SCHOLARLY INTERESTS

I am primarily interested in numerical analysis and applied mathematics. The title of the volume my first published paper appears in says it best: Math Is For Solving Problems (Editors L. Pamela Cook and Victor Roytburd, SIAM, 1996).

My thesis work (under Julian D. Cole and Don W. Schwendeman) was in theoretical aerodynamics/ scientific computing. There I showed that it was possible to compute solutions of the Transonic Small Disturbance equations that in the case of flow around a slender body of revolution without fore-aft symmetry possessed vanishingly small shocks. This was a very tricky calculation because it involves trying to obtain the solution accurately as near as one can get to a rather nasty singularity. In 1998, I was invited to present these results at the 2nd Theoretical Fluid Mechanics meeting at the annual American Institute of Aeronautics and Astronautics conference. Along the way we stumbled upon an interesting way to discretize the radial derivatives in our partial differential equation. I later discovered that Professor Ron Mickens of Clark Atlanta University had been writing about these same kinds of "nonstandard finite difference" schemes for years.

I have always been interested in applying mathematics to unusual phenomena so when I had a sabbatical coming up in Spring 2000 I suggested to my colleague David Edwards that we come up with a new mathematical model for "how movies make money." I call this area "cinematic box-office dynamics" and have written two papers on the topic (Edwards and Buckmire, 2001; Edwards, Buckmire and Ortega-Gingrich, 2013) and conducted a lot of research (which sadly has not resulted in publication, yet).

Most of my mathematical work has been investigating nonstandard finite difference schemes, particularly Mickens discretizations of the radial operators in the cylindrical and spherical Laplacian operators.

A famous old problem, called the Bratu problem, is a nonlinear version of Helmholtz equation where the right hand side of
Laplacian u = - f is exponential. In cylindrical and planar coordinates there are known exact solutions to the Bratu problem, making it ideal for numerical benchmarking of my nonstandard finite difference scheme(s).

More recently I have expanded my work to venture into the area of data science. Specifically, applying machine learning teachniques to produce mathematical models of the  commitment decisions of students who are admitted to colleges.

AUTOMATHOGRAPHY

• I often assign an "mathematical autobiography" assignment that I call an automathography in my classes. During my 2022-2023 sabbatical, I wrote a paper with Colton Sawyer that summarizes the utility of this idea. More details can be found here: https://sites.oxy.edu/ron/research/automathography/

DATA SCIENCE  & SOCIAL JUSTICE

• In Summer 2022 I attended the ICERM Program Data Science and Social Justice

DATA SCIENCE  / MACHINE LEARNING

• April  2019 "Application of Machine Learning to Student College Decision Making" (contributed talk at Southern California Applied Mathematics Symposium at Caltech, April 27, 2019)
• March  2019 "Predictive Models of Student College Commitment Decisions Using Machine Learning" (with K. Basu, T. Basu and N. Lal), Data, Volume 4, Number 2, Article 65, 2019.

NONSTANDARD FINITE DIFFERENCES

• March 2016 "Different Differences" (seminar talk at Pomona College)
• January 2013 "AMS Special Session on Nonstandard Finite Difference Discretizations and Nonlinear Oscillations (in Honor of Ronald Mickens 70th Birthday)" (co-organized mini-symposium at JMM in San Diego)
• October 2012, "Different Differences" (invited presentation at Michigan Technological University and Bucknell University)
• February 2012, "Different Differences" (invited seminar talk at Howard University)
• May 2009, "Numerical Studies of a Nonlinear Heat Equation with Square Root Reaction Term" (co-authored with student Karl McMurtry and Professsor Ronald S. Mickens of Clark Atlanta University)
• November 2008, "Numerical Studies of a Nonlinear Heat Equation with Square Root Reaction Term" (poster presentation at the 2008 Blackwell-Tapia Conference in Raleigh, NC)
• April 2005, "Applications of Mickens Finite Differences to Several Related Boundary Value Problems" (Published January 2006)
• October 2004 and November 2004 "Different Differences," colloquium, California State University at Long Beach and Occidental College
• June 2003 Contributed Presentation, "Application of Mickens finite differences to single-variable Bratu problems," SIAM Annual Meeting, Montreal, Quebec, Canada
• June 2003 "On exact and numerical solutions to the one-dimensional planar Bratu problem"
• May  2003 "Application of a Mickens finite difference to the cylindrical Bratu-Gelfand problem"
• September 2002 "Investigations of Nonstandard, Mickens-type, Finite-Difference Schemes for Singular Boundary Value Problems in Cylindrical or Spherical Coordinates"
• July 1995 "A New Finite-Difference Scheme for Singular Differential Equations in Cylindrical or Spherical Coordinates"

