Abstract-This paper proposes a new framework for linear active circuits that can encompass both circuit analysis and synthesis. The framework is based on a definition of port equivalence for admittance matrices. This is extended to cover circuits with ideal active elements through the introduction of a special type of limit-variable called the infinity-variable ( -variable). A theorem is developed for matrices containing -variables that may be utilized in both circuit analysis and synthesis. The notation developed in this framework can describe nonideal elements as well as ideal elements and therefore the framework encompasses systematic circuit modeling.

This book provides a solid foundation to a number of important topics in mathematics of interest to science and engineering students. The authors' approach is simple and direct, the emphasis being on the analytical structure and applications of the material. The text is virtually self-contained, assuming only that the student has received a good basic course in ancillary mathematics. Each chapter contains a large number of worked examples, and concludes with problems for solution, with answers given in the back of the book. There is no comparable text that covers this material in such a concise form. This book will be of great value to undergraduates in physics, chemistry, theoretical biology, and in all engineering disciplines, as a source book of advanced mathematical methods, and also to postgraduate students as a revision text.

The connection between the Painleve property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrodinger and mKdV equations. Those equations which do not possess the Painleve property are easily seen not to have self-truncating Hirota expansions. The Backlund transformations derived from the Painleve analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painleve analysis and the eigenfunctions of the AKNS inverse scattering transform.

A framework for symbolic analysis and synthesis of linear active circuits has previously been proposed which is based on the use of admittance matrices and infinity-variables. The notation has the important advantage that it can describe both ideal circuit elements, for which an infinite limit is implied, and nonideal circuit elements for which matrix elements are considered finite. The nullor is a very important circuit element because it can represent the ideal operational amplifier and the ideal transistor. For the nonideal case, the use of finite matrix elements implies that the operational amplifier and transistor are both modelled as a voltagecontrolled current source, which is fine if the transistor is a field effect transistor or if the operational amplifier is of the transconductance type, but not otherwise. The purpose of this paper is to apply the -variable framework in order to derive alternative models for the nullor that can be used to model voltage, current and transresistance operational amplifiers and bipolar junction transistors. We also show that the -variable description of an ideal transistor can include a factor to represent transistor geometry.