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Electric circuit analysis under non-sinusoidal, non-linear situations has been a hot topic for a long time. Many scientific communities hold opposing viewpoints on the additional analysis tool and domain, resulting in a variety of standards and definitions. With the advent of Power Electronic equipment, converters, and Renewable Energy sources, the electric power system has become increasingly sophisticated since its inception. Electronic equipment has transformed the electrical system and provided a slew of advantages to industrial applications. Unfortunately, this comes at the expense of power system distortion (voltage and current). Understanding power flow in non-sinusoidal linear and non-linear circuit circumstances is required for this. As a result, a novel mathematical framework to analyze the circuit in such an environment is always required. Finally, in sinusoidal conditions, a consensus can be reached on norms that comply with well-known, established standards. The work provided here compares the use of harmonic domain and geometric algebra in circuits with disturbances for sinusoidal and non-sinusoidal excitations in order to demonstrate the accuracy of geometrical algebra in power flow calculations.

ВВ криптографических системах шифрования информации используются гиперкомплексные числа: кватернионы и октонионы. В качестве ключа применяется кватернион, который производит вращения группы выборок информации. Кватернионы и бикватернионы являются частными случаями геометрической алгебры Клиффорда. Использование векторов и мультивекторов геометрической алгебры для шифрования информации позволяет расширить разнообразие этих векторов. Для шифрования информации, представленной совокупностью векторов геометрической алгебры, эти векторы умножаются на мультивектора, которые осуществляют операцию ротор (rotor). В качестве ключа используется мультивектор (ротор). Для дешифрования информации применяется операция, которая соответствует обратному ротору. Алгоритмы геометрической алгебры повышают безопасность шифрования информации за счет повышения размерности алгебры. Для повышения производительности шифрования предлагается коэффициенты информационного вектора и мультивектора вращения выбирать из поля Z256. Предлагается вектор информации с коэффициентами из Z256 складывать со случайным вектором с коэффициентами из Z256 и считать эти коэффициенты ключами шифрования. Приведены базисные векторы применяемых геометрических алгебр и таблицы геометрических произведений базисных векторов.

For color image fusion, existing traditional algorithms regard each color image pixel as a scalar which may lose the relationships among spectral channels. In this paper, a novel multi-focus image fusion algorithm using geometric algebra based Discrete Fourier Transform (GA-DFT) is proposed. First, a novel Discrete Fourier Transform based on geometric algebra (GA) is provided. Second, the proposed algorithm represents the color image as a GA multivector by mapping the RGB channels into the GA space. Third, the variance is selected as a contrast measure to fuse the source images in GA space. The experimental results demonstrate that the proposed algorithm can achieve the comparable performance with its counterparts in terms of subjective quality evaluation, and outperform most of the traditional algorithms in terms of objective quality metrics, indicating good performance in image fusion applications.

Adaptive filtering algorithms based on higher-order statistics are proposed for multi-dimensional signal processing in geometric algebra (GA) space. In this paper, the proposed adaptive filtering algorithms utilize the advantage of GA theory in multi-dimensional signal processing to represent a multi-dimensional signal as a GA multivector. In addition, the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) adaptive filtering algorithms are extended to GA space for multi-dimensional signal processing. Both the proposed GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixed-norm (GA-LMMN) algorithms need to minimize cost functions based on higher-order statistics of the error signal in GA space. The simulation results show that the proposed GA-LMF algorithm performs better in terms of convergence rate and steady-state error under a much smaller step size. The proposed GA-LMMN algorithm makes up for the instability of GA-LMF as the step size increases, and its performance is more stable in mean absolute error and convergence rate.

This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra $${\mathcal {G}}_{4,1}$$ that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an $$SU(2)_L$$ relationship among the chiral electron and neutrino states. The required Dirac-like equation in $${\mathcal {G}}_{4,1}$$ results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension.

