Advances in Applied Clifford Algebras最新文献

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.

In this paper, the octonion matrix equation (AXB=C) is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation (AXB=C) by combining these representations with (mathcal {H})-representation of the special matrices. In addition, we also put forward the equivalent condition of existence and general expression of the Hermitian solution to the octonion matrix equation (AXB=C.) Finally, the validity and stability of our method is verified by numerical experiments.

Let X be a two-sided quaternionic Banach space and let (A, B, C: X longrightarrow X) be bounded right linear quaternionic operators such that (ACA=ABA). Let q be a non-zero quaternion. In this paper, we investigate the common properties of ((AC)^{2}-2Re(q)AC+|q|^2I) and ((BA)^{2}-2Re(q)BA+|q|^2I) where I stands for the identity operator on X. In particular, we show that

where (sigma ^{S}_{{mathcal {F}}}(.)) is a distinguished part of the spherical spectrum.

In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of (*)-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions.