Office: St. Mary's 327
email: vogta@georgetown.edu
Phone: 202-687-6254 (ring six times)
FAX: 202-687-6067
My research of late concerns the electromagnetic field generated by a single charged point particle,
the Lorentz force law, and the mutual interactions of two charged point particles in an arbitrary frame.
In sampling theory one question of interest is how to build up general probability distributions from simple ones
such as the uniform distribution or the standard normal. For example, how do you pick a point arbitrarily from the
interior of a regular pentagon? One way is to think of it as five equally likely triangles, rearrange each triangle
into a rectangle, and each of the latter can be sampled by a product of two independent uniform distributions.
Some older items of possible interest:
A Sobolev-type upper bound for rates of approximation by linear combinations of Heavisides, Journal of Approximation Theory Volume 147, Issue 1, July 2007, Pages 1-10, coauthored with Paul Kainen and Vera Kurkova
Global and hierarchical linear regression in two-stage sampling, Proceedings of the Joint Statistical Meetings JSM 2006, coauthored with Dhiren Ghosh (electronic)
Mathematical analysis of the Mandelstam_Tamm time-energy uncertainty principle, coauthored with John Gray, J. Math. Physics, 46 5, 2005.
Prigogine’s Theories. Proceedings of the Washington Academy of Sciences, Capital Science 2004 [electronic]
Story telling and the story of human intelligence, Proceedings of the Washington Academy of Sciences, Capital Science 2004 [electronic]
Mathematical Clarification of Dirac's Hole Theory and Feynman's Reformulation, ICMP 2000 (International Congress on Mathematical Physics), Imperial College, London, July 22, 2000
Heaviside Activation Functions in Neural Nets, joint work with Paul Kainen and Vera Kurkova, WCNA-2000 (The Third World Congress on Nonlinear Analysis), Catania, Sicily, July 22, 2000
Should Activation Functions be Affinely Recursive?, IJCNN'2000 (International Joint Conference on Neural Networks), Como, Italy, July 24, 2000
Disguised Linearities, WCNA-2000 (The Third World Congress on Nonlinear Analysis), Catania, Sicily, July 26, 2000
Quantum Computing: Common Themes, ECHO IV (Fourth International Conference on Emergence), Odense, Denmark, August 2, 2000
The State Space Approach to Evolution, CACYS'2000 (Computing Anticipatory Systems), Liege, Belgium, August 9, 2000
I was a co-author of several other presentations on Neural Networks and of the following statistical presentations:
Sampling on curves and surfaces: some examples, with Dhiren Ghosh, ASA meetings (American Statistical Association), San Francisco, August 2003
Sampling Methods related to Bernoulli and Poisson Sampling, with Dhiren Ghosh, ASA meetings, Atlanta, August 2002
Determining an optimal split for a lengthy questionnaire, with Dhiren Ghosh, ASA meetings, Indianapolis, August 2000
Frank Reifler and I wrote an article that shows that many wave functions cannot vanish on measurable rectangles of positive measure (or on open sets) in space-time. Examples include free particle Schrodinger and Dirac waves, Schrodinger waves whose Hamiltonians have analytic eigenfunctions (such as the harmonic oscillator), and positive energy solutions of the ordinary wave equation. Results of this type were intuitively known to Dirac, have also been discovered by Hegerfeldt, and are mentioned by Rauch (To catch a lion, put a cage someplace and wait: there is a finite probability that the lion will be in it.)
Frank Reifler and Andrew Vogt, Unique continuation of some dispersive waves, Communications in Partial Differential Equations, 19, July 1994, 1203-1215.
I wrote an article reviewing Freeman Dyson's speculations on the origins of life. Dyson argues that protein-life came first, followed by replicative DNA-life. He makes specific claims about the numbers of species of molecules that participated in this early life, the total numbers of molecules, and the time required for monomers to assemble into polymers. My article reviews his ideas and explains how Markov chains are used in his analysis.Of particular interest is how one estimates the time required to approach an equilibrium distribution, and the time required to transit from the neighborhood of one maximum ("Death," "Disorder," "Nonlife") to another maximum ("Order," "Life") when the equilibrium distribution is bimodal. Dyson's model is speculative (as he acknowledges) but entertaining.
Andrew Vogt, Dyson's model of prebiotic evolution, Actes du Symposium ECHO, Amiens, France, August 1996, 168-173.
Other work on statistics with Dhiren Ghosh, approximation theory and neural networks with Paul Kainen and Vera Kurkova, accident modeling with Joe Bared, and graph theory with Richard Squier and Bruce Torrence is also available.
If you are interested in reprints or preprints in any of the areas mentioned,
contact me by e-mail.