September 2012 The 4xn Multistate Lights Out Game, Mathematical Sciences International Research Journal, Vol 1, N Number 1, Pages 10-13.
This paper includes a discussion of solutions for 4xn Lights Out games in which buttons can take on more than two states. Games with buttons states which are a power of 3 are explored in detail.
2. May 2012 Developing Curiosity in Science with Service, accepted to The Journal for Civic Commitment, accepted December 28, 2012
In 2008, my sabbatical focused on analyzing the impact of the Elon Traveling Science Center. This paper discusses the experience and how it impacts those involved. This paper follows several presentations on the subject and has led to several consulting opportunities.
3. May 2012 Multistate Lights Out, with Rachel Wilson, submitted to Mathematics Magazine.
The two state Lights Out game has been studied in detail. In this paper we explore the multistate Lights Out, in which rectangular Lights Out games have buttons that can take on more than two states. A linear algebra approach is discussed and several classes of games that have solutions are explored.
4. February 2012 Pascal’s Triangle in Higher Dimensions, with Amanda Coe, accepted with revisions to Involve, revisions submitted November 2012.
This paper is in the field of number theory and is an extension of results of interest in the field including an extension of Pascal’s Tetrahedron and Star of David Theorem to multinomial coefficients.
5. December 2011 Seriation Algorithms for Determining the Evolution of the Star Husband Tale, with J. Todd Lee and Cheryl Borden, accepted to Involve.
The Star Husband Tale is an American Indian tale shared by over 80 tribes across North America. As the tale traveled from one tribe to the next it was altered and morphed but still each version shared some characteristics of commonality. This paper looks at the characteristic present, or lacking, in each of the versions of the tale and uses applied linear algebra techniques to determine the path through which the tale evolved. It is interesting to note that the mathematical results of this paper match up nicely with historians’ views of where the tale began.
6. Turning Lights Out, UMAP/ILAP/BioMath Modules 2010: Tools for Teaching, with J. Todd Lee and Brianna Yoho, edited by Paul J. Campbell. Bedford, MA: COMAP, Inc., Pages 1-26.
This is a reprint of Turning all Lights Out. Editor Campbell chooses a few modules to highlight in his Tools for Teaching. The editor recognized the thoroughness and impact of this article and asked the authors to reprint the results. The UMAP Modules presents new results in applied mathematics to an audience with a style toward teaching the audience about the result, including suggested exercises for the reader.
7. Turning all Lights Out, with J. Todd Lee and Brianna Yoho, The UMAP Journal, Vol 31.1.
Lights Out is a hand-held game of electric buttons and adjacent connections. The goal of the game is, given a random initial condition of on and off buttons, to press buttons in a sequence that will turn all of the lights off. In this paper, we explore a general grid of buttons and connections where all of the lights start on. Several approaches are taken to prove that one can always turn the lights out.
8. Notes on Leximorphic Spaces, with J. Todd Lee, and Ellen Mir, The Rocky Mountain Journal of Mathematics, Volume 40, Number 4, Pages 1257-1274.
There are many ways to order objects or numbers. Lexicographic ordering is similar to dictionary ordering in that if (a,b) < (a’,b’) then a<a’ and if a=a’ then b <b’. Leximorphic spaces are spaces related to these orderings and in this paper the leximorphic spaces and their properties are further explored.
9. Collecting Cards, with J. Todd Lee and Ellen Mir, UMAP Journal, Vol 28.4.
There are so many collector cards set out there today for kids to try to collect so in this paper we explore the mathematical theory of collecting all of the cards in a set through the collection process. This paper the probability of collecting all of the cards in a set is discussed as well as an estimation to the number of cards that would have to be collected in order to collect an entire set of cards.
10. Parameter estimation for a drying system in a porous medium, with Diego Murio, Computers & Mathematics with Applications, Volume 51, Issues 9-10, Pages 1519-1528.
This paper introduces a numerical solution to the Luikov model for drying a porous medium. This model is an application of heat and mass transfer. Initial conditions are known on a boundary of the medium and the behavior of the medium is estimated under the introduced model.
11. Numerical solutions of inverse spatial Lotka-Volterra systems, with Diego Murio, Mathematical and Computer Modelling, Volume 42, Issue 13, Pages 1411-1420.
Lotka-Volterra systems are used to model predator prey biological systems. These are well know models however this paper looks at inverse systems were predator and prey density populations are known, within a window of error, in a particular location of a habitat and these initial densities are used to predict the population densities throughout a habitat over time.
12. August 2004 Parameter Identification Using Mollification for Predator-Prey Models in Spatially Heterogeneous Environments, Computers & Mathematics with Applications, Volume 48, Issues 3-4, Pages 505-515.
This paper concentrates on a Lotka-Volterra model in which spatially dependent diffusion effects and bias dispersal are added. Bias dispersal allows the species to use their perception to move toward favorable regions. The numerical algorithm described in this paper allows for the recovery of population densities and diffusion coefficients in the model.
13. December 2002 Parameter Estimation in the 1-D Transport Equation with Advection Computers & Mathematics with Applications, Volume 44, Issue 12, Pages 1493-1502.
This paper looks at an ecological model in which the habitat is spatially non-uniform and which random and non-random dispersion are present. The dispersion coefficient and population density are recovered. I was informed by the editor of this journal that this paper was among the most downloaded from January 2003 to March 2003.
14. December 2001 Simultaneous Space Diffusivity and Source Term Reconstruction in 2D IHCP, with Diego Murio, Computers & Mathematics with Applications, Volume 42, Issue 12, Pages 1549-1564.
This paper focuses on a two dimensional inverse heat conduction problem in which an external source term is present. In this model, the external source term is dependent on both space parameters as well as time. It is assumed that this term can be written as two independent functions, one as a function of space, the spatially dependent source, and the other as a function of time, the temporal dependent source. The diffusivity coefficient and the spatially dependent source are estimated simultaneously.
15. Automatic Numerical Solution of the Generalized 2-D IHCP by Discrete Mollification, with S. Zhan and D.A. Murio, Computers and Mathematics with Applications, Volume 41, Issues 1-2, Pages 15-38.
16. Identification of Parameters in the Two Dimensional Inverse Heat Conduction Problem, with D. A. Murio, Computers and Mathematics with Applications, Volume 40, Issues 8-9, Pages 939-956.