Annotated Bibliography

 

The brief annotated bibliography below highlights my scholarly activities over the last 10 years.  For a complete listing of all these activities please see my CV.  Because I am active in both research and pedagogy, I have chosen to subdivide this bibliography accordingly.  Moreover, because invited presentations are usually about research as opposed to pedagogy I have only included a section describing my research presentations.

 

Research Papers

 

The five papers listed below illustrate the breadth of my scholarly interests in applied mathematics.  Items (1) and (4) are both in population genetics, items (2) and (3) in computer science, and item (5) is in neuroscience.  What unites these five papers is the underlying mathematics.  All of them use methods in dynamical systems to prove fundamental results about the phenomenon being modeled.  I think it is also important to note that three of these papers are co-written with undergraduate students.

 

1.      H.G. Spencer, T. Dorn, and T. LoFaro (2006) Population Models of Genomic Imprinting II.  Maternal and Fertility Selection.  Genetics, 173:2391-2398.

 

This paper is co-authored with Timothy Dorn (class of 2005).  Much of the mathematics in this paper was done as part of TimÕs Honors Thesis.  We show that in this model of genomic imprinting a Hopf bifurcation of a stable equilibrium occurs and leads to stable oscillatory solutions.  This implies that in situations where the genotype of the mother affects the fitness of her offspring it is possible for allele frequencies to oscillate over time.

 

2.      A Farahat, T. LoFaro, J. C. Miller, G. Rae and L. A. Ward (2006) Authority rankings from HITS, PageRank, and SALSA: existence, uniqueness, and effect of initialization. SIAM Journal on Scientific Computing. 27:1181-1201.

 

Two of the co-authors were undergraduate students at Harvey Mudd College.  This paper provides the theoretical foundations that describe when various web ranking algorithms will converge to a unique and meaningful ranking vector.   Web ranking algorithms are the main tool of search engines such as Google, Yahoo!, etc. and this paper provides the theoretical underpinnings for the convergence of these methods to reasonable and useful results.

 

3.      A. Farahat et. al. (2001) Modifications of KleinbergÕs HITS algorithm using matrix exponentiation and web log records (2001), in Proceedings of the 24th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval.

 

Three of the co-authors were undergraduate students at Harvey Mudd College.  This paper discusses a modification to the HITS (Hypertext Induced Topic Search) web ranking algorithm that addresses some of the convergence issues exhibited by the original algorithm.   In particular, there are certain classes of web topologies that cause this particular algorithm to either converge to ranking that depend on initial data or to converge to an essentially meaningless ranking.  After categorizing these topologies we propose a modification to this algorithm that both eliminates this problem and results in a more meaningful ranking when the algorithm is applied to a non-degenerate structure.

 

4.      T. LoFaro and R. Gomulkiewicz (1999) Adaptation versus migration in demographically unstable populations, Journal of Mathematical Biology,38:571-584.

 

This paper uses methods of dynamical systems and ergodic theory to establish necessary conditions for the initial spread of a rare allele.  In particular, we model a scenario where the fitness associated with each genotype is density dependent and show that it is necessary for the geometric mean of absolute fitness of the heterozygote to exceed 1 if the novel allele is to spread from rarity.  This is in contrast to models that ignore density dependence where relative fitness is supposed to be the key factor that determines an initial spread from rarity.

 

5.      T. LoFaro and N. Kopell (1999) Timing regulation in a network reduced from voltage-gated conductance equations, Journal of Mathematical Biology, 38:479-533.

 

This paper mathematically describes a model of coupled neurons and shows that the presence of a certain ionic current in one of these neurons can serve to regulate the rhythms generated by the network.  One of the key developments in this work was a method for reducing the dynamics of a high-dimensional system of differential equations to a low-dimensional discrete dynamical system that can be more easily analyzed.   This method has been utilized in subsequent models of coupled neurons to better understand how certain ionic currents help regulate the rhythms generated by these networks.

