Advances in Applied Clifford Algebras最新文献

In this paper, we revisit Kravchenko’s method for analyzing the radial static Maxwell system in a three-dimensional inhomogeneous isotropic medium:

where the coefficient function (varepsilon ) is assumed to be a radial analytic function. By introducing a new class of modified normalized systems of functions with respect to the Dirac operator, we construct a transmutation operator that maps vector-valued monogenic functions into solutions of the system.

Machine learning is well-suited for forecasting typhoon intensity due to its capacity to model complex nonlinear relationships. However, current deep learning methodologies often treat vector field components independently, neglecting the geometric relationships between them. This oversight results in a loss of information and less accurate typhoon intensity forecasts. In contrast, geometric algebra considers multidimensional variables holistically, preserving internal correlations and relevant inductive biases pertinent to wind field data. To address this limitation, this study develops a geometric algebra-based method for typhoon intensity forecasting. Initially, wind field data, encompassing both longitudinal and latitudinal components across various isobaric levels, are represented as multivector inputs. Geometric algebra convolution is subsequently employed to capture the spatial features of typhoon wind speed data. Following this, a geometric algebra-based spatial attention mechanism is introduced to dynamically focus on regions with significant wind speed variations. This is followed by a geometric algebra convolutional fusion, which enhances the representation of typhoon features by integrating data across different stages. Finally, a Wide and Deep framework is utilized to incorporate both two-dimensional and three-dimensional typhoon characteristics, modeling the interrelationships between these variables and typhoon intensity, thereby establishing a forecasting model. A comparative analysis utilizing best track and reanalysis datasets from the North-Western Pacific region (2015–2018) demonstrates that our model not only enhances prediction accuracy but also reduces the number of required parameters. This study offers new insights and advances in the application of geometric algebra for feature extraction and prediction of multidimensional correlated geoscientific data.

Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation for these algebras and the notion of determinant in them. We give a criteria for checking if an element has a multiplicative inverse or not and, for the first time, explicit formulas for multiplicative inverses in the case of arbitrary dimension. The new theorems involve only operations of conjugation and do not involve matrix operations. We also consider notions of trace and other characteristic polynomial coefficients and give explicit formulas for them without using matrix representations.

On August 22, 2023 we lost our dear friend and esteemed colleague Prof. Yuri M. Grigor’ev has passed away. Throughout his scientific career, he made remarkable contributions to the field of hypercomplex analysis, particularly in advancing applications of quaternionic analysis to various problems of mathematical physics. In this paper, we aim to honour his memory by presenting an overview of his contributions to quaternionic analysis.

In this paper, in order to study the B–P formula of weighted inframonogenic functions, two important lemmas are first given, which solve the difficulty caused by the non commutativity of Clifford valued functions in multiplication operations. Then, using the conclusion of the above lemmas and the relationship between  (textrm{d}sigma )  under non-Euclidean distances and  (textrm{d}mu _{r})  under Euclidean distances, the B–P formula for weighted inframonogenic functions is obtained by digging out singularities to satisfy the conditions for using the Stokes formula, and introducing new operators to simplify the calculation steps. Furthermore, the Cauchy integral formula for weighted inframonogenic functions is obtained.

A model elliptic pseudo-differential equation is considered in a plane with cuts along coordinate axes. Using special wave factorization for an elliptic symbol one can describe the kernel of the pseudo-differential equation in Sobolev–Slobodetskii space. To annihilate the kernel they use some boundary conditions on cuts sides. A unique solvability for obtained boundary value problem is reduced to the unique solvability for a system of certain linear integral equations.

The mathematical tools of physics, based on group theory, are in permanent evolution. Major covariance groups are the orthogonal, unitary and symplectic groups. These groups are generally expressed in terms of real and complex matrices. Here we shall develop a new representation of the unitary symplectic groups USp(n) in terms of Clifford algebras constituted by tensor products of quaternion algebras called hyperquaternions. Concise expressions of the generators are obtained and a concrete example USp(4) is provided. Isomorphic quaternion matrix representations will also be used in the applications. The first application concerns classical mechanics. The Hamiltonian formalism, Poisson brackets and canonical transforms are related to the unitary symplectic groups. The 1D and 2D harmonic oscillators are examined within that framework. The second application concerns quantum mechanics. The Schrödinger and Heisenberg equations are derived in a new hyperquaternionic unitary symplectic way, the complex imaginary i being replaced by the quaternion k in phase space. The 1D and 2D quantum harmonic oscillators are treated within that formalism. Allowing a representation of both classical and quantum mechanics, it is hoped that the hyperquaternion algebras might deepen our mathematical comprehension of the foundational principles of physics.

A Jordanian deformation of the (N=2) supersymmetry algebra and its first-order noncommutative differential calculus is obtained from the q-deformed Hopf supersymmetry algebra via a singular limit of a linear transformation. We show that the Jordanian (N=2) supersymmetry algebra carries a Hopf superalgebra structure. It is enhanced to a twisted Hopf star superalgebra by four inequivalent families of (star )-structures. It is demonstrated that these star operations induce four types of (star )-involutions on the differential one-forms and partial derivatives. We introduce an appropriate Jordanian super-Hopf algebra that includes two even and two odd generators equipped with four different types of (star )-structures and its corresponding supergroup on the Hopf (N=2) supersymmetry algebra. It is shown that the noncommutative differential calculus over the Jordanian (N=2) supersymmetry algebra is left-covariant with respect to the Jordanian supergroup. The (star )-structures of the supergroup are (star )-preserving on the supersymmetry algebra only for zero values of their parameters.

Weighted Bergman projection operators on the Clifford monogenic Bergman spaces over the real ball of ({mathbb R}^n) have been already studied in a paper of Ren and Malonek. We extend and generalize their results by considering more general Bergman nonprojection operators and stating a necessary and sufficient condition for them to be bounded on weighted Lebesgue spaces. Sharp estimates for the weighted monogenic Bergman kernel are also given.

Let E(n) be the (2^{n+1})-dimensional Hopf algebra generated by anti-commuting elements (g,x_1, ldots , x_n), with g grouplike, each (x_i) skew-primitive, and (g^2=1), (x_i^2=0). In this article we prove that E(n)-coactions over a finite-dimensional algebra A are classified by tuples ((varphi , d_1, ldots , d_n)) consisting of an involution (varphi ) and a family ((d_i)_{i=1,ldots ,n}) of (varphi )-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements ((c, u_1, ldots , u_n)), whenever A is a semisimple Clifford algebra.