Search

Clifford analysis is a higher dimensional function theory and a refinement of harmonic analysis. We will extend Clifford analysis to the infinite‐dimensional setting. To do that, we have to introduce infinite‐dimensional Clifford algebras and the infinite‐dimensional Dirac operator. The paper's main result is the Wiener chaos decomposition in Clifford analysis. We apply the decomposition to white noise and Brownian motion. [ABSTRACT FROM AUTHOR]

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with, but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)‐module. The main aim of the paper is to explain that a good possibility is to take the so‐called generalized Cauchy–Riemann equations proposed a long time ago by Stein and Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)‐modules. This is an important step in developing the corresponding function theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider general multivector elements of Clifford algebras Cl(p,q), p+q<3, and study multivector factorization into products of exponentials and idempotents, where the exponents are blades of grades zero (scalar) to n (pseudoscalar). [ABSTRACT FROM AUTHOR]

This work provides a comprehensive characterization of coincidence site lattices that arise from the compositions of the reflections of cubic lattices. For this purpose, we apply Cartan's theorem and use reflections as basic isometries, which allows the use of Clifford algebra as a powerful tool to handle reflections and products of reflections, leading to explicit expressions for coincidence indices and the basis of coincidence lattices. The three possible cases were studied: simple reflections, the product of two reflections (rotations), and the product of three reflections, where a factorization that produces improper rotations was provided. In the case of rotations, it is formally proven that Ranganathan's well-known generating function of coincidence indices for cubic systems under rotations yields all possible coincidence lattices. Overall, this study provides valuable insights into the mathematical properties of coincidence site lattices and the use of Clifford algebras to describe orthogonal transformations. [ABSTRACT FROM AUTHOR]

This document is an editorial summarizing the Applied Geometric Algebras in Computer Science and Engineering (AGACSE) conference held in Brno, Czech Republic in September 2021. The conference aimed to promote the use of geometric algebra in fields such as image processing, robotics, and quantum computing. The conference proceedings were published in the journal Mathematical Methods in the Applied Sciences. The editorial provides a list of accepted papers, covering topics such as applied geometry, technological applications, algebra, and quantum phenomena. One specific paper explores the use of geometric algebra in teaching rotations through neural networks. The document is a compilation of research papers showcasing the applications of geometric algebra in various fields, including robotics, control systems, image processing, cryptography, and physics. Each paper presents a specific problem or application and proposes a unique approach or solution using geometric algebra. The authors compare their methods with existing techniques and provide mathematical analysis to support their claims. Overall, the papers demonstrate the versatility and effectiveness of geometric algebra in different domains. [Extracted from the article]

We study the embedding of octonions in the Clifford geometric algebra for space‐time STA Cl(1,3)$$ Cl\left(1,3\right) $$, as suggested by Anthony Lasenby at AGACSE 2021. As far as possible, we extend the approach to similar octonion embeddings for all three‐ and four‐dimensional Clifford geometric algebras Cl(p,q)$$ Cl\left(p,q\right) $$, n=p+q=3,4$$ n=p+q=3,4 $$. Noticeably, the lack of a quaternionic subalgebra in Cl(2,1)$$ Cl\left(2,1\right) $$ seems to prevent the construction of an octonion embedding in this case and necessitates a special approach in Cl(2,2)$$ Cl\left(2,2\right) $$. As examples, we present for Cl(3,0)$$ Cl\left(3,0\right) $$ the nonassociativity of the octonionic product in terms of multivector grade parts with cyclic symmetry and show how octonion products and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra Cl(3,0)$$ Cl\left(3,0\right) $$ and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra. [ABSTRACT FROM AUTHOR]

Different ways of representing the group SU(3)$$ SU(3) $$ within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for SU(3)$$ SU(3) $$ and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of SU(3)$$ SU(3) $$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing SU(3)$$ SU(3) $$, such as G2$$ {G}_2 $$ and SU(8)$$ SU(8) $$, are also considered. [ABSTRACT FROM AUTHOR]

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum Stot=3ℏ/2, (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the "spinor" or the "vector" form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space. [ABSTRACT FROM AUTHOR]

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras. [ABSTRACT FROM AUTHOR]

This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game. [ABSTRACT FROM AUTHOR]

Closed form expressions in real Clifford geometric algebras Cl0,3,Cl3,0,Cl1,2$$ C{l}_{0,3},C{l}_{3,0},C{l}_{1,2} $$, and Cl2,1$$ C{l}_{2,1} $$ are presented in a coordinate‐free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components ai$$ {a}_i $$ and ajk$$ {a}_{jk} $$ in a form (ai−ajk)2,i≠j≠k$$ {\left({a}_i-{a}_{jk}\right)}^2,i\ne j\ne k $$. After disentanglement, the obtained formulas simplify to the well‐known de Moivre‐type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics. [ABSTRACT FROM AUTHOR]

