The present paper deals with the Richards' transformation which converts a multi-variable positive (real) function to another multi-variable positive (real) function. This transformation is based on the fact that the network function reduces to a constant for all zeros of an irreducible polynomial h (p₁, p₂, ・・・, p_n) (1deg_pi h). And its application to the cascade synthesis or the Bott-Duffin type synthesis of a class of multi-variable driving-point impedances, is discussed. Finally, we present the corresponding transformation for the multi-variable positive (real) matrices.

In Codd's relational data model, several dependencies have been introduced to specify the intensional properties of a relation. Fagin and independently, Zaniolo introduced the notion of a multivalued dependency (MVD). The definition of MVD's refers to an underlying set of attributes of a relation. The embedded multivalued dependency (EMVD), which is also introduced by Fagin, is an MVD that holds for a projection of an original relation on the subset of attributes of the relation. The properties of EMVD's are not well known although EMVD's play an important role in designing relation schemata by Fagin's decomposition approach. Our study in this paper is motivated from the following problems: (a) Since the validity of an MVD depends on an underlying set of attributes, it is not so easy to specify valid" MVD's for a relation with many attributes. (b) There has not been a useful tool to analyze whether or not a set of relation schemata obtained by Fagin's decomposition approach can represent the same data and the same dependencies of an initial relation schema. Our standpoint is to handle these problems by studying the properties of EMVD's. The main results of this paper are the following: (1) A basic theorem about the interaction between MVD's and EMVD's is provided. Several useful inference rules for MVD's and EMVD's are derived from this theorem. (2) A marked Hasse diagram called a dependency diagram is introduced to investigate the interections between MVD's and EMVD's. (3) We provide some conditions for an MVD or a set of MVD's to be invariant under the addition or deletion of attributes. (4) Some useful equivalence relationships between two dependency sets including MVD's and EMVD's are provided. We also provide some conditions to represent a given dependency set in a reduced form.

We give a simple formula which represents the relationship between incident matrices of two transformation semigroups X and Y and the incident matrix of their wreath product X Y. The structure generating function of the wreath product can be easily obtained by using this incident matrix.

This paper describes a novel theory analysing the baseband transfer function of mode-coupled multi-mode optical fiber. The treatment of this method is based on the scattering matrix. This theory is direct, correct and self-consistent. Most transmission characteristics of the multimode optical fiber can be analyzed.