Abstract: Abstract The main purpose of the present paper is to develop the theory of center constructions on Hom–Hopf algebras. Let H be a Hom–Hopf algebra, we first introduce the notions of nth Yetter–Drinfeld modules and mth Drinfeld codouble for H. Also we prove that the category \({\mathcal {YD}}_H^H(n)\) of nth Yetter–Drinfeld modules of H is a braided autonomous category. Finally, we show that \({\mathcal {YD}}_H^H(n)\) and \(Corep^{i,j}(CD_m(H))\) (i.e., the corepresentation category of the Drinfeld codouble of H) are braided isomorphic as the full subcategories of \(Corep^{i,j}(H)\) . PubDate: 2019-03-16

Abstract: Unfortunately, the communicating editor was wrongly published as Dr. Cristina Elena Flaut instead of Prof. Rafał Abłamowicz PubDate: 2019-03-04

Abstract: Abstract This paper proposes a \({2\times 2}\) real matrix isomorphic representation form of split quaternions and a \({2m\times 2n}\) real matrix isomorphic representation form of \({m\times n}\) split quaternion matrices. In particular, we highlighted the inner relationships among the existing different \({2\times 2}\) matrix representation forms of split quaternions. Furthermore, we studied the inner relationships among different \({2\times 2}\) matrix representation forms of Hamilton quaternions as an application. The forms and relationships discussed in this paper can simplify the computation and make split quaternions get extensive applications. PubDate: 2019-03-01

Abstract: Abstract Some algebraic structures that can be defined on the spaces of paravectors and k-paravectors are studied. Firstly, a version of the exterior and interior products resembling those in the exterior algebra of k-vectors but according to the k-paravector grading is defined. Secondly, a new Clifford algebra is constructed from the operations of exterior and interior products of paravectors and k-paravectors such that if the original vector space has a metric of signature (n, 0), then the metric of this new Clifford algebras has a metric of signature (1, n). The noticeable difference between this new Clifford algebra and usual ones is the necessity of the conjugation operation in its definition. Thirdly, since the space of k-paravectors is not an invariant space under the Clifford product by a paravector, another product is defined in such a way to make the space of k-paravectors invariant under this product by a paravector. The algebra defined by this product is shown to be a DKP algebra. PubDate: 2019-02-26

Abstract: Abstract We address the \(\ell _1 \) -norm minimization problem, which plays an important role in the compressed sensing (CS) theory. While most previous approaches are designed for signals of low dimensions, we present in this paper an algorithm using geometric algebra (GA) for solving the problem of \(\ell _1 \) -norm minimization for multi-dimensional signals with and without noise contamination by converting it to second-order cone programming. The algorithm represents the multi-dimensional signal as a multivector in the GA form to process it in a holistic way without losing the correlation among different dimensions of the multi-dimensional signal. Numerical experiments are provided to demonstrate the effectiveness of the proposed algorithm and its robustness to noise. The proposed algorithm can also be used to guide perfect recovery of multi-dimensional signals, and may find potential applications when CS theory meets the multi-dimensional signal processing. PubDate: 2019-02-26

Abstract: Abstract This article combines two independent theories: firstly, the algorithm of Faddeev–Leverrier which calculates characteristic polynomials of matrices; secondly, the Descent Theory which, in particular, lets many properties of matrix algebras descend down to Azumaya algebras, especially the characteristic polynomials. The algorithm of Faddeev–Leverrier is completely revisited. The details of the descent are explained as far as they are needed for Clifford algebras over fields. PubDate: 2019-02-19

Abstract: Abstract Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space \(\mathbb {R}^{p',q'}\) , \(n'=p'+q'=2m\) , we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over \(\mathbb {R}^{p,q}\) , \(n=p+q=p'+q'+1=2m+1\) . Explicit examples are provided for dimensions \(n'=2,4,6\) , and the resulting inverses for \(n=n'+1=3,5,7\) . The general result for \(n=7\) appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), \(n=p+q=7\) , only involving a single addition of multivector products in forming the determinant. PubDate: 2019-02-19

Abstract: Abstract We construct two novel quaternion-valued smooth compactly supported symmetric orthogonal wavelet (QSCSW) filters of length greater than existing ones. In order to obtain their filter coefficients, we propose an optimization-based method for solving a specific kind of multivariate quadratic equations. This method provides a new idea for solving multivariate quadratic equations and could be applied to construct much longer QSCSW filters. PubDate: 2019-02-18

