Physical Sciences and Mathematics Commons^™

Color Models As Tools In Teaching Mathematics, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

In this paper we discuss various situations how color models and patterns can be used to simplify the study of abstract mathematics and serve as tools in understanding mathematical ideas. We illustrate the realization of such models through the development of advanced computer technology. In particular, we present how a computer algebra software such as Mathematica, or a dynamic geometry environment, can be utilized to facilitate the study of transformation geometry and group theory.

Operator Monotone Functions And Löwner Functions Of Several Variables, Jim Agler, John E. Mccarthy, N J. Young

Mathematics Faculty Publications

We prove generalizations of Loewner's results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.

Bio-Logic Builder: A Non-Technical Tool For Building Dynamical, Qualitative Models, Tomáš Helikar, Brian Kowal, Alex Madrahimov, Manish Shrestha, Jay Pedersen, Kahani Limbu, Ishwor Thapa, Thaine Rowley, Rahul Satalkar, Naomi Kochi, John Konvalina, Jim A. Rogers

Mathematics Faculty Publications

Computational modeling of biological processes is a promising tool in biomedical research. While a large part of its potential lies in the ability to integrate it with laboratory research, modeling currently generally requires a high degree of training in mathematics and/or computer science. To help address this issue, we have developed a web-based tool, Bio-Logic Builder, that enables laboratory scientists to define mathematical representations (based on a discrete formalism) of biological regulatory mechanisms in a modular and non-technical fashion. As part of the user interface, generalized “bio-logic” modules have been defined to provide users with the building blocks for many …

Sustainable Technology For Person-Centered Accessible Integrated Multimodal Information Systems, Lawrence J. Harman, Uma Shama, Heather Standring, Sabitha Gopalsamy, Anil Sadhu, Mateusz Pacha-Sucharzewski

Mathematics Faculty Publications

This paper reports on a mobility management technology project conducted by the GeoGraphics Laboratory at Bridgewater State University in Bridgewater, Massachusetts, in the Northeastern United States (U.S.). This study is a part of a much larger mobility management technology deployment by the Cape Cod Regional Transit Authority (CCRTA) that deployed integrated intermodal intelligent transportation system (ITS) to support the mobility of a metropolitan region that has a high proportion of elderly residents and persons with disabilities and is a significant tourist destination for national and international travelers. This paper reports on a research project that is developing smartphone applications to …

The Cell Collective: Toward An Open And Collaborative Approach To Systems Biology., Tomáš Helikar, Brian Kowal, Sean Mcclenathan, Mitchell Bruckner, Thaine Rowley, Alex Madrahimov, Ben Wicks, Manish Shrestha, Kahani Limbu, Jim A. Rogers

Mathematics Faculty Publications

Background: Despite decades of new discoveries in biomedical research, the overwhelming complexity of cells has been a significant barrier to a fundamental understanding of how cells work as a whole. As such, the holistic study of biochemical pathways requires computer modeling. Due to the complexity of cells, it is not feasible for one person or group to model the cell in its entirety. Results: The Cell Collective is a platform that allows the world-wide scientific community to create these models collectively. Its interface enables users to build and use models without specifying any mathematical equations or computer code - addressing …

Maximum Likelihood Estimation Of Individual Inbreeding Coefficients And Null Allele Frequencies, Nathan Hall, Laina Mercer, Daisy Phillips, Jonathan Shaw, Amy D. Anderson

Mathematics Faculty Publications

In this paper, we developed and compared several expectation-maximization (EM) algorithms to find maximum likelihood estimates of individual inbreeding coefficients using molecular marker information. The first method estimates the inbreeding coefficient for a single individual and assumes that allele frequencies are known without error. The second method jointly estimates inbreeding coefficients and allele frequencies for a set of individuals that have been genotyped at several loci. The third method generalizes the second method to include the case in which null alleles may be present. In particular, it is able to jointly estimate individual inbreeding coefficients and allele frequencies, including the …

Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu

Mathematics Faculty Publications

In this paper, we use Hermite cubic finite elements to approximate the solutions

of a nonlinear Euler-Bernoulli beam equation. The equation is derived

from Hollomon’s generalized Hooke’s law for work hardening materials with

the assumptions of the Euler-Bernoulli beam theory. The Ritz-Galerkin finite

element procedure is used to form a finite dimensional nonlinear program

problem, and a nonlinear conjugate gradient scheme is implemented to find

the minimizer of the Lagrangian. Convergence of the finite element approximations

is analyzed and some error estimates are presented. A Matlab finite

element code is developed to provide numerical solutions to the beam equation.