We extensively survey applications of Clifford Geometric algebra in recent years (mainly 2019–2022). This includes engineering; electric engineering; optical fibers; geographic information systems; geometry; molecular geometry; protein structure; neural networks; artificial intelligence; encryption; physics; signal, image, and video processing; and software. [ABSTRACT FROM AUTHOR]

For noncuspidal irreducible admissible representations of GSp(4, k) over a local non‐Archimedean field k, we determine the exceptional poles of the spinor L‐factor attached to anisotropic Bessel models by Piatetski–Shapiro. This completes the classification of spinor L‐factors for GSp(4, k). [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

Analytic signal is a useful mathematical tool. It separates qualitative and quantitative information of a signal in form of the local phase and local amplitude. Clifford Fourier transform (CFT) plays a vital role in the representation of multidimensional signals. By generalizing the CFT to Clifford linear canonical transform (CLCT), we present a new type of Clifford biquaternionic analytic signal. Due to the advantages of more freedom, the envelop detection problems of 3D images, with the help of this new analytic signal, can get a better visual appearance. Synthesis examples are presented to demonstrate these advantages. [ABSTRACT FROM AUTHOR]

The paper by Gopalan [(2020). Acta Cryst. A76, 318–327] presented an enumeration of the 41 physical quantity types in non‐relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1′, and wedge reversion, 1†. A consideration of the set of all seven antisymmetries (1, 1′, 1†, 1′†, 1†, 1′, 1′†) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a wide class of space‐fractional and time‐fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space‐time lattice hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty)$ (h>0$h>0$), resembling to fractional semidiscrete counterparts of the so‐called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup exp(−teiθ(−Δh)α)t≥0$\left\lbrace \exp (-te^{i\theta }(-\Delta _h)^{\alpha })\right\rbrace _{t\ge 0}$, carrying the parameter constraints 0<α≤1$0<\alpha \le 1$ and |θ|≤απ2$|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty)$. [ABSTRACT FROM AUTHOR]

We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate‐free Clifford quotient between a multidifference pseudo‐vector and a pseudo‐scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non‐degenerate simplex contracting to that point. Such result allows to fix an issue with a defective definition of pseudo‐scalar field in Sobczyck's simplicial calculus. Then, we provide some consequences and conjectures implied by the foregoing results. [ABSTRACT FROM AUTHOR]

In this paper, we consider general multivector elements of Clifford algebras Cl(2,1)$$ Cl\left(2,1\right) $$ and look for possibilities to factorize multivectors into products of blades, idempotents, and exponentials, where the exponents are frequently blades of grades zero (scalar) to n$$ n $$ (pseudoscalar). We will succeed mostly, with a minor open case remaining. [ABSTRACT FROM AUTHOR]

A common type problem of optimization is to find simultaneously n$$ n $$ parameters that globally minimize an objective function of n$$ n $$ variables. Such problems are seen in signal and image processing and in various applications of mathematical analysis of several complex variables and Clifford algebras. Objective functions are usually assumed to be Lipschitzian with maybe unknown Lipschitz constants. A number of methods have been established to discard the sets called "bad sets" in a partition that is impossible to contain any optimal point, as well as to treat the unknown Lipschitz bound problem along with the algorithm. In the present paper, a simple criterion of eliminating bad sets is proposed for the first time. The elimination method leads to a concise and rigorous proof of convergence. The algorithm, on the range space side, converges to the global minimum with an exponential rate, while on the domain space side, converges with equal accuracy to the set of all the global minimizers. To treat the unknown Lipschitz constant dilemma, we propose a practical pseudo‐Lipshitz bound process. The methodology is of fundamental nature with straightforward mathematical formulation applicable to multivariate objective functions defined on any compactly connected manifolds in higher dimensions. The method is tested against an extensive number of benchmark functions in the literature. The experimental results exhibit considerable effectiveness and applicability of the algorithm. [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

