Benford Applied Mathematics

Philosophy, Gallery & Links

On Being an Applied Mathematician. All mathematicians agree that mathematics is simultaneously an extremely beautiful subject of study, and an immensely useful intellectual tool for the solution of “real world” problems. The difference between “pure” and “applied” mathematicians is the relative weight they give to these two reasons to study mathematics.

A pure mathematician gets to choose a particular sub area of mathematics in which to specialize. An applied mathematician does not have this luxury. Instead, the particular areas of mathematics the applied mathematician studies are determined by the particular real world problems that he or she chooses to address. For example, when I was doing the research for my Harvard dissertation on island biogeography I studied stochastic processes, linear algebra, and numerical analysis. When my research concerned the analysis and design of environmental regulations, I studied optimization theory and game theory. When I worked on the model of red pine, I studied the numerical and statistical aspects of non-linear least squares. More recently, my studies have been split between econometrics and time series analysis on one hand, and the mathematics of quantum mechanics on the other.

Consequently, the characteristic trait of an applied mathematician is versatility. The applied mathematician must feel able and willing to master whatever fields of mathematics are required by the situation. In short, being an applied mathematician requires a very broad base of mathematical knowledge, plus a level of confidence in one’s abilities that might pass for arrogance. Well, I accept the challenge! It justifies my natural inclination to be a perpetual graduate student.


This Lissajous curve has parameters a = 3, b = 4.
A projection of a hypercube (sometimes called a tesseract). Click here to see an animated three dimensional projection of a hypercube.
A plot of ζ(z) for z = ½ + iy as y ranges between 5 and 37. ζ(z) is the Riemann zeta function. This figure illustrates the Riemann Hypothesis, which states that all the non-trivial zeros of the zeta function lie on the “critical line” Re(z) = ½ in the complex plane. This conjecture is considered by many mathematicians to be the most important unresolved problem in pure mathematics.
Trajectories of the Lorenz equations are chaotic.  For a nice animation of these trajectories, click here. The subjects of chaos theory and fractals are tightly intertwined.  For a wonderful gallery of fractal images, click here.


I’m a compulsive writer of mathematical expositions. I like explaining mathematics. Mind you, I produce these write-ups mostly for my own continuing education. Consequently, I don’t think that many of my write-ups would be of much use in the classroom, although I could be wrong about that. In any case, here is a list of write-ups that are available over the internet upon request.

The Theory of Projection Operators, with Application in Economics. September, 2006. This monograph might be a useful supplement to the text for a course called “mathematics for economists” taught to economics graduate students.

Notes on Vector Calculus (following Apostol, Schey, and Feynman). March, 2007. I think this write-up could be used to supplement the text for a course in vector calculus. Also, as it includes a lot of physics (Maxwell’s equations, etc.), it could be useful to sufficiently advanced students of physics.

Recently I’ve been working on a massive summary of the mathematical foundations of econometrics. The only readers I can picture for this tome are econometricians who want a single reference source for the mathematical, probability theory, and statistical concepts and results that underlie econometrics.


This is a collection of links (some mathematical, some not) that I think are worthwhile.

MIT Open Courseware

Stanford Encyclopedia of Philosophy

Encyclopedia of Earth

Environmental Economics


The R project for Statistical Computing


Center for the Study of Complex Systems

Pacific Northwest Ecosystem Research Consortium


Resources for the Future

Cooperative Ecosystem Studies Units National Network

American Mathematical Society

The International Society for Ecological Economics

Berkeley Madonna

STELLA Modeling Software

Rocky Mountain Institute

The Santa Fe Institute

Complexity Digest

Wolfram Math World

Stephen J. Miller:

Mark Newbold:

Mark Nigrini:

Junpei Sekino’s Fractal Gallery:

Andrew Gelman:

Benford Applied Mathematics

About Frank Benford

I hold two Ph.D.s: one in Applied Mathematics and the second in Economics. I've worked as a computer programmer (at the Harvard Center for Population Studies), a professor of math, statistics, and econometrics (at several universities), and as an economist (at the Pacific Northwest Research Station of the Forest Service). I enjoy working with people with subject matter expertise to convert their intuitions into mathematical models.

More information about me is available on my LinkedIn public profile:Frank Benford