Some …

Training The Next Generation Of Transportation Professionals In Geographic Data Collection, Spatial Analysis And Customer Information, Uma Shama, Lawrence J. Harman

Mathematics Faculty Publications

Applying advanced technology for transportation research and management has been the focus of Bridgewater State’s GeoGraphics Laboratory since 1994. The laboratory has attracted a broad range of students from many academic disciplines and walks of life to engage in leading edge applications of innovative hardware and software to meet the needs of our transportation systems. The GeoGraphics Lab has provided a laboratory experience for nearly two decades. Geographic technology has changed dramatically over this time. The availability of global positioning systems (GPS) on cell phones is nearly ubiquitous. The increased power and reduced cost of desktop geographic information systems (GIS) …

When The Trivial Is Nontrivial, William Capecchi, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Mean-Field Boolean Network Model Of A Signal Transduction Network, Naomi Kochi, Mihaela Teodora Matache

Mathematics Faculty Publications

In this paper we provide a mean-field Boolean network model for a signal transduction network of a generic fibroblast cell. The network consists of several main signaling pathways, including the receptor tyrosine kinase, the G-protein coupled receptor, and the Integrin signaling pathway. The network consists of 130 nodes, each representing a signaling molecule (mainly proteins). Nodes are governed by Boolean dynamics including canalizing functions as well as totalistic Boolean functions that depend only on the overall fraction of active nodes. We categorize the Boolean functions into several different classes. Using a mean-field approach we generate a mathematical formula for the …

Groups Of Graphs Of Groups, David P. Byrne, Matthew J. Donner, Thomas Q. Sibley

Mathematics Faculty Publications

We classify all groups of color preserving automorphisms (isometries) of edge colored complete graphs derived from finite groups.

Undergraduate Students' Self-Reported Use Of Mathematics Textbooks, Aaron Weinberg, Emilie Wiesner, Bret Benesh, Timothy Boester

Mathematics Faculty Publications

Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook. The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself, as well as students' beliefs about reading and the …

Intersections Of Dilatates Of Convex Bodies, Stefano Campi, Richard J. Gardner, Paolo Gronchi

Mathematics Faculty Publications

We initiate a systematic investigation into the nature of the function ∝_K(L,ρ) that gives the volume of the intersection of one convex body K in Rⁿ and a dilatate ρL of another convex body L in Rⁿ, as well as the function η_K(L, ρ) that gives the (n - 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρ L) of ρL. The focus is on the concavity properties of α_K (L, ρ). Of particular interest is the …

On The Reproducing Kernel Of A Pontryagin Space Of Vector Valued Polynomials, Branko Ćurgus, Aad Dijksma

Mathematics Faculty Publications

We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of d×1 vector polynomials is determined by a generalized Nevanlinna pair of d×d matrix polynomials.

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann

Mathematics Faculty Publications

We consider the class of abelian groups possessing partial decomposition bases in L^δ_∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.

Finite Factors Of Bernoulli Schemes And Distinguishing Labelings Of Directed Graphs, Andrew Lazowski, Stephen M. Shea

Mathematics Faculty Publications

A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the k -block presentation of a Bernoulli scheme X . We determine the automorphism group of G , and thus the distinguishing labelings of G . A labeling of G defines a finite factor of X . We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of X . We use …

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider abelian groups with partial decomposition bases in L^δ_∞ω for ordinals δ. Jacoby, Leistner, Loth and Str¨ungmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Warfield invariants and Ulm-Kaplansky invariants up to ωδ for some ordinal δ, then they are equivalent in L^δ_∞ω. Here we prove that the modified Warfield invariant is expressible in L^δ_∞ω and hence the converse is true for appropriate δ.

