Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

Exact solutions of the 2D Dunkl–Klein–Gordon equation: The Coulomb potential and the Klein–Gordon oscillator

We introduce the Dunkl–Klein–Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein–Gordon (KG) equation. We show that the generalization with Dunkl derivative of the z-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein–Gordon oscillator analytically and from an su(1, 1) algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.

On wave equations for the Majorana particle in (3+1) and (1+1) dimensions

In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain.

Constructions of polynomially complete quasigroups of arbitrary order

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different methods to construct polynomially complete quasigroups of any order [Formula: see text]. First method is based on a starting quasigroup of same order, second method is based on a particular permutation of [Formula: see text] and third and fourth methods are based on products of lower order quasigroups. In the last case, all quasigroups and their isotopes are polynomially complete. We also develop and implement an algorithm to derive a permutation for a given permutation of [Formula: see text] so that they generate whole [Formula: see text].

Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.

On a Weak Conditional

It is well-known that adding to a lattice the usual relative meet complement is not conservative, in the sense that distributivity is implied. In this paper we consider a weak relative meet complement that does not have the mentioned effect. We mostly study the mentioned operation from an algebraic point of view. However, we also provide a Hilbert-style axiomatization for its corresponding assertional logic.

Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces

Point projection is an important geometric need when boundaries described by parametric curves and surfaces are immersed in domains. In problems where an immersed parametric boundary evolves with time as in solidification or fracture analysis, the projection from a point in the domain to the boundary is necessary to determine the interaction of the moving boundary with the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge, we present in this paper an algebraic procedure to project a point on to Non-uniform rational B-spline (NURBS) curves and surfaces. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bézier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain are then used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G 0 continuity—problems where the Newton–Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively.

Isomorphism Theorems for Groupoids and Some Applications

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.