Allen R Miller

Encyclopedia

**Allen R Miller**was an American mathematician.

Miller was a major contributor to the field of special functions

Special functions

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....

, especially confluent hypergeometric function

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

s.

## Biography

Miller began his professional career at the Ballistics Research Laboratory at the Aberdeen Proving GroundAberdeen Proving Ground

Aberdeen Proving Ground is a United States Army facility located near Aberdeen, Maryland, . Part of the facility is a census-designated place , which had a population of 3,116 at the 2000 census.- History :...

where he worked with physicist Jane Dewey. In 1967 he moved to the Naval Research Laboratory where he developed high-fidelity mathematical models of interactions between physical systems and electromagnetic fields. Unlike most pure mathematicians of the time, Miller implemented simulations of his models on early supercomputers. In the 1970s Miller was recognized not only as a leading mathematician but also as an expert on the CRAY

Cray

Cray Inc. is an American supercomputer manufacturer based in Seattle, Washington. The company's predecessor, Cray Research, Inc. , was founded in 1972 by computer designer Seymour Cray. Seymour Cray went on to form the spin-off Cray Computer Corporation , in 1989, which went bankrupt in 1995,...

line of supercomputer.

Miller maintained an academic affiliation with George Washington University

George Washington University

The George Washington University is a private, coeducational comprehensive university located in Washington, D.C. in the United States...

as an adjunct professor after retiring from the Naval Research Laboratory in 1991. He published actively until his death from a heart embolism on August 15, 2010.

## Special Function Theory

Miller made numerous contributions in the area of special functionsSpecial functions

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....

and addressed several significant open problems. Notable among these include his work with Emanuel Vegh in formulating an exact closed-form solution for the specular grazing angle for a sphere, a problem which had been unsolved for almost 50 years.

## Mathematical Models

The Miller-Brown-Vegh model (developed at NRL) is commonly used to describe scattering from the ocean surface. It has been adopted in the US Navy's models for propagation over the ocean and appears in the Radar Handbook.## Computer Security and Information Theory

Miller, with Ira S. Moskowitz, made several contributions to the mathematical analysis of covert channels in computer security. In particular he showed how special functions could simplify the closed form for the Shannon channel capacityChannel capacity

In electrical engineering, computer science and information theory, channel capacity is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel...

and how the difference of divergent series could also be used to express channel capacity in certain physical situations that arise in anonymity networks.