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In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only "authorized" sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness ("unauthorized" sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for instantaneous reactive power analysis in the geometric algebra (GA) environment is presented. The different formulations of the instantaneous reactive power theory (IRPT) proposed, to date, have been developed in three-phase systems. There, an instantaneous power variable, and two/three reactive power variables, all handled independently, were introduced. Thanks to GA, it is possible to carry out a global treatment where an instantaneous power multivector is defined. Thus, in the same multidimensional entity all the power variables are included. From the instantaneous power multivector, the instantaneous power current and the instantaneous reactive current are determined. It should be noted that in this mathematical framework there is no limitation on the number of phases, and the extension of the IRPT to the analysis of multi-phase systems appears in a natural manner. In this study, a systematic approach with the most relevant definitions and theorems corresponding to the proposed methodology has been established. Two practical cases of five-phase and three-phase systems have been included to apply the new established formulation. [ABSTRACT FROM AUTHOR]

Geographic information space is a high-dimensional space combining virtual and real. Geographic information system (GIS) based on European geometry plays an important role in the development of geographic information science. However, because the expression of GIS geographic objects and phenomena in European-style space depends on object coordinates and the adopted coordinate system, it is difficult for the expression, calculation and analysis of geographic objects to form a multi-dimensional unified operation rule and framework, which increases the algorithm’s complexity and reduces the efficiency of algorithm analysis. Geometric algebra is a kind of combination algebra based on dimensional operation. Space can be defined as the operation between vector sets. The dimension of space is directly determined by the algorithm, which realizes the unification of high - dimensional geometric calculation and analysis. Based on the unified calculation and analysis framework of geometric algebra, GIS can better express and analyze high - dimensional objects, and then improve the ability of expressing complex geographical objects and dynamic geographical phenomena, analyzing spatiotemporal multi - scale objects and integration of different scale geographical models. This lays the theoretical foundation for the development of GIS towards holographic maps with real-time dynamics,virtual-real fusion,full perspective, full elements and full content expression. [ABSTRACT FROM AUTHOR]

A new index for measuring the closeness to the singularities of parallel manipulators using geometric algebra is proposed in this paper. Constraint wrenches acting on the moving platform of a parallel manipulator are derived using the outer product and dual operations. Removing the redundant constraint wrenches, a singularity polynomial is obtained when the coefficient of the outer product of all the non-redundant constraint wrenches equals zero. A singularity surface can be drawn using the singularity polynomial. Similarly, an approximate singularity polynomial and approximate singularity surface can be obtained by imposing a threshold to the singular polynomial. Then the singularity volume is calculated as the space between singularity surface and approximate singularity surface. The new index is derived by calculating the ratio of the non-singularity workspace volume (the workspace volume minus the singularity volume) to the workspace volume. The proposed index is coordinate-free and has a clear geometrical and physical interpretation. This index can be a basis for selecting structural parameters, path planning and mechanism design. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric. [ABSTRACT FROM AUTHOR]

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford's geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin- $${\frac{1}{2}}$$ particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra-this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information. [ABSTRACT FROM AUTHOR]

Abstract: The analysis of large volumes of unordered multidimensional data is a problem confronted by scientists and data analysts every day. Often, it involves searching for data alignments that emerge as well-defined structures or geometric patterns in datasets. For example, straight lines, circles, and ellipses represent meaningful structures in data collected from electron backscatter diffraction, particle accelerators, and clonogenic assays. Also, customers with similar behavior describe linear correlations in e-commerce databases. We describe a general approach for detecting data alignments in large unordered noisy multidimensional datasets. In contrast to classical techniques such as the Hough transforms, which are designed for detecting a specific type of alignment on a given type of input, our approach is independent of the geometric properties of the alignments to be detected, as well as independent of the type of input data. Thus, it allows concurrent detection of multiple kinds of data alignments, in datasets containing multiple types of data. Given its general nature, optimizations developed for our technique immediately benefit all its applications, regardless the type of input data. [Copyright &y& Elsevier]

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