Advances in Applied Clifford Algebras最新文献

The theory of generalized partial-slice monogenic functions is considered as a synthesis of the classical Clifford analysis and the theory of slice monogenic functions. In this paper, we introduce a Cauchy integral formula and a Plemelj formula for generalized partial-slice monogenic functions. Further, we study some properties of the Teodorescu transform in this context. A norm estimation for the Teodorescu transform is discussed as well.

In this paper, the two-sided quaternionic Dunkl transform satisfies some uncertainty principles of quaternion algebra. An analog of the Beurling theorem for the two-sided quaternionic Dunkl transform is obtained. As a direct consequence of Beurling’s theorem, other versions of the uncertainty principle, such as Hardy’s, Gelfand–Shilov’s, Cowling–Price’s and Morgan’s theorems are also deduced.

This paper systematically studies Hilbert boundary value problems for monogenic functions on the hyperplane for the solutions being of any integer orders at infinity, where the negative order cases are new even when restricted to the complex plane context. The explicit solution formulas are provided and the solvability conditions are specified. The results are proved using the Clifford symmetric extension method, which reduces Hilbert boundary value problems to Riemann boundary value problems, involving many innovative geometric techniques.

In this paper, Cauchy theorems for solutions to polynomial Dirac equations with (alpha )-weight in superspace are studied using two methods. First, by constructing a new fundamental solution, the first kind of Cauchy theorem is obtained. Then the connection between polynomial Dirac operators with (alpha )-weight and iterative Dirac operators with (alpha )-weight in superspace is obtained. Finally, using this connection, the second kind of Cauchy theorem is obtained.

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.