Superfluous Elements Research Articles

A general matrix factorization method for network synthesis

THIS PAPER considers a new matrix factorization method for the synthesis of RLC two terminal-pair-networks. Equivalent matrices suitable for the synthesis of RLC ladder and parallel ladder networks are developed by linear transformations of matrix multiplication. The application of the method is demonstrated through the synthesis of a general minimum-phase transmission function as ladder networks <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ‾ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> and a general nonminimum-phase transmission function as parallel ladder networks. The advantage of the present method is that superfluous elements in the realized ladder networks are reduced considerably. Furthermore, an unbalanced form can be obtained for realizing a nonminimum-phase transmission function which has generally been realized in the lattice form.1, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5&amp;</sup> #x0203E; <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup>

A General Matrix Factorization Method for Network Synthesis

THIS PAPER considers a new matrix factorization method for the synthesis of RLC two terminal-pair-networks. Equivalent matrices suitable for the synthesis of RLC ladder and parallel ladder networks are developed by linear transformations of matrix multiplication. The application of the method is demonstrated through the synthesis of a general minimum-phase transmission function as ladder networks <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ‾ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> and a general nonminimum-phase transmission function as parallel ladder networks. The advantage of the present method is that superfluous elements in the realized ladder networks are reduced considerably. Furthermore, an unbalanced form can be obtained for realizing a nonminimum-phase transmission function which has generally been realized in the lattice form.1, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5&amp;</sup> #x0203E; <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup>

Reductions of ideals in local rings

This paper contains some contributions to the analytic theory of ideals. The central concept is that of areductionwhich is defined as follows: ifandare ideals and⊆, thenis called a reduction ofifn=n+1for all large values ofn. The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.

Operators and Observables in Isotopic Spin Space

The usefulness of the conventional isotopic spin formalism is somewhat marred by the fact that it contains superfluous formal elements which have no physical counterpart. It is demonstrated that application of the "superselection principle" for total electric charge, recently suggested by Wick, Wightman, and Wigner, is sufficient to eliminate these superfluous elements and thus reduce the content of the theory to that of the more conventional formalism in which neutrons and protons are distinguished ab initio. This superselection principle applied to the conventional state-function representation of quantum mechanics requires that admissable state functions be eigenfunctions of the total charge and that operators representing observables commute with the total charge. The disjointness of the various subspaces in Hilbert space corresponding to different total charge eigenvalues, demanded by the superselection principle, leads to the result that many observables which are outwardly different in form are actually essentially equivalent. The construction of all inequivalent observables compounded from nucleonic isotopic spin operators only is carried out and it is shown that all such observables are simply functions of the square of the total isotopic spin and its $z$ component. The essentially uniqueness of the charge parity operator introduced by Kroll and Foldy is established.

Theorems Regarding the Driving-Point Impedance of Two-Mesh Circuits*

The necessary and sufficient conditions that a driving-point impedance be realizable by means of a two-mesh circuit consisting of resistances, capacities, and inductances are stated in terms of the four roots and four poles (including the poles at zero and infinity) of the impedance. The roots and the poles are the time coefficients for the free oscillations of the circuit with the driving branch closed and opened, respectively. For assigned values of the roots, the poles are restricted to a certain domain, which is illustrated by figures for several typical cases; the case of real poles which are not continuously transformable into complex poles is of special interest. All driving-point impedances satisfying the general conditions can be realized by any one of eleven networks, each consisting of two resistances, two capacities, and two self-inductances with mutual inductance between them; these are the only networks without superfluous elements by which the entire range of possible impedances can be realized; the three remaining networks of this type give special cases only. For each of these eleven networks, formulas are given for the calculation of the values of the elements from the assigned values of the roots and poles.