Mathematics Commons^™

Divergence Diagrams: More Than Cantor Dust Lies At The Edge Of Feigenbaum Diagrams, John H. Rickert, Aaron Klebanoff

Mathematical Sciences Technical Reports (MSTR)

The dynamical system analysis of the logistic map f(x)=ax(1-x) is studied for values of a greater than 4.

Rectangular Groups, Nick Fiala, Crystal Hanscom, Patrick Keenan, Tung Tran

Mathematical Sciences Technical Reports (MSTR)

We provide an overview of results and conjectures relating to rectangular groups.

Symmetries Of Accola-Maclachlan And Kulkarni Surfaces, Sean A. Broughton, E Bujalance, A F. Costa, J M. Gamboa, G Gromadzki

Mathematical Sciences Technical Reports (MSTR)

For all g greater than or equal to 2, there is a Riemann surface of genus g whose automorphism group has order 8g+8, establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and MacLachlan established the existence of such surfaces; we shall call them Accola-MacLachlan surfaces. In this paper we determine the symmetries of surfaces with genus g = 3(mod 4), computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves.

Population Genetics: Estimation Of Distributions Through Systems Of Non-Linear Differential Equations, Nacer E. Abrouk, Robert J. Lopez

Mathematical Sciences Technical Reports (MSTR)

In stochastic population genetics, the fundamental quantity used for describing the genetic composition of a Mendelian population is the gene frequency. The process of change in the gene frequency is generally modeled as a stochastic process satisfying a stochastic differential equation. The drift and diffusion coefficients in this equation reflect such mechanisms as mutation, selection, and migration that affect the population. Except in very simple cases, it is difficult to determine the probability law of the stochastic process of change in gene frequency. We present a method for obtaining approximations of this process, enabling us to study models more realistic …

An Elementary Proof That Finite Groups Lack Unique Product Structures, Matthew Cushman

Mathematical Sciences Technical Reports (MSTR)

We provide an elementary proof that no nontrivial finite group has a unique m-element product structure for any m greater than or equal to 2.

Conjugacy Classes Of Triple Products In Finite Groups, Kevin Hutson, Emily Salvo

Mathematical Sciences Technical Reports (MSTR)

For an underlying finite group G, we establish estimates on the number of triples that bind a certain set to one conjugacy class, or else breaks it into two conjugacy classes.

Square Roots Of Finite Groups - Ii, Matthew Devos, David Mcadams, Rebecca Rapoport

Mathematical Sciences Technical Reports (MSTR)

A subset R of a finite group G is a square root of G if R² = G. If R is a square root of G for which |R|² = G, then R is referred to as a perfect square root of G. It can be shown using character theory that perfect square roots do not exist. The purpose of this paper is to work toward an elementary proof of this result.

Square Roots Of Finite Groups, Kashi Abhyankar, Daniel Grossman

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group of order n². A perfect square root of G is a subset X of G such that |X| = n and X² = G. Neither generalized dihedral groups nor groups of nilpotency class two have perfect square roots.

Finite Groups Can Be Arbitrarily Hamiltonian, Stephen Ahearn, Mark Huber

Mathematical Sciences Technical Reports (MSTR)

Let r be a rational in (0,1]. There exists a finite group G which is the direct product of at most four metacyclic groups and whose proportion of normal subgroups is r. An analogous result holds for three other measures of "Hamiltonianess".

Counting Order Classes Of Triple Products In Finite Groups, Scott Annin, Jennifer Ziebarth

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group and let W_G denote the proportion of triples, (.x:, y , z) , i n G3 for which x yz , x zy, y x z, zx y , yzx , and z y x have the same order. The following results are established.

i) G is abelian if, and only if, W_G = 1.

ii) W_G can be arbitrarily close to 1.

iii) Additional estimates on W_G

Reconstruction Of Multiple Cracks From Experimental, Electrostatic Boundary Measurements, Kurt M. Bryan, Valdis Liepa, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We demonstrate the viability of using Electrical Impedance Tomography (EIT) for the reconstruction of multiple macroscopic cracks in a conductive medium.

Approximation Methods For Singular Diffusions Arising In Genetics, Nacer E. Abrouk

Mathematical Sciences Technical Reports (MSTR)

Stochastic models in population genetics leading to diffusion equations are considered. When the drift and the square of the diffusion coefficients are polynomials, an infinite system of ordinary differential equations for the moments of the diffusion process can be derived using the Martingale property. An example is provided to show how the classical Fokker-Planck Equation approach may not be appropriate for this derivation. A Gauss-Galerkin method for approximating the laws of the diffusion, originally proposed by Dawson (1980), is examined. In the few special cases for which exact solutions are known, comparison shows that the method is accurate and the …

Time-Discretization Of Hamiltonian Dynamical Systems, Yosi Shibberu

Mathematical Sciences Technical Reports (MSTR)

Difference equations for Hamiltonian systems are derived from a discrete variational principle. The difference equations completely determine piecewise-linear, continuous trajectories which exactly conserve the Hamiltonian function at the midpoints of each linear segment. A generating function exists for transformations between the vertices of the trajectories. Existence and uniqueness results are present as well as simulation results for a simple pendulum and an inverse square law system.

