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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]

A brief introduction of the Extended Relativity Theory in Clifford Spaces (C -space) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C -spaces. The most general transformations furnish a full mixing of the grades of the multivector-valued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass ℳ in C -spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E 2 − p ⋅ p = m 2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators. [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]

The space of spinors, defined as the basic representation of a Clifford algebra, can be expressed as the spin representation of an orthogonal Lie algebra. At the same time, these spin representations can also be characterized as finite-dimensional projective representations of the special orthogonal group. From a geometrical perspective, the behavior of spinors under the action of Lie groups can be examined. Thus, one has the advantage of making a concrete and basic explanation about what spinors are in a geometrical sense. In this study, the spinor representations of an orthogonal frame moving on a analytic curve is investigated geometrically. The spinor equations corresponding to a modified orthogonal frame and a modified orthogonal frame with are derived. The relations between modified orthogonal frames and the Frenet frame are established regarding their spinor formulations. Our motivation in this paper is to give spinor representations of the modified orthogonal frame. Consequently, this study has been planned as an interdisciplinary study between Clifford algebras and geometry. [ABSTRACT FROM AUTHOR]

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras : alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure. [ABSTRACT FROM AUTHOR]

Is there more to Dirac's gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac's space-time algebra to Clifford algebra Cl (0 , 6). The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion–antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that all the regular classical Lagrangian terms are of quantum condensation origin, thus possibly addressing the cosmological constant problem provided that we exercise extreme caution in the renormalization procedure that entails multiplications of divergent integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle–antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global U (1) symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies. [ABSTRACT FROM AUTHOR]

Different implementations of Clifford algebra: spinors, quaternions, and geometric algebra, are used to describe physical and technical systems. The geometric algebra formalism is a relatively new approach, destined to be used primarily by engineers and applied researchers. In a number of works, the authors examined the implementation of the geometric algebra formalism for computer algebra systems. In this article, the authors extend elliptic geometric algebra to hyperbolic space-time algebra. The results are illustrated by different representations of Maxwell's equations. Using a computer algebra system, Maxwell's vacuum equations in the space-time algebra representation are converted to Maxwell's equations in vector formalism. In addition to practical application, the authors would like to draw attention to the didactic significance of these studies. [ABSTRACT FROM AUTHOR]

With the rapid development of synthetic aperture radar (SAR) techniques, satellite systems' capabilities to acquire information are continually improving. The PAZ satellite, with its high resolution and wide scanning swath, can provide high-quality data support for SAR applications. Oil tanks serve as energy storage devices, and their identification holds significant value in both military and civilian fields. Challenges in the detection and recognition of oil tanks using classical methods include poor detection, slow computation speed, and multiple windows of correct recognition. This paper centers on the analysis of oil tanks using PAZ data. We employ a sliding-window approach to acquire candidate target windows, process the windows through Weibull distribution modeling and hole filling, and extract target features using the monogenic signal based on regional L2 norm. The results demonstrate that the proposed method effectively improves the accuracy, and the model exhibits strong generalization ability and robustness. [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]

Let D be a quaternion algebra over a number field F , and G be an arbitrary genus of O F -orders of full rank in D. Let K be a quadratic field extension of F that embeds into D , and B be an O F -order in K that can be optimally embedded into some member of G. We provide a necessary and sufficient condition for B to be optimally spinor selective for the genus G , which generalizes previous existing optimal selectivity criterions for Eichler orders as given by Arenas, Arenas-Carmona and Contreras, and by Voight independently. This allows us to obtain a refinement of the classical trace formula for optimal embeddings, which will be called the spinor trace formula. When G is a genus of Eichler orders, we extend Maclachlan's relative conductor formula for optimal selectivity from Eichler orders of square-free levels to all Eichler orders. [ABSTRACT FROM AUTHOR]

