Advances in Applied Clifford Algebras最新文献

In this research, the Clifford–Fourier transform introduced by E. Hitzer, satisfies some uncertainty principles similar to the Euclidean Fourier transform. An analog of the Beurling–Hörmander’s theorem for the Clifford–Fourier transform is obtained. As a straightforward consequence of Beurling’s theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand–Shilov and Cowling–Price theorems are also deduced.

We give a formula for (f(eta ),) where (f:{mathbb {C}} rightarrow {mathbb {C}}) is a continuously differentiable function satisfying (f(bar{z}) = overline{f(z)},) and (eta ) is a dual quaternion. Note this formula is straightforward or well known if (eta ) is merely a dual number or a quaternion. If one is willing to prove the result only when f is a polynomial, then the methods of this paper are elementary.

Within a Geometric Algebra (GA) framework, this article presents a general method for synthesis of sliding mode (SM) controllers in Single Input Single Output (SISO) switched nonlinear systems. The method, addressed as the invariance control method, rests on a reinterpretation of the necessary and sufficient conditions for the local existence of a sliding regime on a given smooth manifold. This consideration leads to a natural decomposition of the SM control scheme resulting in an invariance state feedback controller feeding a Delta–Sigma modulator that, ultimately, provides the required binary-valued switched input to the plant. As application examples, the obtained results are used to illustrate the design of an invariance controller for a switched power converter system. Using the invariance control design procedure, it is shown how well-known second order sliding regime algorithms can be obtained, via a limiting process, from traditional sliding regimes induced on linear sliding manifolds for certain nonlinear switched systems.

We define a new ({{mathbb {Z}}}_2)-graded quantum (2+1)-space and show that the extended ({{mathbb {Z}}}_2)-graded algebra of polynomials on this ({{mathbb {Z}}}_2)-graded quantum space, denoted by ({mathbb F}({{mathbb {C}}}_q^{2vert 1 })), is a ({{mathbb {Z}}}_2)-graded Hopf algebra. We construct a right-covariant differential calculus on ({{mathbb {F}}}({{mathbb {C}}}_q^{2vert 1 })) and define a ({mathbb Z}_2)-graded quantum Weyl algebra and mention a few algebraic properties of this algebra. Finally, we explicitly construct the dual ({{mathbb {Z}}}_2)-graded Hopf algebra of ({{mathbb {F}}}({mathbb C}_q^{2vert 1 })).

In this paper, we consider a family of the hypercomplex rings ({mathscr {H}}=left{ {mathbb {H}}_{t}right} _{tin {mathbb {R}}}) scaled by ({mathbb {R}}), and the dynamical system of ({mathbb {R}}) acting on ({mathscr {H}}) via a certain action (theta ) of ({mathbb {R}}). i.e., we study an analysis on dynamical system induced by ({mathscr {H}}). In particular, we are interested in free-probabilistic information on the dynamical system dictated by our hypercomplex analysis.

In this article we investigate the application of complex split biquaternions and bioctonions to the standard model. We show that the Clifford algebras Cl(3) and Cl(7) can be used for making left-right symmetric fermions. Hence we incorporate right-handed neutrinos in the division algebras-based approach to the standard model. The right-handed neutrinos, or sterile neutrinos, are a potential dark-matter candidate. Using the division algebra approach, we discuss the left-right symmetric fermions and their phenomenology. We describe the gauge groups associated with the left-right symmetric model and prospects for unification through division algebras. We briefly discuss the possibility of obtaining three generations of fermions and charge/mass ratios through the exceptional Jordan algebra (J_3(O)) and the exceptional groups (F_4) and (E_6).

The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a type of graded structure on Lie groups, not evident in their matrix representation. Embracing this graded structure, we prove the invariant decomposition theorem: any composition of k linearly independent reflections can be decomposed into (lceil {k/2}{rceil }) commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi–Chasles’ theorem as its 3D Euclidean special case. To demonstrate its utility, we briefly discuss various examples such as Lorentz transformations, Wigner rotations, and screw transformations. The invariant decomposition also directly leads to closed form formulas for the exponential and logarithmic functions for all Spin groups, and identifies elements of geometry such as planes, lines, points, as the invariants of k-reflections. We conclude by presenting a novel algorithm for the construction of matrix/vector representations for geometric algebras ({mathbb {R}}^{{}}_{pqr}), and use this in (text {E}({3})) to illustrate the relationship with the classic covariant, contravariant and adjoint representations for the transformation of points, planes and lines.

We introduce several modifications of conic fitting in Geometric algebra for conics by incorporating additional conditions into the optimisation problem. Each of these extra conditions ensure additional geometric properties of a fitted conic, in particular, centre point position at the origin of coordinate system, axial alignment with coordinate axes, or, eventually, combination of both. All derived algorithms are accompanied by a discussion of the underlying algebra and computational optimisation issues. Finally, we present examples of use on a sample dataset and offer possible applications of the algorithms.

We study the inhomogeneous equation ({text {curl}}vec {w}+lambda vec {w}=vec {g},,lambda in {mathbb {C}},,lambda ne 0) over unbounded domains in ({mathbb {R}}^{3}), with (vec {g}) being an integrable function whose divergence is also integrable. Most of the results rely heavily on the “good enough” behavior near infinity of the (lambda ) Teodorescu transform, which is a classical integral operator of Clifford analysis. Some applications to inhomogeneous time-harmonic Maxwell equations are developed. Moreover, we provide necessary and sufficient conditions to guarantee that the electromagnetic fields constructed in this work satisfy the usual Silver–Müller radiation conditions. We conclude our work by showing that a particular case of our general solution of the inhomogeneous time-harmonic Maxwell equations coincide with the integral representation generated by the dyadic Green’s function.