of Lyapunov exponents, and generation of …

Clinical Experience And Examination Performance: Is There A Correlation?, Gary L. Beck, Mihaela Teodora Matache, Carrie Riha, Katherine Kerber, Frederick A. Mccurdy

Mathematics Faculty Publications

Context  The Liaison Committee on Medical Education (LCME) requires there to be: ‘…comparable educational experiences and equivalent methods of evaluation across all alternative instructional sites within a given discipline’. It is an LCME accreditation requirement that students encounter similar numbers of patients with similar diagnoses. However, previous empirical studies have not shown a correlation between the numbers of patients seen by students and performance on multiple-choice examinations.

Objective  This study examined whether student exposure to patients with specific diagnoses predicts performance on multiple-choice examination questions pertaining to those diagnoses.

Methods  The Department of Pediatrics at the University of Nebraska Medical …

Perturbations Of Roots Under Linear Transformations Of Polynomials, Branko Ćurgus, Vania Mascioni

Mathematics Faculty Publications

Let P_n be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators T:P_n→P_n for which there exists a constant C > 0 such that for all nonconstant f∈P_n there exist a root u of f and a root v of Tf with |u−v|≤C. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of f and Tf, the roots are never displaced by more than a uniform constant independent on f. We show that such "good" operators T …

A Spectral Order For Infinite Dimensional Quantum Spaces: A Preliminary Report, Joe Mashburn

Mathematics Faculty Publications

In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approximations of which pure states. In a previous paper the author extended the Bayesian order to infinite dimensional spaces of classical states. The order on infinite dimensional spaces retains many of the characteristics important to physics, …

Dynamics Of Asynchronous Random Boolean Networks With Asynchrony Generated By Stochastic Processes, Xutao Deng, Huimin Geng, Mihaela Teodora Matache

Mathematics Faculty Publications

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897–923] for a varying number of parents. In both these papers the authors consider an …

An Unexpected Limit Of Expected Values, Branko Ćurgus, Robert I. Jewett

Mathematics Faculty Publications

Let t⩾0. Select numbers randomly from the interval [0,1] until the sum is greater than t . Let α(t) be the expected number of selections. We prove that α(t)=e^t for 0⩽t⩽1. Moreover, . This limit is a special case of our asymptotic results for solutions of the delay differential equation f^′(t)=f(t)-f(t-1) for t>1. We also consider four other solutions of this equation that are related to the above selection process.

Calculus Students’ Difficulties In Using Variables As Changing Quantities, Susan S. Gray, Barbara J. Loud, Carole Sokolowski

Mathematics Faculty Publications

The study of calculus requires an ability to understand algebraic variables as generalized numbers and as functionally-related quantities. These more advanced uses of variables are indicative of algebraic thinking as opposed to arithmetic thinking. This study reports on entering Calculus I students’ responses to a selection of test questions that required the use of variables in these advanced ways. On average, students’ success rates on these questions were less than 50%. An analysis of errors revealed students’ tendencies toward arithmetic thinking when they attempted to answer questions that required an ability to think of variables as changing quantities, a characteristic …

A "Sound" Approach To Fourier Transforms: Using Music To Teach Trigonometry, Bruce Kessler

Mathematics Faculty Publications

If a large number of educated people were asked, ``What was your most exciting class?'', odds are that very few of them would answer ``Trigonometry.'' The subject is generally presented in a less-than-exciting fashion, with the repeated caveat that ``you'll need this when you take calculus,'' or ``this has lots of applications'' without ever really seeing many of them. This manuscript addresses how the author is trying to change this tradition by exposing casual students from kindergarten to college to Joseph Fourier's secret, that nearly any function can be built out of sine and cosine curves. And music serves as …

The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover

Mathematics Faculty Publications

In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the primary part of the Farrell cohomology of a few pure mapping class groups.

The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms...