The exterior algebra E$E$ on a finite‐rank free module V$V$ carries a Z/2$\mathbb {Z}/2$‐grading and an increasing filtration, and the Z/2$\mathbb {Z}/2$‐graded filtered deformations of E$E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on V$V$. We extend this result to A∞$A_\infty$‐algebra deformations A$\mathcal {A}$, showing that they are classified by formal functions on V$V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho–Hong–Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such A$\mathcal {A}$. By applying these ideas to a related construction of Cho–Hong–Lau, we prove a local form of homological mirror symmetry: the Floer A∞$A_\infty$‐algebra of a monotone Lagrangian torus is quasi‐isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential. [ABSTRACT FROM AUTHOR]

Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and Depenbrock's FBD transformations can be derived by imposing orthogonality on the voltage and current vectors defined in a Euclidean space by using GA. This paper presents these transformations as spatial‐like rotations and projections by means of the use of special algebraic objects named GA rotors. We prove that there is no need to use complex numbers nor matrices to perform the mentioned transformations and to provide geometrical intuition for electrical quantities and their transformations. Furthermore, power properties can be described using GA terminology by means of the proposed geometric power. The manipulation of the geometric power allows a useful current decomposition for a variety of applications such active filtering, frequency estimation, or electrical machinery. We provide in this paper an alternative approach focused on power systems under the paradigm of GA. [ABSTRACT FROM AUTHOR]

Melanoma is a deadly skin disease. Availability of digital skin lesion datasets ease the exploration of ample classification studies. Both theoretical and heuristics improvements are achieved thanks to these new datasets. Being one of many high‐level feature‐driven classification methods, support vector machines (SVMs) are widely used in the literature as melanoma classifiers. Almost all of these studies are using a limited set of predefined kernels. In this study, we propose a newly developed Clifford kernel for the classification of dermoscopic skin lesions. We develop Clifford‐based linear, polynomial, and exponential kernels in the Clifford algebra (CA) Cℓ5, 0 0‐, 2‐, and 4‐vector subspaces. CAs are noncommutative but associative and distributive over addition. We showed that the newly developed Clifford kernels are embedded into SVM classifiers to successfully identify malignant skin lesions in a binary classification settings. Clifford kernel results are compared with mostly used gaussian and polynomial kernels with real‐valued SVM classifiers. Accuracy of all classifiers are assessed with cross‐validation using imbalanced and balanced datasets of 112, 162, and 192 lesions. SVM kernels in comparison are parameterized to scan wide range of possibilities. We show that Clifford‐based polynomial kernels outperforms in all, balanced and imbalanced, datasets having average accuracy of 83%. The consistence of high accuracies obtained with Clifford polynomial kernel shows that skin lesion features are logically designed and Clifford‐based SVM is able to model class separations in the feature space. [ABSTRACT FROM AUTHOR]

In this paper, the equation ex∧y=1, such that x,y∈ℝn, is proved using defined group homomorphism and Euler's formula, where x ∧ y is the outer product between x and y. Firstly, this equation is verified for n=2,3,4,5 using Clifford product. Then, it turns out that it is difficult to maintain the proof in this way since the outer product is anticommutative, the size increases, and the calculation becomes a lot of work. For this reason, a group of homomorphism from the group Cℓn, 0 to the group ℝn,0 is described and used Euler's formula. Eventually, it is proved for any n∈ℤ+n≥2 by generalizing this equation. [ABSTRACT FROM AUTHOR]

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5). [ABSTRACT FROM AUTHOR]

The magnetotelluric impedance tensor is analysed in the context of Clifford algebras. In this framework, the tensor is broken down into different parts, each one with a particular geometric algebra meaning, the simplicity of which allows us to deduce a number of known properties and opens up many other possibilities. As examples to show its capabilities, some of the algebraic relationships involving the impedance tensor, such as rotations, Mohr diagrams and phase tensor, are shown under this theory. Rotations are analysed in Clifford algebra Cl3 and Mohr diagrams, and phase tensor are expressed in Clifford algebra Cl2. The galvanic distortion matrix and its transformations are also seen in Clifford algebra Cl2, where a number of relationships allow us to recognize the galvanic distortion in a measured two‐dimensional/three‐dimensional impedance tensor. These relationships are useful as constraints to determine the galvanic distortion parameters. [ABSTRACT FROM AUTHOR]