Bounds On Squares Of Two-Sets, Dan Slilaty, Jeff Vanderkam

Mathematical Sciences Technical Reports (MSTR)

For a finite group G, let p_i(G) denote the proportion of (x,y) in GxG for which the set {x²,xy,yx,y²} has cardinality i. In this paper we develop estimates on the p_i(G) for various i.

Tracking Plasma Lactate Concentration In Vivo With A Catheter-Tip L-Lactate Sensor, Brett T. Weinzapfel, Mark D. Ball, Lee R. Waite, Nacer E. Abrouk, Shun P. Lim

Mathematical Sciences Technical Reports (MSTR)

To circumvent the problems of repeated blood sampling for in vitro analysis, a catheter-tip L-lactate sensor has been developed. The sensor was tested in anesthetized pigs (n=6). The sensor in vivo tracked the lactate concentration non-linearly, seeming to obey Michaelis-Menten kinetics. Calibration time was short, typically 1.5 min per lactate standard. Furthermore, time drift was small, typically -1.3% to -3.3% per hour of in vivo use.

Hypergraph Representations And Orders Of Cwatsets, Julie Kerr

Mathematical Sciences Technical Reports (MSTR)

We determine upper bounds on the order of cwatsets of odd order.

When Is The Number Of P-Subgroups Of A Group Satisfying A Property Congruent To 1 (Mod P)?, Jason Fulman, Jeff Vanderkam

Mathematical Sciences Technical Reports (MSTR)

Let T be a property which holds for a group independent of whether or not this group is embedded in a group G or in a p-Sylow subgroup of G. Using a generalization of Sylow's second Theorem, we prove that if for any p-group P the number of subgroups of P satisfying T is congruent to 1 (mod p), then for any group G, the number of p-subgroups satisfying T is also congruent to 1 (mod p). As an application, we give simple proofs of several theorems, including the well-known Frobenius theorem.

Vertex Coloring For Weighted Graphs With Application To Timetabling, Lynn Kiaer, Jay Yellen

Mathematical Sciences Technical Reports (MSTR)

This paper applies the methods of vertex coloring to timetabling problems

Divisibility By |G| For Powers Of Ordered K-Sets, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

It is shown that the number of ordered k-sets of a group G whose nth power contains exactly i elements is always a multiple of IGI. An elementary proof of the fact that the number of ordered pairs ( x , y ) such that x² = y² is equal to k_r lGI is also given.

Cubing Ordered 2-Sets, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

Given a group G, we define p_i as the probability that, given an ordered pair X = (x,y), there are exactly i elements in X³ = {x₁x₂x₃ l x_i in X}. We show that P₂( G) = 0 if, and only if, IGI is odd, and that p₃(G) = 0 if, and only if, IGI is not divisible by three. The groups for which p₄ ( G) = 0 and p₅ ( G) = 0 are also determined.

Counting Nilpotent Pairs, Jason Fulman, Michael Galloy, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

In this paper , we consider the probability that two elements chosen at random from a finite group G generate a subgroup of a given nilpotency class. It is shown that in solvable non­-nilpotent groups, the probability that two elements generate a nilpotent subgroup is <= l/p,, where p, is the smallest prime dividing the order of the group, and it is also shown that there exist groups such that the probability of two elements generating a subgroup of class i approaches one (and other groups for which it approaches zero) for all i =>2. It is also shown …

Simultaneous Rational Approximations And Related Diophantine Equations, John Rickert

Mathematical Sciences Technical Reports (MSTR)

In this paper we consider simultaneous approximations to algebraic numbers a₁,...,a_m .

The 2/3 Bound For Rewritable N-Tuples, Lawren Smithline, Catherine Sugar

Mathematical Sciences Technical Reports (MSTR)

We study the number of rewritings of an n-tuple in a finite a group, for certain classes of groups.

Some Upper Bounds For Commutativity And Cyclicity Measures In Finite Groups, David Patrick, Catherine Sugar, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

The number of commuting and cyclic n-tuples in a group are enumerated for certain classes of groups.

Counting Nilpotent Pairs In Finite Groups: Some Conjectures, H Dubose-Schmidt, Michael D. Galloy, D,L, Wilson

Mathematical Sciences Technical Reports (MSTR)

The number of nilpotent pairs is determined for a number of small groups.

Some Facts About Cycels And Tidy Groups, Kevin O'Bryant, D. Patrick, Lawren Smithline, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

No abstract provided.

A4 Rewriteability, Eric Wepsic, Kevin O'Bryant, Lawren Smithline

Mathematical Sciences Technical Reports (MSTR)

The concept of 4-rewriteability with permutation coming form the alternating group A₄ is explored.

Rewriteability, Commutators, And Fundamental N-Rewritings, Lawren Smithline

Mathematical Sciences Technical Reports (MSTR)

We consider the relationship between commutators and rewriteability.

Undergraduates, The Right Questions, And Cayley Produce Results, Gary J. Sherman

Mathematical Sciences Technical Reports (MSTR)

During the summers of 1989, 1990, and 1991, eighteen undergraduates participated in a National Science Foundation Research Experiences for Undergraduates program at Rose-Hulman for which the author was the principal investigator. This paper provides some examples of the mathematics discovered during these three summers and discusses the philosophy, environment and process which made these discoveries possible.

More Upper Bounds On The 3-Rewriteability Of Non-3-Rewriteable Groups, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

We find an upper bound on the probability that a randomly selected triple in a group is 3-rewriteable, and a bound for the core set rewriteability.