Mathematical Competitions In Hungary: Promoting A Tradition Of Excellence & Creativity, Julianna Connelly Stockton

Mathematics Faculty Publications

Hungary has long been known for its outstanding production of mathematical talent. Extracurricular programs such as camps and competitions form a strong foundation within the Hungarian tradition. New types of competitions in recent years include team competitions, multiple choice competitions, and some exclusively for students who are not in a special mathematics class. This study explores some of the recent developments in Hungarian mathematics competitions and the potential implications these changes have for the very competition-driven system that currently exists. The founding of so many new competitions reflects a possible shift in the focus and purpose of competitions away from …

Controlling Nanoparticles Formation In Molten Metallic Bilayers By Pulsed-Laser Interference Heating, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

The impacts of the two-beam interference heating on the number of core-shell and embedded nanoparticles and on nanostructure coarsening are studied numerically based on the non-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20 nm) metallic bilayers. The model incorporates thermocapillary forces and disjoining pressures, and assumes dewetting from the optically transparent substrate atop of the reflective support layer, which results in the complicated dependence of light reflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillary effect is due to the local thickness-dependent, steady- state temperature profile in the liquid, which is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Linear stability analysis of the model equations set for Ag/Co bilayer predicts the dewetting length scales in the qualitative agreement with experiment.

A Note On Acoustic Propagation In Power-Law Fluids: Compact Kinks, Mild Discontinuities, And A Connection To Finite-Scale Theory, Dongming Wei

Mathematics Faculty Publications

Acoustic traveling waves in a class of viscous, power-lawfluids are investigated. Both bi-directional and unidirectional versions of the one-dimensional (1D), weakly-nonlinear equation of motion are derived; traveling wave solutions (TWS)s, special cases of which take the form of compact and algebraic kinks, are determined; and the impact of the bulk viscosity on the structure/nature of the kinks is examined. Most significantly, we point out a connection that exists between the power-law model considered here and the recently introduced theory of finite-scale equations.

Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton

Mathematics Faculty Publications

This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi

Mathematics Faculty Publications

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Holloman’s power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and …

On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei

Mathematics Faculty Publications

In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.

Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay

Mathematics Faculty Publications

In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.

An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton

Mathematics Faculty Publications

We study geodesics of the H¹ Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L² metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H¹ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C^∞ in the Banach topology C¹ ([0,1], R²), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, …

Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton

Mathematics Faculty Publications

For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.

Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi

Mathematics Faculty Publications

In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …

Triangles And Groups Via Cevians, Árpád Bényi, Branko Ćurgus

Mathematics Faculty Publications

For a given triangle T and a real number ρ we define Ceva’s triangle Cρ(T) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratioρ: (1 – ρ). We identify the smallest interval MT⊂R such that the family Cρ(T),ρ∈MT, contains all Ceva’s triangles up to similarity. We prove that the composition of operators Cρ,ρ∈R, acting on triangles is governed by a certain group structure on R. We use this structure to prove that two triangles have the same Brocard angle if and …

Existence And Uniqueness Conditions For A Class Of (K+4j)-Point N-Th Order Boundary Value Problems, Paul W. Eloe, Johnny Henderson, Rahmat Ali Khan

Mathematics Faculty Publications

No abstract provided.

A Study Of The Gam Approach To Solve Laminar Boundary Layer Equations In The Presence Of A Wedge, Rahmat Ali Khan, Muhammad Usman

Mathematics Faculty Publications

We apply an easy and simple technique, the generalized ap- proximation method (GAM) to investigate the temperature field associated with the Falkner-Skan boundary-layer problem. The nonlinear partial differ- ential equations are transformed to nonlinear ordinary differential equations using the similarity transformations. An iterative scheme for the non-linear ordinary differential equations associated with the velocity and temperature profiles are developed via GAM. Numerical results for the dimensionless ve- locity and temperature profiles of the wedge flow are presented graphically for different values of the wedge angle and Prandtl number.