Maeve L. McCarthy

Mathematics Instruction and the Tablet PC.

The use of tablet PCs in teaching is a relatively new phenomenon. A cross between a notebook computer and a personal digital assistant (PDA), the tablet PC has all of the features of a notebook with the additional capability that the screen can also be used for input. Tablet PCs are usually equipped with a stylus that allows the user to write on the screen. Handwriting recognition software converts this input into text for use with software such as internet browsers and email programs. As an educational tool, two of the most important features of the tablet PC are annotation and wireless communication. The annotation feature allows the user to write on almost any document much as one would annotate a printout of the same document. The wireless communication feature allows tablet PCs to share information with one another. The advantages of these features and their impact on the Murray State University (MSU) classroom will be discussed in the evaluation section.

Analysis of real-time numerical integration methods applied to dynamic clamp experiments

Real-time systems are frequently used as an experimental tool, whereby simulated models interact in real time with neurophysiological experiments. The most demanding of these techniques is known as the dynamic clamp, where simulated ion channel conductances are artificially injected into a neuron via intracellular electrodes for measurement and stimulation. Methodologies for implementing the numerical integration of the gating variables in real time typically employ first-order numerical methods, either Euler or exponential Euler (EE). EE is often used for rapidly integrating ion channel gating variables. We find via simulation studies that for small time steps, both methods are comparable, but at larger time steps, EE performs worse than Euler. We derive error bounds for both methods, and find that the error can be characterized in terms of two ratios: time step over time constant, and voltage measurement error over the slope factor of the steady-state activation curve of the voltage-dependent gating variable. These ratios reliably bound the simulation error and yield results consistent with the simulation analysis. Our bounds quantitatively illustrate how measurement error restricts the accuracy that can be obtained by using smaller step sizes. Finally, we demonstrate that Euler can be computed with identical computational efficiency as EE.

Optimal control of insects through sterile insect release and habitat modification

This paper develops an optimal control framework for an ordinary differential equation model to investigate the introduction of sterile mosquitoes to reduce the incidence of mosquito-borne diseases. Existence of a solution given an optimal strategy and the optimal control is determined in association with the negative effects of the disease on the population while minimizing the cost due to this control mechanism. Numerical simulations have shown the importance of effects of the bounds on the release of sterile mosquitoes and the bounds on the likelihood of egg maturation. The optimal strategy is to maximize the use of habitat modification or insecticide. A combination of techniques leads to a more rapid elimination of the wild mosquito population.

The identification of a time dependent sorption parameter from soil column experiments

Soil column studies are used frequently in seeking to understand the behavior of a particular contaminant in a saturated homogeneous soil of a given type. The concentration of the contaminant is modeled by a parabolic partial differential equation. We seek to identify the sorption partitioning coefficient as a function of time from limited boundary data. We discuss an output least squares formulation of the problem with Tikhonov regularization. We explicitly characterize a source condition that determines the rate of convergence of the method. Numerical examples are presented.

Identification of a chemotactic sensitivity in a coupled system

Chemotaxis is the process by which cells behave in a way that follows the chemical gradient. Applications to bacteria growth, tissue inflammation and vascular tumours provide a focus on optimization strategies. Experiments can characterize the form of possible chemotactic sensitivities. This paper addresses the recovery of the chemotactic sensitivity from these experiments while allowing for non-linear dependence of the parameter on the state variables. The existence of solutions to the forward problem is analysed. The identification of a chemotactic parameter is determined by inverse problem techniques. Tikhonov regularization is investigated and appropriate convergence results are obtained. Numerical results of concentration-dependent chemotactic terms are explored.

Isospectral membranes: a connection between shape and density

A connection is established via conformal maps between vibrating membranes that are isospectral with respect to shape and those that are isospectral with respect to density. In particular, inhomogeneous circular membranes are constructed that are isospectral to polygonal membranes of uniform density via the Schwarz–Christoffel mapping. Although some corners of the polygons lead to singularities in the constructed densities, the densities are shown to be integrable.

Diffusing wild type and sterile mosquitoes in an optimal control setting

This paper develops an optimal control framework to investigate the introduction of sterile type mosquitoes to reduce the overal moquito population. As is well known, mosquitoes are vectors of disease. For instance the WHO lists, among other diseases, Malaria, Dengue Fever, Rift Valley Fever, Yellow Fever, Chikungunya Fever and Zika. [http://www.who.int/mediacentre/factsheets/fs387/en/ ] The goal is to establish the existence of a solution given an optimal sterilization protocol as well as to develop the corresponding optimal control representation to minimize the infiltrating mosquito population while minimizing fecundity and the number of sterile type mosquitoes introduced into the environment per unit time. This paper incorporates the diffusion of the mosquitoes into the controlled model and presents a number of numerical simulations.

SIAM Education Subcommittee Report on Undergraduate Degree Programs in Applied Mathematics

The SIAM Education Committee has released a report called Undergraduate Degree Programs in Applied Mathematics. The report describes the general and specific features of undergraduate education in applied mathematics, based on interviews with 12 diverse but representative programs, and offers commentary based on the experience of the committee. This article summarizes the key findings of the SIAM report, focusing on curricular requirements, the role of industry, undergraduate research, student recruitment, and starting a new program. The goal of the report and this article is to provide guidance to new programs, existing programs, and the development of policy.

A model of inter-cohort cannibalism and paedomorphosis in Arizona Tiger Salamanders, Ambystoma tigrinum nebulosum

Cannibalism is widespread in size-structured populations. If cannibals and victims are in different life stages, dominant cohorts of cannibals can regulate recruitment. Arizona Tiger Salamanders, Ambystoma tigrinum nebulosum, exhibit facultative paedomorphosis in which salamander larvae either metamorphose into terrestrial adults or become sexually mature while still in their larval form. Although many salamanders exhibit cannibalism of larvae, the Arizona Tiger Salamander also exhibits cannibalism of young by the aquatic adults. We formulate a differential equations model of this system under the assumption that the terrestrial adults do not impact the system beyond their contribution to the birth of young larvae. We establish non-negativity, boundedness and persistence of the salamander population under certain assumptions. We consider the equilibrium states of the system in the presence or absence of a birth contribution from the terrestrial or metamorph adults. Constant per capita paedomorphosis leads to asymptotically stable equilibria. The per capita paedomorphosis rate of the larvae must be density dependent in order for periodic solutions to exist. Furthermore, the stage transition rate must be sufficiently decreasing in order to guarantee the existence of an unstable equilibrium. Periodic solutions are only possible in the presence of a unique nontrivial unstable equilibrium. Our results conform to previous theory on paedomorphosis which suggests general applicability of our results to similar systems.

BioMaPS: A Roadmap for Success

The manuscript outlines the impact that our National Science Foundation Interdisciplinary Training for Undergraduates in Biological and Mathematical Sciences program, BioMaPS, has had on the students and faculty at Murray State University. This interdisciplinary program teams mathematics and biology undergraduate students with mathematics and biology faculty and has produced research insights and curriculum developments at the intersection of these two disciplines. The goals, structure, achievements, and curriculum initiatives are described in relation to the effects they have had to enhance the study of biomathematics.