A method for obtaining the galvanic distortion matrix is presented so that the regional impedance tensor (free of distortion) is recovered. The method is a constrained stochastic heuristic method, which consists in randomly exploring the space of the distortion parameters. Constraints are imposed on the shortest periods of the regional impedance tensor that, at these short periods, tends to be two dimensional (or one dimensional). Depending on the constraints used, two different methods to recover the regional impedance tensor in this 2D/3D case are presented. Method 1 needs to find the strike of the short periods and Method 2 applies to the measurement directions. Twist, shear and anisotropy parameters are obtained. Thus, the regional impedance tensor is recovered with the only exception being the vertical shift due to the gain, which is equal for all the components of the tensor. Examples with synthetic impedance tensors from 2D/3D models perturbed with galvanic distortion are presented to illustrate how the algorithm works. The presence of noise in data is considered and rules for proceeding are provided. The same examples perturbed by Gaussian noise together with experimental data illustrate the capabilities of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Fueter's theorem to the higher spin setting. To do so, we consider an alternative proof for the celebrated theorem that uses the Fischer decomposition. This decomposition is then extended to spaces of polynomials that depend on wedge variables, after which we can finish the proof of our higher spin Fueter theorem. [ABSTRACT FROM AUTHOR]

Due to the huge volume and complex structure, simplification of point clouds is an important technique in practical applications. However, the traditional algorithms often lose geometric information and have no dynamic expanding structure. In this paper, a new simplification algorithm is proposed based on conformal geometric algebra. First of all, a multiresolution subdivision is constructed by the sphere tree, which computes the minimal bounding spheres with the help of k‐means clustering, and then 2 kinds of simplification methods with full advantages of distance computing convenience are applied to carry out self‐adapting simplification. Finally, several comparisons with original data or other algorithms are implemented from visualization to parameter contrast. The results show that the proposed algorithm has good effect not only on the local details but also on the overall error rate. [ABSTRACT FROM AUTHOR]

The triple conformal geometric algebra (TCGA) for the Euclidean R 2‐plane extends CGA as the product of 3 orthogonal CGAs and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3‐vectors that linearize the representation of nonlinear curves, and the entities are inner product null spaces with respect to all points on the represented curves. Each inner product null space entity also has a dual geometric outer product null space form. Orthogonal or conformal (angle preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line, or circle are possible. The TCGA defines commutator‐based differential operators in the coordinate directions that can be combined to yield a general n‐directional derivative. [ABSTRACT FROM AUTHOR]

Protein structure calculations using nuclear magnetic resonance (NMR) experiments are one of the most important applications of distance geometry. The chemistry of proteins and the NMR data allow us to define an atomic order, such that the distances related to the pairs of atoms {i−3,i},{i−2,i},{i−1,i} are available, and solve the problem iteratively using a combinatorial method, called branch‐and‐prune. The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i−3,i−2,i, with radius given by the atomic distances di−3,i,di−2,i,di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i may not be precise or even not be available. Using conformal Clifford algebra, in addition to take care of NMR uncertainties, which implies that we have to calculate sphere intersections considering that their centers and radius may not be fixed anymore, we consider a more flexible atomic order, where distances di−3,i are replaced by dj,i, where j⩽i−3. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We show the advantages and disadvantages of specific geometric algebras and propose practical implementations in colorimetry. The colour space CIEL∗a∗b∗ is endowed by an Euclidean metric; the neighbourhood of a point is therefore a sphere, and the choice of a conformal geometric algebra is thus obvious. For the colour space CMC(l:c), the neighbourhood is an ellipsoid and thus we choose the quadric geometric algebra to linearize the metric by means of the scalar product. We discuss the distance problems in colour spaces with these particular geometric algebras applied. [ABSTRACT FROM AUTHOR]

In this paper, we introduced geometric algebra to develop a new multisource multisink optimal evacuation route planning method. A dynamically updatable data structure and a matrix‐based greedy searching algorithm were developed to support the dynamic evacuation route searching for multiple evacuees. Unlike most existing methods, which iteratively search the optimal path for each evacuee, our method can search all the possible evacuation routes synchronously. The dynamic updating of the network topography, weights, and constraints during the route searching is direct and flexible, thus can support the evacuation in dynamic cases. The method is demonstrated and tested by an evacuation case in the city of Changzhou, China. The simulation experiments suggest that the method can well support the dynamic route searching in large scales with dynamic weight changes. [ABSTRACT FROM AUTHOR]

We provide a non‐linear image transformation by means of conformal geometric algebra (CGA) elements and operations. We show the correspondence to the fisheye correction algorithms; more precisely, we prove the proportionality of classical models with CGA algorithm and provide exact formula in terms of CGA. Consequently, we show that the geometric construction allows to determine the inverse model quite universally. [ABSTRACT FROM AUTHOR]

We construct bases of polynomials for the spaces of square‐integrable harmonic functions that are orthogonal to the monogenic and antimonogenic R 3‐valued functions defined in a prolate or oblate spheroid. [ABSTRACT FROM AUTHOR]

