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Using the powerful language of geometric algebra, we present an observationally symmetric derivation of the strong correlations predicted by the entangled singlet state in a deterministic and locally causal model, usually also referred to as a local-realistic model, in which the physical space is assumed to be a quaternionic 3-sphere, or S 3 , available as the spatial part of a solution of Einstein’s field equations of general relativity, and compare it in quantitative detail with Bell’s local-realistic model for the singlet correlations set within a flat Euclidean space I R 3 . Since the quantitatively detailed expressions of relative-angle-dependent probabilities of observing measurement outcomes for Bell’s local model do not seem to have been fully articulated before, our novel analysis exploiting the non-commutative properties of quaternions, in addition to allowing the comparison with the quaternionic 3-sphere model, may also provide useful comparisons for other less compelling local-realistic models, such as those relying on retrocausality or superdeterminism. Apart from the conservation of zero spin angular momentum, the key attribute underlying the strong singlet correlations within S 3 in comparison with Bell’s local model turns out to be the spinorial sign changes intrinsic to quaternions that constitute the 3-sphere. In addition, we also discuss anew a macroscopic experiment that can, in principle, test our 3-sphere hypothesis. [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]

Copyright of Turkish Journal of Remote Sensing & GIS is the property of Halil Akinci and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The instantaneous reactive power theory (IRPT) has long been utilized for load compensation through active power line conditioners (APLCs). IRPT effectively divides the current vector into two components: the instantaneous power current and the instantaneous reactive current. Among these components, only the former is necessary to transfer instantaneous real power to the load. However, the commonly adopted approach faces challenges when extended to multi-phase systems and their compensation strategy design. This challenge is particularly pertinent today due to the consensus that treating a three-phase system with a neutral wire as a four-conductor system is more appropriate. This paper introduces a formulation of IRPT tailored for multi-phase systems within the framework of geometric algebra (GA). GA is a mathematical structure that defines a single power variable, the instantaneous power multivector, encompassing both instantaneous real power and instantaneous reactive power within a unified mathematical entity. The current components can be directly derived from this power multivector. Additionally, this paper establishes a connection with the original p-q formulations and lays the foundations for time-instantaneous compensation (TIC) and time-average compensation (TAC). Finally, to validate the proposed model, simulation and experimental results from a three-phase four-wire industrial system using a novel approach are presented.

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]

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]

If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points. In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a 'best fit' rotor between these sets and compare our results with other methods. [ABSTRACT FROM AUTHOR]

An algorithm for finding a control of a linear switched system by means of Geometric Algebra is designed. More precisely, we develop a switching path searching algorithm for a two‐dimensional linear dynamical switched system with a non‐singular matrix whose integral curves are formed by two sets of centralized ellipses. It is natural to represent them as elements of Geometric Algebra for Conics and construct the switching path by calculating switching points, i.e., intersections and contact points. For this, we use symbolic algebra operations or, more precisely, the wedge and inner products that are realizable by sums of products in the coordinate form. Therefore, no numerical solver to the system of equations is needed. Indeed, the only operation that may bring in an inaccuracy is vector normalization, i.e., square root calculation. The resulting switching path is formed by pieces of ellipses that are chosen, respectively, from the two sets of integral curves. The switching points are either intersections in the first or final step of our algorithm, or contact points. This choice guarantees the optimality of the switching path with respect to the number of switches. Two examples are provided to demonstrate the search for the intersections of the conics and, consequently, a description is presented of the construction of a switching path in both cases. [ABSTRACT FROM AUTHOR]

Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis‐angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high‐noise scenarios. [ABSTRACT FROM AUTHOR]

Geometric algebras of dimension n<6$$ n<6 $$ are becoming increasingly popular for the modeling of 3D and 3 + 1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature‐agnostic analysis of these common operations in all geometric algebras of dimension n<6$$ n<6 $$ and gives efficient numerical implementations in the most popular algebras ℝ4,ℝ3,1,ℝ3,0,1$$ {\mathbb{R}}_4,{\mathbb{R}}_{3,1},{\mathbb{R}}_{3,0,1} $$, and ℝ4,1$$ {\mathbb{R}}_{4,1} $$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers. [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]

