Algebraic number

About: Algebraic number is a research topic. Over the lifetime, 20611 publications have been published within this topic receiving 315606 citations.

An enumeration of knots and links, and some of their algebraic properties

Abstract: Publisher Summary This chapter describes knots and links, and some of their algebraic properties. An edge-connected 4-valent planar map is called a polyhedron, and a polyhedron is basic if no region has just 2 vertices. The term region includes the infinite region, which is regarded in the same light as the others. Knot diagrams can be obtained from polyhedra by substituting tangles for their vertices, for instance, tangles 1 or −1 could always be substituted. A knot diagram K can be obtained by substituting algebraic tangles for the vertices of some nonbasic polyhedron P. There is a polyhedron Q with fewer vertices than P obtained by shrinking some 2-vertex region of P, and K can simply be obtained by substituting algebraic tangles for the vertices of Q. Any knot diagram can be obtained by substituting algebraic tangles for the vertices of some basic polyhedron P in fact P, and the manner of substitution is essentially unique.

Complete solution classification for the perspective-three-point problem

Abstract: We use two approaches to solve the perspective-three-point (P3P) problem: the algebraic approach and the geometric approach. In the algebraic approach, we use Wu-Ritt's zero decomposition algorithm to give a complete triangular decomposition for the P3P equation system. This decomposition provides the first complete analytical solution to the P3P problem. We also give a complete solution classification for the P3P equation system, i.e., we give explicit criteria for the P3P problem to have one, two, three, and four solutions. Combining the analytical solutions with the criteria, we provide an algorithm, CASSC, which may be used to find complete and robust numerical solutions to the P3P problem. In the geometric approach, we give some pure geometric criteria for the number of real physical solutions.

How algebraic Bethe ansatz works for integrable model

Abstract: I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.

On explicit algebraic stress models for complex turbulent flows

Abstract: Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope who based his analysis on the Launder, Reece, and Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models -- as well as anistropic eddy visosity models -- is theoretically established. The need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.