In recent years, Hopfield neural networks using Clifford algebra have been studied. Clifford algebra is also referred to as geometric algebra, and is useful to deal with geometric objects. There are three kinds of Clifford algebra with degree 2; complex, hyperbolic, and dual-numbered. Complex-valued Hopfield neural networks have been studied by many researchers. Several models of hyperbolic Hopfield neural networks have also been proposed. It has been difficult to construct dual-numbered Hopfield neural networks. In this work, we propose dual-numbered Hopfield neural networks by modification of hyperbolic Hopfield neural networks with the split activation function. The stability condition and Hebbian learning rule are also provided. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

In this work, we obtain the fundamental solution (FS) of the multidimensional time-fractional telegraph Dirac operator where the 2 time-fractional derivatives of orders α∈]0,1] and β∈]1,2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters α and β. Finally, using the FS, we study some Poisson and Cauchy problems. [ABSTRACT FROM AUTHOR]

We define and study an action of the symmetric group on the Yokonuma-Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the Yokonuma-Hecke algebras, and in turn with an algebra defined by Aicardi and Juyumaya known as the algebra of braids and ties. The second one can be seen as new deformations of complex reflection groups of type [ABSTRACT FROM AUTHOR]

Starting from the representation of the ( n − 1) + n − dimensional Lorentz pseudo-sphere on the projective space [ABSTRACT FROM AUTHOR]

In recent years, applications of neural networks with Clifford algebra have become widespread. Clifford algebra is also referred to as geometric algebra and is useful in dealing with geometric objects. Hopfield neural networks with Clifford algebra, such as complex numbers and quaternions, have been proposed. However, it has been difficult to construct Hopfield neural networks by Clifford algebra with positive part of the signature, such as hyperbolic numbers. Hyperbolic numbers are useful algebra to deal with hyperbolic geometry. Kuroe proposed hyperbolic Hopfield neural networks and provided their continuous activation functions and stability conditions. However, the learning algorithm has not been provided. In this paper, we provide two quantized activation functions and the primitive learning algorithm satisfying the stability condition. We also perform computer simulations and compare the activation functions. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in [ABSTRACT FROM AUTHOR]

In this paper, we show that a higher order Borel-Pompeiu (Cauchy-Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of a Euclidean space, can be extended to the framework of generalized Clifford analysis. Furthermore, in lower dimensional cases, as well as for combinations of standard structural sets, explicit expressions of the kernel functions are derived. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [Z. Z. Tang and W. J. Yan, 'Isoparametric foliation and Yau conjecture on the first eigenvalue', J. Differential. Geom. 94 (2013) 521-540; Z. Z. Tang, Y. Q. Xie and W. J. Yan, 'Isoparametric foliation and Yau conjecture on the first eigenvalue, II', J. Funct. Anal. 266 (2014) 6174-6199]. Eells and Lemaire [Selected topics in harmonic maps, C.B.M.S. Regional Conference Series in Mathematics 50 (American Mathematical Society, Providence, RI, 1983)] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to Sn. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung ['Minimal submanifolds in a sphere II', Bull. London Math. Soc. 23 (1991) 387-390] (posed in 1991) on minimal submanifolds in unit spheres. Note that these conjectures of Leung have been proved in the case that the normal connection is flat [T. Hasanis and T. Vlachos, 'Ricci curvatures and minimal submanifolds', Pacific J. Math. 197 (2001) 13-24]. [ABSTRACT FROM AUTHOR]

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector e j is split into a forward and backward basis vector: [ABSTRACT FROM AUTHOR]

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford-Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A correction is provided for Proposition 3.5 in the above-mentioned article. The correction is: if $S$ is a skew polynomial ring on finitely many generators of degree 1 that are normal elements in $S$, and if $I$ is a homogeneous ideal of $S$ that is generated by a normalizing sequence, then $\dim _{\Bbbk } (S/I)$ is finite if and only if $S/I$ has no point modules and no fat point modules. A similar correction is provided for Corollary 3.6 of the same article. [ABSTRACT FROM PUBLISHER]

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space [ABSTRACT FROM AUTHOR]

Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space [ABSTRACT FROM AUTHOR]

Appell sequences in Clifford analysis are defined as polynomial families on which the Heisenberg algebra acts through a raising and a lowering operator satisfying the canonical Heisenberg relation. Recently, these sequences have gained new interest, as they are connected to the topic of special functions (such as harmonic or monogenic Gegenbauer polynomials) and branching rules for certain irreducible representations of the spin group. In this paper, we will explain how Jacobi polynomials appear quite naturally in the setting of Appell sequences related to certain branching problems. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]