Advances in Applied Clifford Algebras最新文献

This paper investigates the distribution function and nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for ( mathbb{B}mathbb{C})-valued functions, which characterizes valuable insights into the behavior and structure of (mathbb{B}mathbb{C})-valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties.

In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces (L^p) and (L^q.) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras.

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for (A_8), (D_8) and (E_8) for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface curvature estimates. The paper presents a comprehensive comparison of the average surface curvature of a tuned QMLP model and a tuned real-valued MLP model on both a regression task and a classification task. The results provide strong support for the improved optimization performance observed in QMLPs across various problem domains.

We introduce new types of fractional generalized elliptic operators on a compact Riemannian manifold with Clifford bundle. The theory is applicable in well-defined differential geometry. The Connes-Moscovici theorem gives rise to dimension spectrum in terms of residues of zeta functions, applicable in the presence of multiple poles. We have discussed the problem of scalar fields over the unit co-sphere on the cotangent bundle and we have evaluated the associated Dixmier traces as Wodzicki residues. It was observed the emergence of different types of elliptic operators, including inverse square, fractional and higher-order operators which are practical in various fields including cyclic cohomology and index problems in theoretical physics.

In this paper, we introduce a new multidimensional fractional S transform (S_{phi ,varvec{alpha },lambda }) using a generalized fractional convolution (star _{varvec{alpha },lambda }) and a general window function (phi ) satisfying some admissibility condition. The value of (S_{phi ,varvec{alpha },lambda }f) is also written in the form of inner product of the input function f with a suitable function (phi _{textbf{t},textbf{u}}^{varvec{alpha }_{lambda }}). The representation of (S_{phi ,varvec{alpha },lambda }f) in terms of the generalized fractional convolution helps us to obtain the Parseval’s formula for (S_{phi ,varvec{alpha },lambda }) using the generalized fractional convolution theorem. Then, the inversion theorem is proved as a consequence of the Parseval’s identity. Using a generalized window function in the kernel of (S_{phi ,varvec{alpha },lambda }) gives option to choose window function whose Fourier transform as a compactly supported smooth function or a rapidly decreasing function. We also discuss about the characterization of range of (S_{phi ,varvec{alpha },lambda }) on (L^2(mathbb {R}^N, mathbb {C})). Finally, we extend the transform to a class of quaternion valued functions consistently.

In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by Clifford cohomology. We show that Clifford cohomology controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold M which admits a (Spin^{c}) structure.

The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal’s formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek for several versions of the uncertainty inequalities associated with the proposed quaternion ambiguity function.

Let (Gamma ) be a d-summable surface in (mathbb {R}^m), i.e., the boundary of a Jordan domain in ( mathbb {R}^m), such that (int nolimits _{0}^{1}N_{Gamma }(tau )tau ^{d-1}textrm{d}tau <+infty ), where (N_{Gamma }(tau )) is the number of balls of radius (tau ) needed to cover (Gamma ) and (m-1<d<m). In this paper, we consider a singular integral operator (S_Gamma ^*) associated with the iterated equation ({mathcal {D}}_{underline{x}}^k f=0), where ({mathcal {D}}_{underline{x}}) stands for the Dirac operator constructed with the orthonormal basis of ( mathbb {R}^m). The fundamental result obtained establishes that if (alpha >frac{d}{m}), the operator (S_Gamma ^*) transforms functions of the higher order Lipschitz class (text{ Lip }(Gamma , k +alpha )) into functions of the class (text{ Lip }(Gamma , k +beta )), for (beta =frac{malpha -d}{m-d}). In addition, an estimate for its norm is obtained.

For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg’s inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.