It is well-known that the Schwarz Theorem for one-variable function has been applied to the theory of one-variable positive real function. In this short note, the Theorem is generalized to the case of multi-variable function, and its application is shown.

Usually, for the definitions on n-variable positive real functions, the properties on 2n-dimentional domain are used as the conditions of the definition, as examplified below. On the domain D {(p1, p2, , pn)|Re p10, Re p20, , Re pn0)}, |Re w(p1, p2, , pn) must be positive." In the present paper, the approach is simplified by the following two items. (i) Fixing all variables except for one on the axis (real axis or imaginary axis), the problems on multi-variables are reduced to the one's on singlevariable. (ii) The problems are discussed only on one-dimentional domain, that is, on the axes. Simplifying the problems concerning with multi-variable positive real function by the two items stated above, the definitions of several classes of positive real function are presented. Clearly, these definitions may simplify extremely the verification of positive realness of any given function, and may give an effective means for the studies of multi-variable positive real function.

This paper presents a method for the study of multivariable positive real rational matrices. The method is based on the following two items. (i) Each port of the network can be represented by a specific element and this element identifies the port conversely. (ii) A Hurwitz polynomial which corresponds to a given multi-variable positive real rational matrix in a one-to-one manner can be found. In short, the followings are shown in this paper. 1) The well defined Hurwitz polynomial corresponds to a given n-variable reactance scattering (or paraunitary) mm matrix in a one-to-one manner. 2) The concept of degeneration (or degree-down)" and degree-up" of immittance functions are clarified. 3) If each element of network is of different kind, no degeneration can occur. 4) For the realization of more than three-variable reactance function, it is generally necessary to degree-up the given immittance function. 5) For the practical realization of two-variable reactance function, it is not always convenient to apply directly the method which is used for the proof of realizability. Here, a new practical realization method for two-variable reactance functions based on junction matrix approach, is presented. The new method needs neither factorization nor degree-up operation.

The three types of definitions for the multivariable reactance function are presented, and the equivalence among these definitions are also shown. Viewed from the other standpoint, the new two definitions can be considered as new theorems on the function which indicate the important properties of multi-variable positive real function.

The present paper deals with the realization of a class of homogeneous positive real function, that is, the class of function with denominator composed of linear factors only. The theories are stated by the following steps. (a) Firstly, two basic theorems on the homogeneous positive real functions are presented. (b) Secondly, the theory of partial fraction expansion of the given function with denominator composed of linear factors, is presented. Here, any partial fraction is a homogeneous positive real function of degree two. (c) Lastly, the realization procedures of the partial fractions are presented.