MATHEMATICAL MODELLING

• January 2016 "My PIC Math Experience: Teaching An Industrial Mathematics Course At A Small Liberal Arts College" (contributed presentation at JMM in Seattle)
• October 2015 "Cinematic Box-Office Dynamics: An Overview of a Particular Application of Ordinary Differential Equations to the Time Evolution of Theatrical Film Grosses" (Presentaion to the AMS Southern California Sectional Meeting)
  
• June 2013 "A mathematical model of cinematic box-office dynamics with geographic effects" (with Professor David Edwards and student Jacob Ortega-Gingrich)
• May 2011 "Mathematical Modfels and Movies: A Sneak Preview" (seminar talk at the University of Delaware)
• January 2012 "A Mathematical Investigation of the Financial Performance of Movie Sequels" (contributed presentatio at JMM in Boston)
• August 2010 "Film Franchises and the Edwards-Buckmire Model: An Examination of the Financial Performance of Movie Sequels" (summer undergraduate research talk by Jacob Ortega-Gingrich)
• Summer 2003 "A Revised Mathematical Model for Cinematic Box-Office Prediction"
  October 2000 "A Differential Equation Model of North American Cinematic Box-Office Dynamics" (with Professor David Edwards)

CLASSROOM VOTING / PEER INSTRUCTION

• January 2016 "An investigation of the effects on teaching and learning of different pedagogical practices in an introductory differential equations course"
  
• August 2013 "Investigating the effects of classroom voting and peer instruction on teaching and learning in an introductory differential equations course"
• November 2011  "Student Surveys: What Do They Think?" in Teaching Mathematics with classroom Voting (With and Without Clickers), MAA Notes 79.
• January 2010 "Use of Classroom Voting in Liberal Arts College Classes (Small and Large)" (contributed presentation at JMM 2010)
• May 2009 "Adventures in Assessment: Using clickers as tools for formative and summative assessment" (at the MSRI Critical Issues in Education conference)

DIVERSITY, EQUITY & INCLUSION IN MATHEMATICS

• February  2020 "Who Does The Math? On the Diversity and Demographics of the Mathematics Community." Eleectronic Seminar in Matthematics Education
• February 2020  "The Intersection of Mathematics and LGBT Identity is Not Empty." University of California Irvine seminar
• July 2019 "Who Does The Math? On the Diversity and Demographics of the Mathematics Community." International Congress of Industrial and Applied Mathematics, Valencia, Spain

COMPUTATIONAL AERODYNAMICS

• July 1998  "On The Design of Shock-Free, Transonic Slender Bodies of Revolution" (AIAA Paper 98-2686)
• July 1994 The Design of Shock-Free Transonic Slender Bodies  (Ph.D. Thesis)

GENDER, SEXUALITY and THE LAW

• May 1997 "You Can't Get There From Here: The Impact of California's Proposition 209 on Same-Sex Marriage"

VISUALIZATION OF CONCEPT MAPS

• October 2001: I have become interested in doing mathematical education research on how students visualize Concept Maps in Linear Algebra. Here are some of the Concept Maps that I have produced so far.

  1. Determinants

  2. Eigenvalues and Eigenvectors

  3. Lines and Planes

  4. Matrices

  5. Matrix Vector Spaces

  6. Systems of Linear Equations

  7. Vectors

UNDERREPRESENTATION IN MATHEMATICS

• 2005-present: I have been involved in a long-term project to improve mathematics (primarily algebra) instruction in northeast Los Angeles schools. I co-created and co-taught a  class Mathematics, Education and Access To Power multiple times to give students a community-based learning experience in mathematics.
• April 2010: "A Black, Gay mathematics department chair? Or, How I Got Here From There" (Friday April 2, 2010, Seminar on Underrepresentation in the Mathematical Sciences, Harvey Mudd College)

Ron Buckmire

Last modified: January 25, 2023