A novel least-mean kurtosis adaptive filtering algorithm based on geometric algebra (GA-LMK) is proposed for multidimensional signal processing. First, taking advantage of geometric algebra (GA) in terms of the representation of multidimensional signal, the GA-LMK algorithm represents a multidimensional signal as a GA multivector. Second, we extend the original least mean kurtosis (LMK) algorithm in GA space for multidimensional signal processing. The proposed GA-LMK algorithm minimizes the cost function of negated kurtosis of the error signal in GA space, and provides a way to make tradeoff problem between convergence rate and steady-state error. Third, we study the steady-state behavior of the GA-LMK algorithm under Gaussian noises to acquire conditions of misadjustment. The simulation results show that our proposed GA-LMK adaptive filtering algorithm can outperform significantly existing the state-of-the-art algorithms in terms of convergence rate and steady-state error.

Multispectral images can provide more faithful representations for real scenes than the traditional images and improve the performance of image restoration tasks. In this paper, we propose a novel multivector sparse representation model for multispectral images using geometric algebra (GA), with the truth that GA is now well used in image processing and it gives a formidable way to represent multispectral images. The proposed model represents a multispectral image as a GA multivector by fully considering the spatial and spectral information, where a GA dictionary learning algorithm is presented using the K-GA-singular value decomposition (GASVD) (generalized K-means clustering for GASVD) method. Consequently, with the complete consideration of the relationship between spectral channels in multispectral images, artifacts and blurring effects can be successfully avoided. The experimental results demonstrate that the proposed sparse model surpasses the existing methods for multispectral images reconstruction and denoising tasks by capturing correlations between spectral channels thoroughly and shows its usefulness and effectiveness for multispectral images processing.

Although the theories of geometric algebra (GA) are widely applied in engineering design and analysis, the studies on their formalization have been scarcely conducted. This paper proposes a relatively complete formalization of GA in HOL Light. Both algebraic and geometric parts of the GA theories are formalized successively. For the algebraic part, a uniform abstract product is proposed to facilitate the formalization of the three basic products based on the formal definition of multivectors with three types of metrics. For the geometric part, the formal formulation is provided for the blades and versors and their relations at first. Then, several commonly used specific spaces are formally represented in the theoretical framework of GA. The novelty of the present paper lies in two aspects: (a) the multivector type, (P,Q,R)geomalg, is defined and the definition provides the most important foundation for the formalization of geometric algebra, and (b) a procedure is developed for automatically proving the properties of GA operations. The present work improves the function of HOL Light and makes the GA-based formal analysis and verification more convenient.

In this paper, theoretical proofs and performance analysis of geometric algebra based singular value decomposition (GA-SVD) are provided for multispectral image. The multispectral image is represented as a multivector based on geometric algebra (GA). According to the GA multivector, its real representation is constructed. Then the SVD problem for multispectral image is solved by defining a relationship between the singular values given by the GA-SVD of the GA multivector and the real SVD of its real representation. The multispectral image denoising application is carried out by selecting several largest singular values and the corresponding unitary matrices. The denoising experiments with Gaussian noise and salt-and-pepper noise are implemented to validate its rationality and effectiveness. The higher PSNR value also shows its potential as an effective and homogeneous tool in multispectral image processing.

Knowledge graph (KG) embedding aims at embedding entities and relations in a KG into a lowdimensional latent representation space. Existing KG embedding approaches model entities andrelations in a KG by utilizing real-valued , complex-valued, or hypercomplex-valued (Quaternionor Octonion) representations, all of which are subsumed into a geometric algebra. In this work,we introduce a novel geometric algebra-based KG embedding framework, GeomE, which uti-lizes multivector representations and the geometric product to model entities and relations. Ourframework subsumes several state-of-the-art KG embedding approaches and is advantageouswith its ability of modeling various key relation patterns, including (anti-)symmetry, inversionand composition, rich expressiveness with higher degree of freedom as well as good general-ization capacity. Experimental results on multiple benchmark knowledge graphs show that theproposed approach outperforms existing state-of-the-art models for link prediction., This paper is accepted by COLING2020

Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in quantum theory, in such a way that preserves their algebraic properties. They are that one must be able to arbitrarily transform the basis of the Clifford algebra, via multiplication by elements of the algebra that act trivially on the state space; all such elements must be neighbored by operators corresponding to factors in the original expression and not the state vectors. We explore the consequences of these criteria for a physics of dynamical multivector fields., Comment: This is a pre-print of an article published in Advances in Applied Clifford Algebras. The final authenticated version is available online at: https://doi.org/10.1007/s00006-019-1031-7

Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Li\'enard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism., Comment: 32 pages, 26 figures

Sparse representations have been extended to color image processing. However, existing sparse models treat each color image pixel either as a scalar which loses color structures or as a quaternion vector matrix with high computational complexity. In this paper, we propose a novel sparse representation model for color image that bears multiple channels based on geometric algebra. First, a novel theory of reduced geometric algebra (RGA) is provided, including commutative sparse basis and the geometric operations. Second, taking advantage of the RGA theory, the model represents color image with three-channel as a multivector with the spatial and spectral information in RGA space. Third, the dictionary learning algorithm is provided using the K-RGA-based singular value decomposition (K-RGASVD) (generalized K-means clustering for RGASVD) method. The comparison results demonstrate the proposed model can remove the data redundancy and reduce the computational complexity, and can meanwhile effectively preserve the inherent color structures. The result suggests its potential as a homogeneous and efficient tool in various applications of color image analysis.

Electromagnetic problems governed by Maxwell's equations are solved by using a Clifford algebra valued boundary element method (BEM). The well-known Maxwell's equations consist of eight pieces of scalar partial differential equations of the first order. They can be rewritten in terms of the language of Clifford analysis as a nonhomogeneous k-Dirac equation with a Clifford algebra valued function. It includes three-component electric fields and three-component magnetic fields. Furthermore, we derive Clifford algebra valued boundary integral equations (BIEs) with Cauchy-type kernels and then develop a Clifford algebra valued BEM to solve electromagnetic scattering problems. To deal with the problem of the Cauchy principal value, we use a simple Clifford algebra valued k-monogenic function to exactly evaluate the Cauchy principal value. Free of calculating the solid angle for the boundary point is gained. The remaining boundary integral is easily calculated by using a numerical quadrature except the part of Cauchy principal value. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In the numerical implementation, we introduce an oriented surface element instead of the unit outward normal vector and the ordinary surface element. In addition, we adopt the Dirac matrices to express the bases of Clifford algebra C l 3 ( ℂ ) . We also use an orthogonal matrix to transform global boundary densities into local boundary densities for satisfying boundary condition straightforward. Finally, two electromagnetic scattering problems with a perfect spherical conductor and a prolate spheroidal conductor are both considered to examine the validity of the Clifford algebra valued BEM with Cauchy-type kernels.

International audience; This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++ / Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. Garamon is designed to produce easy to install , easy to use, effective and numerically stable libraries. The design of the libraries is based on a data structure using precomputed functions for low dimensions and a smooth transition to the new recursive products for higher dimensions. Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.

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.

Generally, a multivector in Cl3 can be rewritten aswhere for vectors and complex vectors we are using the bold format here. We have already seen that the unit pseudoscalar j commutes with all elements of the algebra and squares to −1, and therefore, it is an ideal replacement for the ordinary imaginary unit (there are many “imaginary units” in GA). A pseudoscalar with such properties will appear again in Cl7, Cl11, … (see E10). Here we often use a very useful form of a multivector

Sustainable smart buildings play an essential role in terms of more efficient energy. However, these buildings as electric loads are affected by an important distortion in the current and voltage waveforms caused by the increasing proliferation of nonlinear electronic devices. Overall, buildings all around the world consume a significant amount of energy, which is about one-third of the total primary energy resources. Optimization of the power transfer process of such amount of energy is a crucial issue that needs specific tools to integrate energy-efficient behaviour throughout the grid. When nonlinear loads are present, new capable ways of thinking are needed to consider the effects of harmonics and related power components. In this manner, technology innovations are necessary to update the power factor concept to a generalized total or a true one, where different power components involved in it calculation, properly reflect each harmonic interaction. This work addresses an innovative theory that applies the Poynting Vector philosophy via Geometric Algebra to the electromagnetic energy transfer process providing a physical foundation. In this framework, it is possible to analyse and detect the nature of disturbing loads in the exponential growth of new globalized buildings and architectures in our era. This new insight is based on the concept of geometric objects with different dimension: vector, bivector, trivector, multivector. Within this paper, these objects are correlated with the electromagnetic quantities responsible for the energy flow supplied to the most common loads in sustainable smart buildings. Besides, it must be considered that these phenomena are characterized by a quality index multivector appropriate even for detecting harmonic sources. A numerical example is used to illustrate the clear capabilities of the suggested index when it applies to industrial loads for optimization of energy control systems and enhance comfort management in smart sustainable buildings.

