Advances in Applied Clifford Algebras最新文献

In this work, using the quaternion linear canonical transform, we establish an analogue of the classical Titchmarsh theorem and Younis’ theorem for higher-order differences of quaternion-valued functions satisfying certain Lipschitz conditions in the space ( L^{2}( {mathbb {R}}^{2},{mathbb {H}}),) where ({mathbb {H}}) is a quaternion algebra.

This paper presents a significant extension of the classical Bedrosian identity to the quaternionic domain for functions of two variables. By leveraging the Quaternion Fourier transform, we develop a rigorous theoretical framework for the Quaternion Partial and Total Hilbert transforms. The core advantage of this Hilbert-based approach, as opposed to one using the rotational-invariant Riesz transform, is the simplicity of its Fourier multiplier. This property is fundamental and uniquely enables the derivation of Bedrosian-type identities, which are proven to be unattainable for the Riesz transform. We establish sufficient conditions for these identities to hold, providing a powerful multiplicative law for quaternionic signals under specific spectral conditions. Building upon this foundation, we delineate the necessary and sufficient conditions for the Quaternion Analytic Signal (QAS). Furthermore, as a key application of the Bedrosian theorems, we derive the precise criteria that ensure that the product of two holomorphic QASs remains a quaternion holomorphic function. The practical superiority of this framework is demonstrated through calculated examples and applications in two-dimensional image processing, where it offers a computationally effective and theoretically sound alternative to the monogenic signal, particularly for images with strong directional or lattice structures. This work provides essential theoretical tools for advancing hypercomplex signal processing and opens new avenues for sophisticated image analysis.

In this paper, we intend to bring together the hyperbolic spinors, which are useful frameworks from mathematics to physics, and non-null framed curves in Minkowski 3-space (mathbb {R}_1^3), which are new type attractive frames and a very crucial issue for singularity theory especially. First, we obtain new adapted frames for framed curves in (mathbb {R}_1^3). Then, we investigate the hyperbolic spinor representations of non-null framed curves of the general and adapted frames. Also, we find some geometric results and interpretations with respect to them, and we obtain illustrative and numerical examples with figures in order to support the given theorems and results.

This work provides an overview of the algebraic properties of primitive idempotents, which are fundamental in defining spinor spaces within the Clifford algebra framework. In addition to the key concepts, we also present novel results. In particular, we show that the primitive idempotent can be expressed as a polynomial in a specific special bivector. More generally, we demonstrate that every endomorphism on the spinor space can be represented as a polynomial in this special bivector. We also establish that the primitive idempotent, interpreted as a zero projection, represents a special case of this broader polynomial framework. By combining established insights with new contributions, this article offers a fresh perspective on these fundamental structures.

We investigate commutative analogues of Clifford algebras—algebras whose generators square to (pm {1}) but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces—we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to ‘multi split-complex space’ (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques.

To find an optimal current in a three-phase four-wire power system we have to solve a quadratic programming problem with a positive definite quadratic form with an equality constraint. We offer an approach which solves this and similar problems using an apparatus of geometric algebras, namely Projective geometric algebra. We add dimensions to encode parts of a quadratic function and reformulate the problem to seeking an orthogonal projection of the origin to an intersection of hyperplanes.

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers (left{ mathbb {D}_{t}right} _{tin mathbb {R}}) are constructed as sub-structures of scaled hypercomplex numbers (left{ mathbb {H}_{t}right} _{tin mathbb {R}}) under the scales (or, the moments) of the set (mathbb {R}) of real numbers. We show that if (t<0), then the classical free probability theory covers our free probability on (left{ mathbb {D}_{t}right} _{t<0}); if (t>0), then our free probability on (left{ mathbb {D}_{t}right} _{t>0}) is represented by the free probability over the classical hyperbolic numbers (mathcal {D}=mathbb {D}_{1}); and if (t=0), then the free probability on (mathbb {D}_{0}) is actually over the dual numbers (textbf{D}=mathbb {D}_{0}). Since the usual free probability theory is over (mathbb {C}), we here concentrate on establishing our free probability theory on (mathcal {D}), or that on (textbf{D}). Our approaches are motivated by the Speicher’s combinatorial free probability. As applications, the (mathbb {D}_{t})-free-probabilistic versions of semicircular elements and circular elements are considered.

We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Cayley Heisenberg group. As a consequence we deduce some qualitative uncertainty principles associated with this transform.

This paper investigates the Lorentz invariance of the multidimensional Dirac–Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac–Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1, n) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action.

In this paper, by utilizing the complex adjoint matrices, we transform the generalized right eigenvalue problem of the split quaternion matrix pencil into an equivalent generalized complex eigenvalue problem. This transformation enables us to propose an effective algebraic method for solving generalized eigenvalues and their corresponding eigenvectors. Additionally, we investigate the corresponding generalized right least squares eigenvalue problem for the split quaternion matrix pencil, providing a comprehensive framework for these types of problems. Secondly, we define the standard generalized right eigenvalues for the split quaternion matrix pencil. We rigorously prove that a split quaternion matrix pencil of order n has exactly n standard generalized right eigenvalues, all of which are complex numbers. Thirdly, we introduce the Rayleigh quotient for the split quaternion matrix pencil and study its fundamental properties. The definition and analysis of the Rayleigh quotient contribute to the theoretical understanding and potential applications of generalized split quaternion eigenvalue problems.