Abstract: Abstract Let \(\mathbb R_{0, m+1}^{(s)}\) be the space of s-vectors ( \(0\le s\le m+1\) ) in the Clifford algebra \(\mathbb R_{0, m+1}\) constructed over the quadratic vector space \(\mathbb R^{0, m+1}\) , let \(r, p, q\in \mathbb N\) with \(0\le r\le m+1\) , \(0\le p\le q\) and \(r+2q\le m+1\) and let \(\mathbb R_{0, m+1}^{(r,p,q)}=\sum _{j=p}^q\bigoplus \mathbb R_{0, m+1}^{(r+2j)}\) . Then a \(\mathbb R_{0, m+1}^{(r,p,q)}\) -valued smooth function F defined in an open subset \(\Omega \subset \mathbb R^{m+1}\) is said to satisfy the generalized Moisil–Teodorescu system of type (r, p, q) if \(\partial _x F=0\) in \(\Omega \) , where \(\partial _x\) is the Dirac operator in \(\mathbb R^{m+1}\) . To deal with the inhomogeneous generalized Moisil–Teodorescu systems \(\partial _x F=G\) , with a \(\sum _{j=p}^{q} \bigoplus {\mathbb {R}}^{(r+2j-1)}_{0,m+1}\) -valued continuous function G as a right-hand side, we embed the systems in an appropriate Clifford analysis setting. Necessary and sufficient conditions for the solvability of inhomogeneous systems are provided and its general solution described. PubDate: 2019-02-14

Abstract: Abstract The Monte Carlo (MC) simulations are considered the gold-standard method for calculating the transport and interaction of radiation with the matter. A fundamental component of any MC simulation is the geometrical modeling. Current implementations of the geometrical modeling are based only on the Euclidean representations. However, Euclidean representations may not be the best option for speed up the geometric debugging-modeling computations of radiation transport, due to the number of operations involved in the estimation of position and direction of particles within each geometry shape. In this work, it is proposed for the first time, the use of Conformal Geometric Algebra (CGA), for geometric modeling in MC simulation for radiation transport. In this context, we present some elemental CGA equations for the microscopic modeling of positions and rotations of a radiation particle and the macroscopic modeling of geometrical shapes. It is shown that it is possible to take advantage of the expression power of CGA to create and debug geometry modeling with a triboelectric X-ray application. Additionally, some advantages of the CGA for the microscopic geometric computations are explored. PubDate: 2019-02-08

Abstract: Abstract Analogous to real functions, zeon functions are defined as zeon-valued functions of a zeon variable. In this paper, formal criteria for continuity and differentiability of zeon functions are put on a rigorous footing and the “usual” differentiation rules are formally established. As special cases, zeon extensions of real functions and zeon functions of one real variable are considered. PubDate: 2019-02-08

Abstract: Abstract In this work a generalization of a Born–Infeld theory of gravity with a topological \(\beta \) -term is proposed. These type of Born–Infeld actions were found from the theory introduced by MacDowell and Mansouri. This theory known as MacDowell–Mansouri (MM) gravity was one of the first attempts to construct a gauge theory of gravitation, and within this framework it was introduced in the action a topological \(\beta \) -term relevant for quantization purposes in an analogous way as in Yang–Mills theory. By the use of the self-dual and antiself-dual actions of MM gravity, we further define a Born–Infeld gravity generalization corresponding to MM gravity with the \(\beta \) -term. PubDate: 2019-02-05

Abstract: Abstract Geometric algebra (GA) is a powerful mathematical tool that offers intuitive solutions for image-processing problems, including colour edge detection. Rotor-based and Prewitt-inspired Sangwine (RBS and PIS) filters are amongst the efficient algorithms based on GA operators for solving colour edge detection problem. Algorithms in GA framework have enormous computational load that limits the general-purpose processors’ ability to execute them in reasonable time. Recently, some specialized hardware architectures, called full-hardware implementations, are proposed. These architectures, such as ConformalALU co-processor, are able to execute GA algorithms in acceptable time with the moderate use of computational resources. So far, all colour edge detection hardwares in GA framework exploited RBS filters. Nevertheless, this novel work presents a full-hardware architecture for efficient execution of PIS filters. PIS filters consume less computational resources and are faster to execute. For comparison, the hardware obtained by Gaalop pre-compiler uses twice as much resources with the same speed as the proposed hardware. As an evidence of faster operation, the proposed hardware is able to execute the edge detection algorithm almost 315 times faster than a GA co-processor, with only 2.5 times of its resources. PubDate: 2019-02-05