On A Convex Operator For Finite Sets, Branko Ćurgus, Krzysztof Kołodziejczyk

Mathematics Faculty Publications

Let S be a finite set with m elements in a real linear space and let be a set of m intervals in . We introduce a convex operator which generalizes the familiar concepts of the convex hull, , and the affine hull, , of S . We prove that each homothet of that is contained in can be obtained using this operator. A variety of convex subsets of with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral …

Continued Fractions With Multiple Limits, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincar´e type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan’s with three limits to a continued fraction with …

A Transform Method In Discrete Fractional Calculus, Ferhan M. Atici, Paul W. Eloe

Mathematics Faculty Publications

We begin with an introduction to a calculus of fractional finite differences. We extend the discrete Laplace transform to develop a discrete transform method. We define a family of finite fractional difference equations and employ the transform method to obtain solutions.

A Comparison Of Three Topologies On Ordered Sets, Joe Mashburn

Mathematics Faculty Publications

We introduce two new topologies on ordered sets: the way below topology and weakly way below topology. These are similar in definition to the Scott topology, but are very different if the set is not continuous. The basic properties of these three topologies are compared. We will show that while domain representable spaces must be Baire, this is not the case with the new topologies.

Some Observations On Khovanskii's Matrix Methods For Extracting Roots Of Polynomials, James Mclaughlin, B. Sury

Mathematics Faculty Publications

In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a …

Ramanujan And Extensions And Contractions Of Continued Fractions, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

If a continued fraction K∞n=1an/bn is known to converge but its limit is not easy to determine, it may be easier to use an extension of K∞n=1an/bn to find the limit. By an extension of K∞n=1an/bn we mean a continued fraction K∞n=1cn/dn whose odd or even part is K∞n=1an/bn. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if K∞n=1an/bn is the even part); (ii) Find the limit of the …

Symmetry And Specializability In The Continued Fraction Expansions Of Some Infinite Products, James Mclaughlin

Mathematics Faculty Publications

Let f(x) ∈ Z[x]. Set f0(x) = x and, for n ≥ 1, define fn(x) = f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product Y∞ n=0 ( 1 + 1 fn(x) ) has a specializable continued fraction expansion of the form S∞ = [1; a1(x), a2(x), a3(x), . . . ], where ai(x) ∈ Z[x] for i ≥ 1. When the infinite product and the continued fraction are specialized by letting x take integral values, we get infinite classes of real numbers whose regular continued fraction expansion is predictable. We also show that, under some …

Some More Long Continued Fractions, I, James Mclaughlin, Peter Zimmer

Mathematics Faculty Publications

In this paper we show how to construct several infinite families of polynomials D(¯x, k), such that p D(¯x, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.

Some Properties Of The Distribution Of The Numbers Of Points On Elliptic Curves Over A Finite Prime Field, Saiying He, James Mclaughlin

Mathematics Faculty Publications

Let p ≥ 5 be a prime and for a, b ∈ Fp, let Ea,b denote the elliptic curve over Fp with equation y 2 = x 3 + a x + b. As usual define the trace of Frobenius ap, a, b by #Ea,b(Fp) = p + 1 − ap, a, b. We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums P t∈Fp ap, t, b, P t∈Fp ap, a, t, Pp−1 t=0 a 2 p, t, b, Pp−1 t=0 a 2 p, a, t and Pp−1 …

S-Toeplitz Composition Operators, Valentin Matache

Mathematics Faculty Publications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ.

Distances Between Composition Operators, Valentin Matache

Mathematics Faculty Publications

Composition operators Cϕ induced by a selfmap ϕ of some set S are operators acting on a space consisting of functions on S by composition to the right with ϕ, that is Cϕf = f ◦ ϕ. In this paper, we consider the Hilbert Hardy space H2 on the open unit disk and find exact formulas for distances kCϕ − Cψk between composition operators. The selfmaps ϕ and ψ involved in those formulas are constant, inner, or analytic selfmaps of the unit disk fixing the origin.