Re vie w o f zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED Multidimensional Sysfems Theory, edited zyxwvutsrqponmlkjihgfedcb by N. zyxwvutsrqponmlkjihg K. Bose* Eduardo D. Sontag Department of Mathematics Rutgers University New Brunswick, New Jersey 08903 THE AREA Few parts of application-oriented mathematics have benefited from the interaction with modern algebraic and analytic geometry as much as the area usually referred to as multidimensional systems theory. This field consists of the study of various topics in the theory of functions of several complex variables, motivated mostly by problems in network design and synthesis and by signal-processing applications. Because of finite realizability constraints, the focus is often on rational functions; this accounts for the strong algebraic flavor of papers in the area, and in particular the use of techniques and results from commutative algebra. A linear-algebraic component is introduced by the need to consider matrices whose entries are analytic or rational functions. In “ classical” systems and control theory, one studies ordinary differential equations y(“)(t)+aly(“-‘j(t)+ .-a +a.y(t)=b,u(“-‘j(t)+ --. +b,u(t), (1) where a, ,..., a,,, b, ,..., b, are fixed real numbers, and u( .), y( *) are respec- tively the input and output signals. The (scalar) independent variable t is interpreted as time. Multidimensional systems appear when dealing instead with partial differential (or difference) equations. The independent variables may now represent different space coordinates (as in image-processing appli- cations), or perhaps mixed time and space variables (as in seismic data processing). Multidimensional models are also useful when studying certain types of functional differential equations in one independent variable, as delay-differential systems. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM *D. Reidel, 1985, xv+264 pp. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM LINEAR ALGEBRA AND ITS APPLICATIONS 87~273-278 (1987) 273 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 0024-3795/87/$3.50