Aims: This study aims to describe the prevalence of monogenic diabetes in an Australian referral cohort, in relation to Exeter maturity-onset diabetes of the young (MODY) probability calculator (EMPC) scores and next-generation sequencing with updated testing where relevant. Methods: State-wide 5-year retrospective cohort study of individuals referred for monogenic diabetes genetic testing. Results: After excluding individuals who had cascade testing for a familial variant (21) or declined research involvement (1), the final cohort comprised 40 probands. Incorporating updated testing, the final genetic result was positive (likely pathogenic/pathogenic variant) in 11/40 (27.5%), uncertain (variant of uncertain significance) in 8/40 (20%) and negative in 21/40 (52.5%) participants. Causative variants were found in GCK, HNF1A, MT-TL1 and HNF4A. Variants of uncertain significance included a novel multi-exonic GCK duplication. Amongst participants with EMPC scores ≥ 25%, a causative variant was identified in 37%. Cascade testing was positive in 9/10 tested relatives with diabetes and 0/6 tested relatives with no history of diabetes. Conclusions: Contemporary genetic testing produces a high yield of positive results in individuals with clinically suspected monogenic diabetes and their relatives with diabetes, highlighting the value of genetic testing for this condition. An EMPC score cutoff of ≥ 25% correctly yielded a positive predictive value of ≥ 25% in this multiethnic demographic. This is the first Australian study to describe EMPC scores in the Australian clinic setting, albeit a biased referral cohort. Larger studies may help characterise EMPC performance between ethnic subsets, noting differences in the expected probability of monogenic diabetes relative to type 2 diabetes. [ABSTRACT FROM AUTHOR]

Discrete versions of the Laplace and Dirac operators have been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper, we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equations. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras. [ABSTRACT FROM AUTHOR]

We consider a finite dimensional strongly G -graded algebra A with self-injective 1-component B , and in our main result we prove that the induction from B to A of a basic support τ -tilting pair of B -modules is a support τ -tilting pair (M , P) of A -modules if and only if M is G -invariant. A similar statement holds for the restriction from A to B , so our results may be viewed as Clifford and Maschke type theorems for 2-term silting complexes. We also give applications to semibricks and the associated wide subcategories. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Riemann–Lebesgue theorem and some real Paley–Wiener-type theorems for the fractional Clifford–Fourier transform (FrCFT). Furthermore, because of the non-commutativity of Clifford algebra, we study some basic properties from linearity to modulation of the FrCFT about the left-multiplied functions and the right-multiplied functions. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of X the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if dim ⁡ X ≥ 11. This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties. [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]

In this paper, we discuss the fields described by Dirac wave equation written in Clifford algebra based on Macfalane quaternions. It is shown that the strengths of these fields are nonzero only in the area of sources and the interaction of such fields occurs by overlapping. We consider both the simple spherically symmetric models of sources, which demonstrate attractive and repulsive interaction, and more complicated core-shell systems, which provide the bound states formation. [ABSTRACT FROM AUTHOR]

Consider G to be a finite group and p to be a prime divisor of the order | G | in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ ∈ I r r (G) have a positive height, and let there be an anchor group A ϕ . We prove that if the normalizer N G (A ϕ) is p-constrained, then O p ´ (N G (A ϕ)) ≠ { 1 G } , where O p ´ (N G (A ϕ)) is the maximal normal p ´ subgroup of N G (A ϕ) . We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results. [ABSTRACT FROM AUTHOR]

We introduce a certain subalgebra F ‾ of the Clifford vertex superalgebra (bc system), which is completely reducible (as a L V i r (− 2 , 0) –module) and C 2 –cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF (1)). We show that there is an interesting duality between SF (1) and F ‾ , given by the fact that F ‾ can be equipped with the structure of a SF (1) –module and vice versa. Using the decomposition of F ‾ and a free-field realization from [3] , we decompose L k (osp (1 | 2)) at the critical level k = − 3 / 2 as a module for L k (sl (2)). The decomposition of L k (osp (1 | 2)) is exactly the same as that of the N = 4 superconformal vertex algebra with central charge c = − 9 , denoted by V (2). Using the duality between F ‾ and SF (1) , we prove that L k (osp (1 | 2)) and V (2) are in a duality of the same type. As an application, we construct and classify all irreducible L k (osp (1 | 2)) –modules in the category O and the category R , which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra N − 3 / 2 (osp (1 | 2)) as a N − 3 / 2 (sl (2)) –module. Extending this example, we introduce for each p ≥ 2 a non-conformal vertex algebra A n e w (p) and show that A n e w (p) is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra V n e w (p) , which is isomorphic to the logarithmic vertex algebra V (p) as a module for sl ˆ (2). We also conjecture the existence of the conformal vertex algebra V (sp (2 n)) which is in a duality with the affine vertex algebra L − n − 1 / 2 (o s p (1 | 2 n)). [ABSTRACT FROM AUTHOR]

