Physical Sciences and Mathematics Commons^™

Controllability And Local Accessibility—A Normal Form Approach, Wei Kang, Mingqing Xiao, Issa Amadou Tall

Articles and Preprints

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a …

Quantum Invariants Of Templates, Louis H. Kauffman, Masahico Saito, Michael C. Sullivan

Articles and Preprints

We define invariants for templates that appear in certain dynamical systems. Invariants are derived from certain bialgebras. Diagrammatic relations between projections of templates and the algebraic structures are used to define invariants. We also construct 3-manifolds via framed links associated to tamplate diagrams, so that any 3-manifold invariant can be used as a template invariant.

Irreducible Polynomials Over Gf(2) With Three Prescribed Coefficients, Robert W. Fitzgerald, Joseph L. Yucas

Articles and Preprints

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of x^n-1, x^n-2 and x^n-3 are specified in advance. Formulas for the number of elements in GF(2ⁿ) with the first three traces specified are also given.

A Characterization Of Primitive Polynomials Over Finite Fields, Robert W. Fitzgerald

Articles and Preprints

No abstract provided.

Vertex-Magic Labeling Of Trees And Forests, I. D. Gray, J. Macdougall, John P. Mcsorley, Walter D. Wallis

Articles and Preprints

A vertex-magic total labeling of a graph G(V,E) is a one-to-one map λ from E ∪ V onto the integers {1, 2, . . . , |E| + |V|} such that

λ(x) + Σ λ(xy) where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings …

Feedback Classification Of Nonlinear Single-Input Control Systems With Controllable Linearization: Normal Forms, Canonical Forms, And Invariants, Issa Amadou Tall, Witold Respondek

Articles and Preprints

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analyzing, step by step, the action of homogeneous transformations on the homogeneous part of the same degree of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms (resp., their dual canonical forms) coincide. We give an explicit construction of transformations bringing …