Integrating dynamic analysis models into geographic information system (GIS)-based evacuation simulations is important yet complex. Different models must be smoothly assembled according to the data processing flow to obtain a dynamic, data-forced evacuation simulation. However, because of the diversity of data types and dynamic data updating among different models, closely integrated evacuation simulations are complex and inefficient. In this study, geometric algebra (GA) is introduced to develop a dynamic evacuation simulation framework for a hazardous gas diffusion scheme. In the framework, geospatial data are first integrated into a unified virtual scene with different forms of multivector representation. The major simulation models of gas diffusion, risk assessment, and dynamic evacuation routing compose the major steps of the evacuation simulation. On the basis of the generalized multivector structure, dynamic exchange and updating geospatial data at different evacuation steps can be performed seamlessly with the multivector structure and GA operators. The framework is tested with a case study of a three-dimensional residential area, which shows that our framework can support the integration of dynamic evacuation processes and the model integration is direct and smooth. This framework may also provide a new solution for the integration and dynamic data updating in spatiotemporal GIS.

This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper.

To be a qualified engineer, or to pursue further studies in electronics and information engineering, undergraduate students should attempt to broaden their horizons in their professional fields. Particularly important are emerging mathematical theories or tools that produce relevant mathematical formalism for modelling the nature of physical space as simply and compactly as possible. In this article, we illustrate that geometric algebra provides just such a simplified and elegant representation or description of physical space. The basics of geometric algebra are explained and concrete examples in the plane and in 3D space are given. This article also provides two examples where geometric algebra has been found to provide an improved description of coordinate rotations and the Maxwell equations.

Three Clifford algebras are sufficient to describe all interactions of modern physics: The Clifford algebra of the usual space is enough to describe all aspects of electromagnetism, including the quantum wave of the electron. The Clifford algebra of space-time is enough for electro-weak interactions. To get the gauge group of the standard model, with electro-weak and strong interactions, a third algebra is sufficient, with only two more dimensions of space. The Clifford algebra of space allows us to include also gravitation. We discuss the advantages of our approach.

We consider a Clifford extension of the Grassmann algebra in which operators are constructed from products of Grassmann variables and derivatives with respect to them. We show that this algebra contains a subalgebra isomorphic to a matrix algebra and that it additionally contains operators of a generalized matrix algebra that mix states with different numbers of Grassmann variables. We show that these operators are extensions of spin-tensors to the case of superspace. We construct a representation of Dirac matrices in the form of operators of a generalized matrix algebra.

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p -form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl 1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl 1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford's geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit i is an element of C is replaced by a multivector square root of -1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator.

This chapter gives a detailed outline of geometric algebra and explains the related traditional algebras in common use by mathematicians, physicists, computer scientists, and engineers.

In geometric algebra, there exist specific operators named versors to model rotations, translations, and dilations, and are called rotors, translators and dilators respectively. In general, a versor \({\varvec{G}}\) is a multivector which can be expressed as the geometric product of non-singular vectors $$\begin{aligned} G = \pm {\varvec{v}}_1 {\varvec{v}}_2 ... {\varvec{v}}_k. \end{aligned}$$

There is a steadily increasing interest in applying Geometric Algebra (GA) in diverse fields of science and engineering. Consequently, we need better software implementations to accommodate such increasing demands that widely vary in their possible uses and goals. For large-scale complex applications having many integrating parts, such as Big Data and Geographical Information Systems, we should expect the need for integrating several GAs to solve a given problem. Even within the context of a single GA space, we often need several interdependent systems of coordinates to efficiently model and solve the problem at hand. Future GA software implementations must take such important issues into account in order to scale, extend, and integrate with existing software systems, in addition to developing new ones, based on the powerful language of GA. This work attempts to provide GA software developers with a self-contained description of an extended framework for performing linear operations on GA multivectors within systems of interdependent coordinate frames of arbitrary metric. The work explains the mathematics and algorithms behind this extended framework and discusses some of its implementation schemes and use cases. If properly implemented, the extended framework can significantly reduce the memory requirements for implementing Geometric Algebras with larger dimensions, especially for systems based on the symbolic processing of multivector scalar coefficients.