Spinors play a fundamental role in geometry and physics. The Clifford algebra is the natural linear algebraic setting where spinors and spin groups are formulated. In this study, spinors, which have many uses in Clifford algebras, have been given with a different representation in E3. Firstly, the directional q -frame has been considered, and this frame is represented with a spinor in spinor space. Later, considering the relations between the directional q -frame and Frenet frame, the relations between spinors corresponding to these frames have been obtained. In this way, a different construction of the spinors by using in Clifford algebras has been shown. [ABSTRACT FROM AUTHOR]

We report a theoretical derivation of the Cabibbo–Kobayashi–Maskawa (CKM) matrix parameters and the accompanying mixing angles. These results are arrived at from the exceptional Jordan algebra applied to quark states, and from expressing flavor eigenstates (i.e., left chiral states) as a superposition of mass eigenstates (i.e., the right chiral states) weighted by the square root of mass. Flavor mixing for quarks is mediated by the square root mass eigenstates, and the mass ratios used are derived from earlier work from a left–right symmetric extension of the standard model. This permits a construction of the CKM matrix from first principles. There exist only four normed division algebras, and they can be listed as follows: the real numbers R , the complex numbers C , the quaternions H and the octonions O. The first three algebras are fairly well known; however, octonions as algebra are less studied. Recent research has pointed towards the importance of octonions in the study of high-energy physics. Clifford algebras and the standard model are being studied closely. The main advantage of this approach is that the spinor representations of the fundamental fermions can be constructed easily here as the left ideals of the algebra. Also, the action of various spin groups on these representations can also be studied easily. In this work, we build on some recent advances in the field and try to determine the CKM angles from an algebraic framework. We obtain the mixing angle values as θ 12 = 11.093 ∘ , θ 13 = 0.172 ∘ , θ 23 = 4.054 ∘ . In comparison, the corresponding experimentally measured values for these angles are 13.04 ∘ ± 0.05 ∘ , 0.201 ∘ ± 0.011 ∘ , 2.38 ∘ ± 0.06 ∘ . The agreement of theory with experiment is likely to improve when the running of quark masses is taken into account. [ABSTRACT FROM AUTHOR]

We will study applications of numerical methods in Clifford algebras in ℝ 4 , in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in ℝ 4 . In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can be placed on both sides of the powers. Then there are even five different classes of zeros. All results can be extended to other noncommutative algebras in ℝ 4 . In the paper by R. Lauterbach and G. Opfer (2014), the authors constructed an exact Jacobi matrix for functions defined in noncommutative algebraic systems without the use of any partial derivative. We applied this technique to find the eigenvalues of the companion matrix as zeros of the companion polynomial by Newton's method. [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]

Wireless ad hoc sensor networks have recently emerged as a premier research topic. They have great long-term economic potential and ability to transform our lives and pose many new system building challenges. Sensor networks also pose a number of new conceptual and optimization problems. Most of researches in wireless sensor networks are focused in obtaining better target coverage in order to reduce energy and cost of the network. The problem of planar target analysis is one of the crucial problems that should be considered while studying coverage problem of sensor networks. By combining computational geometry and graph theoretic techniques, specifically the Voronoi diagram and graph search algorithms, this paper introduces a novel sensor network coverage model that deals with plane target problem based on Clifford algebra which is a powerful tool that is coordinate free. Also, the calculations of the node coverage rate for the plane target in the sensor network using Clifford algebra are presented. Then, the maximum clearance path (worst-case coverage) of the sensor network for a plane target is proposed. The optimality and reliability of the proposed algorithm have been proved using simulation. Also, a comparison between the breach weight of the point target and the plane target is provided. [ABSTRACT FROM AUTHOR]

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli's theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications. [ABSTRACT FROM AUTHOR]

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras. [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]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

