Consider a real linear system of m equations in n variables given by a matrix equation Ax = b. The system is said to be sign-solvable if one can determine the signs of the entries of the solution x knowing only the signs of the entries of A and of b. For example, it is easy to see that both entries must be positive for the solution to a system such as
when a, b, c, d and e are all positive. The economist P.A. Samuelson in his classic work Foundations of Economic Analysis appears to have been the first to recognize that it might be beneficial to consider such systems. Sign-solvability is now part of a larger area of study in which one seeks to determine when general properties of a matrix can be predicted from the combinatorial arrangement of the positive, negative and zero entries of certain matrices. Sign-nonsingular matrices, L-matrices, S-matrices, sign-stable matrices and several other interesting qualitative classes of matrices have arisen from such investigations.
Professor Gibson has his Bachelor's, Master's and Ph.D. degrees from North Carolina State University. He has been a member of the UAH faculty since 1967, and served as department chairman from 1985 to 1993. Prior to joining UAH, he held appointments on the applied mathematics staff at the U.S. Naval Research Laboratory and in the mathematics department at North Carolina State University. He has also had visiting appointments in the mathematics department at the University of Wisconsin-Madison and in the Institute for Mathematics and its Applications at the University of Minnesota. Professor Gibson has directed the research of one Ph.D. student (D.K. Baxter) and several Master's students.