Classical Algebraic Theory Research Articles

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter \(\eta \in \left({- \infty , + \infty}\right)\). These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson’s [1] listing of characteristic equations for integer spins out to I = 15, inclusive.

Lexicographical expressions of Boolean functions with application to multilevel synthesis

This paper proposes a new type of expression for Boolean functions called lexicographical expressions. The basic idea is to impose an input ordering for factoring logical expressions. Several algebraic properties are presented and relations with classical algebraic theory are established. The main result is that all elementary factorizations defined by (Cokernel, Kernel) pairs with an input order are all algebraically compatible, i.e., are all parts of a single factorization of the function. Thus for a given input order a unique factorization is defined. This leads to fast division procedures. Basic techniques for obtaining lexicographical factorizations are presented. First, a precedence matrix and an updating procedure are defined and used later to select an input order and a corresponding compatible factorization. Second, a factorization technique respecting a fixed order is detailed. This method is then applied to multilevel synthesis using standard cells which was the original motivation of this work. The goal is to reduce wiring complexity. A lexicographical factorization leads to a wiring area reduction due to the structuring of the logic into layers in which the inputs enter the layout in the order given by the factorization. Experimental results comparing this approach to classical ones are given. These results include routing ratio measurements, routing structure observation, global area measurement and critical path estimation. All these results are analyzed after place and route, using an industrial tool (COMPASS Design Automation tool). >

Strictly Observable Linear Systems

A theory of strictly observable linear systems is developed in a module theoretic framework which is consistent with the classical algebraic theory of linear time invariant realization. The theory incorporates in a unified framework the reduction of linear systems through precompensation, through state feedback, and through dynamic output feedback.

THE DECOMPOSITION OF TOLERANCE AUTOMATA

Tolerance automata are defined and a decomposition theory for such entities is sought. It is shown that two major procedures of the classical algebraic theory produce difficulties in the tolerance case. A weaker approach, employing the idea of inertial tolerance, is presented. Finally, an explicit example is given which illustrates both the difficulties encountered and the theorems proved in the text.