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.

In this paper, we introduce a method to obtain vector generators for the real matrix representation of the real Clifford algebra from the real matrix representation of with for every . This can give a nice tool to classify the Clifford algebra in the sense of generators. Also, we present the inverse of a nonzero vector in and, by using the properties of vector generators in , we obtain some results about determinant and inverse of a vector in . Moreover, we show the relations of solutions for the linear equation for some in and that for in for some , where .

The basis for engineering electromagnetic computations still rely on Gibbs' vector algebra. It is well known that Clifford algebra (geometric algebra) presents several enhancement on the latter. By taking advantage that in the three-dimensional space Clifford algebra is isomorphic to Pauli algebra it is possible to describe all the relevant vector operations occurring in electromagnetic theory in terms of Pauli matrices. In particular it is possible to write Maxwell's equations in a form similar to the Dirac equation. In this way, instead of having six coupled equations from the curls operators, we can deal with just four linear equations. The latter can be further simplified to just two sets of two linear equations by the Weyl decomposition.

In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.

In this paper one proposes to use a new approach of interval arithmetic, the so-called pseudo-intervals (1), (5), (13). It uses a construction which is more canonical and based on the semi-group completion into the group, and it allows to build a Banach vector space. This is achieved by embedding the vector space into free algebra of dimensions higher than 4. It permits to perform linear algebra and differential calculus with pseudo-intervals. Some numerical applications for interval matrix eigenmode calculation, inversion and function minimization are exhibited for simple examples. RESUMO. Neste artigo é proposto o uso de uma nova abordagem de aritmética intervalar, os chamados pseudo-intervalos (1), (5), (13). Essa abordagem utiliza uma construção que é mais canônica e baseada na completude de semi-grupos em um grupo, e permite a construção de espaços vetoriais de Banach. Isso é alcançado através da imersão do espaço vetorial em uma álgebra livre de dimensão maior que 4. Desta forma é possível realizar álgebra linear e cálculo diferencial com pseudo-intervalos. Algumas aplicações numéricas para o cálculo do espectro de matrizes intervalares, inversão e minimização de funções são mostradas como exemplos.

We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article. Throughout, we employ the mathematical language of geometric algebra and calculus., 24 pages

WOS: 000318844600006, The Maxwell-Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of gravitoelectromagnetic Klein-Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.

The present paper discloses that the system of matrices for orthogonal skew-symmetric anticommuting operators is apt to generate a Clifford algebra representation in terms of matrices. The statistical properties of this particular system of operators as applied to a random vector have been examined. The operator system obtained is feasible for producing a set of orthogonal signals in the multiple-access noise-signal communication system.

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3,0 and even Cl 3,1 + subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl 3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.

We introduce Clifford geometric algebra based multivector modeling of quartic and general quadric surfaces, Darboux cyclides, Dupin cyclies, tori and pairs of standard CGA objects. A computer visualization is briefly described.

International audience; Multivectorial algebra is of both academic and technological interest. Its application, however, is not always easy. A distinction must be made between polar and axial vectors and between scalars and pseudo-scalars. Eight element types are often considered even if they are not always identified as multivectors. In some cases, for simplicity’s sake, only vectorial algebra or quaternion algebra is explicitly used for physical and mechanical applications. It would, however, be more convenient to use more complex algebra directly in order to have a wider range of mechanical applications. The aim of this paper is to examine one particular type of Clifford algebra that could solve this problem. The present study focusses on showing how these quantities can be used to model mechanical and engineering problems. First, continuum mechanics in a Cauchy medium is investigated for elastic transformations. Second, a specific type of shot-peening application is studied. Applications are then used to illustrate the scope and efficiency of this type of modeling based on geometric algebra.