The aim of this work is to define quaternion curves and surfaces and their conjugates via operators in Euclidean projective geometric algebra (EPGA). In this space, quaternions were obtained by the geometric product of vector fields. New vector fields, which we call trajectory curves and surfaces, were obtained by using this new quaternion operator. Moreover, dual quaternion curves are determined by a similar method and then their generated motion is studied. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

As an addition to proper points of the real plane, we introduce a representation of improper points, i.e. points at infinity, in terms of Geometric Algebra for Conics (GAC) and offer possible use of both types of points. More precisely, we present two algorithms fitting a conic to a dataset with a certain number of points lying on the conic precisely, referred to as the waypoints. Furthermore, we consider inclusion of one or two improper waypoints, which leads to the asymptotic directions of the fitted conic. The number of used waypoints may be up to four and we classify all the cases. [ABSTRACT FROM AUTHOR]

The goal of this paper is to elucidate the hermitian structure of the algebra of geometric quaternions H (that is, the even algebra of the geometric algebra of the euclidean 3d space), use it to couch a geometric and dynamical presentation of the q-bit, and to see its bearing on the geometric structure of q-registers (arrangements of finite number of q-bits) and with that to pay a brief revisit to the formal structure of q-computations, with emphasis on the algebra structure of H ⊗ n . The main underlying theme is the unraveling of the subtle geometric relations between H and the sphere S 2 in the 3d euclidean space. [ABSTRACT FROM AUTHOR]

Copyright of Light Industry Machinery is the property of Light Industry Machinery Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Projective dualization in geometric algebras can encode them, or conversely, orientation types inform the interpretation of dualization. Oriented elements may be combined using the meet operation, and the dual‐join (which is here introduced for that purpose). We discuss algebra‐based implementation techniques on how to double the representational power of a geometric algebra, without explicitly doubling the algebra. [ABSTRACT FROM AUTHOR]

We present Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present Geometric Algebra Algorithms Optimizer (GAALOP) implementation of our approach. We demonstrate the ability to fully describe and compute with QRA in GAALOP using the geometric product. [ABSTRACT FROM AUTHOR]

3D human motion prediction is one of the essential and challenging problems in computer vision, which has attracted extensive research attention in the past decades. Many previous methods sought to predict the motion state of the next moment using the traditional recurrent neural network in Euclidean space. However, most methods did not explicitly exploit the relationships or constraints between different body components, which carry crucial information for motion prediction. In addition, human motion representation in Euclidean space has high distortion and shows a weak semantic expression when using deep learning models. Based on these observations, we propose a novel Geometric Algebra-based Multiscale Encoder-Decoder network (GAMEDnet) to predict the future 3D poses. In the encoder, the core module is a novel multiscale Geometric Algebra-based multiscale feature extractor(GA-MFE) , which extracts motion features given the multiscale human motion graph. In the decoder, we propose a novel GA-Graph-based Gated Recurrent Unit (GAG-GRU) to sequentially produce predictions. Extensive experiments are conducted to show that the proposed GAMEDnet outperforms state-of-the-art methods in both short and long-term motion prediction on the datasets of Human 3.6M, CMU Mocap. [ABSTRACT FROM AUTHOR]

Given a betweenness relation on a nonempty set E, a certain abelian group T = T E given in terms of generators and relations is investigated. This group controls the given betweenness relation in an algebraic form. That is, the group structure algebraically unfolds geometric relations, and in turn allows us to read off geometric properties from algebraic relations emerging from them. The most important examples for betweenness relations arise from ordered sets on the one side and from intervals in metric spaces on the other side. The structure of T will be determined completely in case of totally ordered sets as well as for several classes of metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science. [ABSTRACT FROM AUTHOR]

The Riemann curvature tensor is constructed using the Clifford-Dirac geometric algebra and division-algebraic operator structure. [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]