 

Research Presentations

 

1.      Crossing the Threshold: The Role of Density Dependence and Demographic Stochasticity in the Evolution of Cooperation, Poster at Dynamical Systems and Topology Conference, Tossa de Mar, Spain, 2008.

 

A poster on current research about a mathematical model of the evolution of cooperation.  This model incorporates both deterministic and stochastic components and illustrates why randomness is necessary for the spread of cooperation.

 

2.      Authority rankings from HITS, PageRank, and SALSA: existence, uniqueness, and effect of initialization, Invited Presentation, Minnesota State University, Mankato, 2006.

 

A presentation on the research described in item 2 in the above section.

 

3.      Crossing the Threshold: The Role of Density Dependence and Demographic Stochasticity in the Evolution of Cooperation, Invited Presentation, University of St. Thomas, 2005.

 

See item 1 above.

 

4.      Modifications of KleinbergÕs HITS algorithm using matrix exponentiation and web log records, Poster at the 24th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, 2001.

 

A poster presentation of the material described in item 3 above.

 

5.      Adaptation versus migration in demographically unstable populations, Invited Presentation, University of Wisconsin, Stevens Point, 2001.

 

A presentation on the material described in item 5 above.

 

Pedagogical Publications

 

I have included this listing of pedagogical publications to illustrate my contributions to the teaching of differential equations and dynamical systems.  It is important to note that each of these projects focuses on a slightly different aspect of undergraduate differential equations education.  

 

1.      T. LoFaro and K.D. Cooper (2003) Nine Projects in Differential Equations with Boundary-Value Problems 6th ed.  by D.G. Zill and M.R. Cullin, Brooks/Cole.

 

Kevin Cooper and I were invited to write nine chapter projects for this classic differential equations text.  These projects appear at the ends of the relevant chapters and are comprehensive modeling projects that require the student to solve differential equations, use computer software to graph solutions, interpret the solutions in the context of the phenomenon being modeled, and finally write up there solutions in a report.  This text book is one of the top selling differential equations textbooks in the country.

 

2.      T. LoFaro and R. Adams (2002) Chasing a polar goose, The UMAP Journal, 23:113-120.

 

In this modeling paper with a former undergraduate student at Harvey Mudd College, we discuss the use of polar coordinates in studying a classic differential equation that describes the path a goose might fly returning to its nest when faced with a cross-wind.  The UMAP Journal presents undergraduate mathematical modeling work and this joint paper was motivated by a question that my co-author Rob asked me during class.  Although this is not groundbreaking research, this paper provides a capsule that connects two different tools that are generally taught in an undergraduate differential equations course.

 

3.      D. Slavit, T. LoFaro and K.D. Cooper (2002) The use of simulations in developing relational understandings of the solutions of differential equations, School Science and Mathematics, 102:380-390.

 

This paper is co-written with a mathematics education professor (David Slavit) and examines how the use of computer simulations in the teaching of differential equations aids students understanding of fundamental concepts in the course.  Unlike the previous entries in this list, this paper is not a modeling project in differential equations but an assessment of how these projects and technology aid student interest and understanding of the material.

 

4.      T. LoFaro, J.M. McDill and A.M. Rash (2001) New visualization software for iterative systems, differential equations and applied calculus, in Proceedings of the 13th Annual International Conference on Technology in Collegiate Mathematics. Addison Wesley Longman.

 

This paper provides a review of various software packages that can be used by students or in classroom demonstrations to illustrate fundamental concepts in differential equations, dynamical systems and calculus.  It is intended to be a resource to instructors who are considering using these tools in their teaching.

 

5.      T. LoFaro (1998) Module on discrete dynamical systems, in ODE Architect, eds. R. Borrelli and C. Coleman, John Wiley and Sons.

 

This was one of the first pieces of software that combined text, simulation, animation and other interactive experiences to illustrate mathematical concepts.   I authored one module of this package and helped design the accompanying simulation software.  In 1998 it won an Invision Award from NewMedia Magazine for the best academic software of the year.