Abstract: Abstract Zeon algebras arise as commutative subalgebras of fermions, and can be constructed as subalgebras of Clifford algebras of appropriate signature. Their combinatorial properties have been applied to graph enumeration problems, stochastic integrals, and even routing problems in communication networks. Analogous to real polynomial functions, zeon polynomial functions are defined as zeon-valued polynomial functions of a zeon variable. In this paper, properties of zeon polynomials and their zeros are considered. Nilpotent and invertible zeon zeros of polynomials with real coefficients are characterized, and necessary conditions are established for the existence of zeros of polynomials with zeon coefficients. Quadratic polynomials with zeon coefficients are considered in detail. A “zeon quadratic formula” is developed, and solutions of \(ax^2+bx+c=0\) are characterized with respect to the “zeon discriminant” of the equation. PubDate: 2019-01-22

Abstract: Abstract It is argued how R \(\otimes \) C \(\otimes \) H \(\otimes \) O-valued Gravity (real-complex-quaterno-octonionic Gravity) naturally can describe a Grand Unified Field theory of Einstein’s gravity with a Yang-Mills theory containing the Standard Model group \(SU(3) \times SU(2) \times U(1)\) . In particular, it leads to a \([SU(4)]^4\) symmetry group revealing the possibility of extending the standard model by introducing additional gauge bosons, heavy quarks and leptons, and an extra fourth family of fermions. We finalize by displaying the analog of the Einstein–Hilbert action for \(\mathbf{R} \otimes \mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O}\) -valued gravity via the use of matrices, and which is based on “coloring” the graviton; i.e. by attaching internal indices to the metric \(g_{\mu \nu }\) . In the most general case, U(16) arises as the isometry group, while U(8) is the isometry group in the split-octonion case. PubDate: 2019-01-22

Abstract: Abstract Using the Dirac (Clifford) algebra \(\gamma ^{\mu }\) as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15 dimensional Lie algebra su*(4) as complex embedding of sl(2, \(\mathbb {H}\) ), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of \(\mathbb {R}^{3}\) . This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac’s ‘square root of \(p^{2}\) ’, the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook. PubDate: 2019-01-08

Abstract: Abstract The modelling of real world objects is not a straightforward subject. There are many different schemes; constructive solid geometry (CSG), cell decomposition, boundary representation, etcetera. Obviously, somehow, any scheme will be related to any other since they have a common goal. The paper shows how to model general polyhedra as an unordered discrete and finite set of geometric numbers of a projective Clifford Algebra or Geometric Algebra (GA). Clearly, not any randomly generated finite set of geometric numbers will have the structure of an object, this set must have some well defined properties. The topological properties extracted from this set are mapped to a boundary representation scheme based on a type of combinatorial map called generalised map or n-gmap. The n-gmaps have different types of orbits (in the mathematical sense) to which an attribute can be attached. When the attribute has a geometrical meaning, it is said that it is the geometrical embedding of the n-gmap. In this way the n-gmap holds explicitly the topology or structure already defined by the discrete geometry. In our proposal, each single element of a n-gmap is consistently embedded into a geometrical number also known as multi-vector. The scheme has been implemented by modifying an open source code [46] of n-gmaps. This representation has interesting properties. GA and n-gmaps complement and reinforce each other. For instance; it improves the robustness when computing the structure from the geometrical information. It is capable of computing lengths, areas and volumes of any polyhedral complex (with or without holes) using the orbits of the n-gmap (some examples are given). Finally the paper gives hints about other potentialities. PubDate: 2019-01-08

Abstract: Abstract We investigate the relation between a triple product system and the super Yangian \(Y(\mathfrak {gl}(2 1))\) . We present the super Yangian covariance structure for a triple product system associated with a rational R-matrix. Moreover, we obtain the ternary Hopf algebraic structure of the super Yangian \(Y(\mathfrak {gl}(2 1))\) . PubDate: 2019-01-02

Abstract: Abstract In this paper a comprehensive analysis of the Horadam quaternion zeros for some new types of bivariate quadratic quaternion polynomial equations is presented. PubDate: 2019-01-02