This article explores a class of Clifford-valued neural networks with leakage delay and proportional delay. By using the non-decomposition method and Banach fixed point theorem, we obtain several sufficient conditions for the existence of weighted pseudo almost periodic solutions for Clifford-valued neural networks with leakage delay and proportional delay. By using the proof by contradiction, we get the global exponential synchronization for Clifford-valued neural networks with leakage delay and proportional delay. Finally, one illustrative example is given to illustrate the feasibility and effectiveness of the main results. [ABSTRACT FROM AUTHOR]

For positive integers a and b, we let [Un] be the Lucas sequence of the first kind defined by U0 = 0, U1 = 1 and Un = aUn-1 + bUn−2 for n ≥ 2, and let π(m) ∶= π(a,b)(m) be the period length of [Un] modulo the integer m ≥ 2, where gcd(b, m) = 1. We define an (a, b)-Wall-Sun-Sun prime to be a prime p such that π(p² ) = π(p). When (a, b) = (1, 1), such a prime p is referred to simply as a Wall-Sun-Sun prime. We say that a monic polynomial f(x) ∈ Z[x] of degree N is monogenic if f(x) is irreducible over Q and {1, θ, θ², . . ., θN−1 } is a basis for the ring of integers of Q(θ), where f(θ) = 0. Let f(x) = x² − ax − b, and let s be a positive integer. Then, with certain restrictions on a, b and s, we prove that the monogenicity of f(xsn ) = x2sn − axsn − b is independent of the positive integer n and is determined solely by whether s has a prime divisor that is an (a, b)-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case b = 1. [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 study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over R n , n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ ∈ R n . Moreover, powerful results on the WVD-CFT have been derived such as Parseval's theorem, convolution theorem, Moyal's formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial. [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]

The behavior of accelerating Unruh-DeWitt detectors coupled with a spinor field in (3+1)-dimensional spacetime is investigated. For a single point-like detector with Gaussian switching function, the transition probability increases with the acceleration and thus the antiUnruh effect effect cannot occur. Due to the spinor structure of the Dirac field, UV divergences are encountered in the calculation of the entanglement between the detectors. After introducing some UV cutoff Λ, the negativity of detectors is shown to behave nonmonotonically with respect to the acceleration. Besides, the negativity increases with the cutoff Λ and decreases with the distance between the detectors. The mutual information between the two detectors is also discussed. [ABSTRACT FROM AUTHOR]

Within the scope of a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model we study the role of a nonlinear spinor field in the evolution of the universe. In doing so, we exploit the FLRW models given in both Cartesian and spherical coordinates. It is found that if the FLRW model is given in the spherical coordinates the energy-momentum tensor (EMT) of the spinor field possesses nontrivial non-diagonal components, which is not the case for Cartesian coordinates. These non-diagonal components do not depend on either the spinor field nonlinearity or the parameter k that defines the type of curvature of the FLRW model. The presence of such components imposes some restrictions on the spinor field. The problem is studied for open, flat and close geometries and the spinor field is used to simulate different types of sources including dark energies. Some qualitative numerical solutions are given. [ABSTRACT FROM AUTHOR]

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions O = C ⊕ C 3 , which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space S of C ℓ 10 Majorana spinors is generated by the C ℓ 6 (⊂ C ℓ 10) volume form, ω 6 = γ 1 ⋯ γ 6 , and is left invariant by the Pati–Salam subgroup of S p i n (10) , G PS = S p i n (4) × S p i n (6) / Z 2 . While the S p i n (10) invariant volume form ω 10 = γ 1 ... γ 10 of C ℓ 10 is known to split S on a complex basis into left and right chiral (semi)spinors, P = 1 2 (1 − i ω 6) is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of G PS that preserves the sterile neutrino (which is identified with the Fock vacuum). The Z 2 -graded internal space algebra A is then included in the projected tensor product A ⊂ P C ℓ 10 P = C ℓ 4 ⊗ C ℓ 6 0 . The Higgs field appears as the scalar term of a superconnection, an element of the odd part C ℓ 4 1 of the first factor. The fact that the projection of C ℓ 10 only involves the even part C ℓ 6 0 of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio m H m W of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to give the basic principles of hyperbolic function theory. First, we discuss the Green identity for H k ¯ H k . Second, we also deal with the integral representations for hyperbolic harmonic function in the Clifford algebra C l n + 1 , 0. [ABSTRACT FROM AUTHOR]