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For some research questions that involve Spin(p, q) representation theory, using symbolic algebra based techniques might be an attractive option for simplifying and manipulating expressions. Yet, for some such problems, especially as they arise in the study of various limits of M-theory (such as dimensional reductions), the complexity of the resulting expressions can become computationally challenging when using popular symbolic algebra packages in a straightforward manner. This work discusses some general properties of Gamma matrices that are computationally useful, down to the level of what one would call "bit-twiddling hacks" in computer science. It is presented in a self-contained way that should be accessible to both physicists and computer scientists. Code is available alongside the TeX source of the preprint version of this article on arXiv. [ABSTRACT FROM AUTHOR]

Multiple-input multiple-output (MIMO) radar is a new type of radar with excellent performance in target detection and parameter estimations. The MIMO radar equipped with electromagnetic vector sensors (EMVSs) can record multi-dimensional information of the electromagnetic (EM) signal. Traditional methods based on long vector model cannot preserve orthogonality between the outputs of each component of the EMVS and have high computational complexity. In this paper, a novel geometric algebra (GA)-based localization method is proposed for direction of arrival (DOA), range, and polarization estimation of near-field (NF) non-circular (NC) sources for a symmetric MIMO array. First, the proposed method converts the output of an EMVS into a multi-vector through a GA matrix transformation. Then, the DOA and range parameters can be estimated by searching the one-dimensional (1-D) spectrum based on the GA-based NC rank reduction (GANC-RARE) algorithm. Finally, the polarization can be estimated by employing estimated DOAs and ranges based on the modified GA-based NC multiple signal classification (MGANC-MUSIC) algorithm. The computational complexity analysis demonstrates that our solution improves memory spaces of the covariance matrix and reduces computation efforts of the eigenvalue decomposition (EVD). The simulation results show the superiority and reliability of our proposed method in model error and coherent noise environment. [ABSTRACT FROM AUTHOR]

Feature analysis plays an important role in many multispectral image applications and scale invariant feature transform (SIFT) has been successfully applied for extraction of image features. However, the existing SIFT algorithms cannot extract features from multispectral images directly. This paper puts forward a novel algorithmic framework based on the SIFT for multispectral images. Firstly, with the theory of the geometric algebra (GA), a new representation of multispectral image including spatial and spectral information is put forward and discussed. Secondly, a new method for obtaining the scale space of the multispectral image is proposed. Thirdly, following the procedures of the SIFT, the GA based difference of Gaussian images are computed and the keypoints can be detected in the GA space. Fourthly, the feature points are finally detected and described in the mathematical framework of the GA. Finally, the comparison results show that the GA-SIFT outperforms some previously reported SIFT algorithms in the feature extraction from a multispectral image, and it is comparable with its counterparts in the feature extraction of color images, indicating good performance in various applications of image analysis. [ABSTRACT FROM AUTHOR]

Abstract: Formal models can be useful in computer graphics as a conceptual framework supporting representation systems. This allows to formally derive properties and algorithms and proof their correctness and validity. This paper describes a formal model based on a geometric algebra. This algebra has been used to obtain specific representation systems and study their equivalence. The representation systems derived in a natural way from this model are based on simplicial coverings and can be applied to non-manifold solids and to solids with holes. Representations have been developed for polyhedral and free-form solids. Algorithms described and proved include boolean operations and representation conversion. The paper covers the three abstraction levels: theoretical model, representations and derived algorithms. As a practical application an experimental modeller for free-form solid has been developed (ESC-MOD system: “Extended Simplicial Chains MOdeller”). [Copyright &y& Elsevier]

Abstract: The main part of the code presented in this work represents an implementation of the split-operator method [J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129–160; R. Heather, Comput. Phys. Comm. 63 (1991) 446] for calculating the time-evolution of Dirac wave functions. It allows to study the dynamics of electronic Dirac wave packets under the influence of any number of laser pulses and its interaction with any number of charged ion potentials. The initial wave function can be either a free Gaussian wave packet or an arbitrary discretized spinor function that is loaded from a file provided by the user. The latter option includes Dirac bound state wave functions. The code itself contains the necessary tools for constructing such wave functions for a single-electron ion. With the help of self-adaptive numerical grids, we are able to study the electron dynamics for various problems in 2+1 dimensions at high spatial and temporal resolutions that are otherwise unachievable. Along with the position and momentum space probability density distributions, various physical observables, such as the expectation values of position and momentum, can be recorded in a time-dependent way. The electromagnetic spectrum that is emitted by the evolving particle can also be calculated with this code. Finally, for planning and comparison purposes, both the time-evolution and the emission spectrum can also be treated in an entirely classical relativistic way. Besides the implementation of the above-mentioned algorithms, the program also contains a large C++ class library to model the geometric algebra representation of spinors that we use for representing the Dirac wave function. This is why the code is called “Dirac++”. [Copyright &y& Elsevier]

Geometric algebra is used in this paper for a rigorous mathematical treatment of power in single-phase circuits under nonsinusoidal conditions, as complex algebra for sinusoidal conditions. This framework clearly displays the multidimensional nature of power, which is represented by a multivector. The power multivector with its three attributes (magnitude, direction and sense) provides the means to encode all the necessary information in a single entity. This property, in conjunction with the fact that there is a one-to-one correspondence between the terms of this multivector, the instantaneous and the apparent power equation, distinguishes it as a highly efficient mathematical tool. In this way one can successfully describe power phenomena and handle practical problems (e.g., power factor improvement). Two simple examples show some of these features. In short, the power multivector under nonsinusoidal situations can be perceived as the generalization of the complex power under sinusoidal situations [ABSTRACT FROM PUBLISHER]