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Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan–Wigner transformation (JWT) and the Bravyi–Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

Protein structure prediction (PSP) is the prediction of the three‐dimensional (3D) folding of a protein (its tertiary structure) starting from its amino acid sequence (its primary structure). The state of the art in PSP is achieved by deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids positively impact the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of geometric algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA‐based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as additional input features to a graph transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle‐based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative and can be (i) closely associated to protein secondary structures and (ii) more easily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way. [ABSTRACT FROM AUTHOR]

We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate‐free Clifford quotient between a multidifference pseudo‐vector and a pseudo‐scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non‐degenerate simplex contracting to that point. Such result allows to fix an issue with a defective definition of pseudo‐scalar field in Sobczyck's simplicial calculus. Then, we provide some consequences and conjectures implied by the foregoing results. [ABSTRACT FROM AUTHOR]

Few-shot learning is challenging in computer vision tasks, which aims to learn novel visual concepts from few labeled samples. Metric-based learning methods are widely used in few-shot learning due to their simplicity and effectiveness. However, comparing the similarity of support samples and query samples in a single metric space appears to be biased. In this work, we design a dual-domain reciprocal metric network (DRM-Net) structure for few-shot classification task which establishes a commutative learning relationship in two feature distributions from different metric domains. Specifically, our reciprocal metric network contains two metric domains, which employ graph neural network (GNN) and geometric algebra graph neural network (GA-GNN) as two metric functions to comprehensively measure the similarity between samples. This structure can help reduce the prediction bias by a single measure. We also construct the reciprocal learning loss between the metric feature distributions from the two branches to promote each other to improve the performance of the overall model. Our extensive experimental results demonstrate that the proposed reciprocal metric learning outperforms existing state-of-the-art few-shot learning methods on various benchmark datasets. [ABSTRACT FROM AUTHOR]

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

The following paper talks about the dynamic modeling and control of robot manipulators using Hamilton's equations in the screw theory framework. The difference between the proposed work with diverse methods in the literature is the ease of obtaining the laws of control directly with screws and co-screws, which is considered modern robotics by diverse authors. In addition, geometric algebra (GA) is introduced as a simple and iterative tool to obtain screws and co-screws. On the other hand, such as the controllers, the Hamiltonian equations of motion (in the phase space) are developed using co-screws and screws, which is a novel approach to compute the dynamic equations for robots. Regarding the controllers, two laws of control are designed to ensure the error's convergence to zero. The controllers are computed using the traditional feedback linearization and the sliding mode control theory. The first one is easy to program and the second theory provides robustness for matched disturbances. On the other hand, to prove the stability of the closed loop system, different Lyapunov functions are computed with co-screws and screws to guarantee its convergence to zero. Finally, diverse simulations are illustrated to show a comparison of the designed controllers with the most famous approaches. [ABSTRACT FROM AUTHOR]

Tensor formalism (and its special case—vector formalism) is a mathematical technique that is widely used in physical and engineering problems. Even though this formalism is fairy universal and suitable for describing many spaces, the application of other special mathematical techniques is sometimes required. For example, the problem of rotation in a 3D space is not very well described in tensor representation, and it is reasonable to use the formalism of Clifford algebra, in particular, quaternions and geometric algebra representations for its solution. In this paper, computer algebra is used to demonstrate the solution of the problem of rotation in a 3D space using both the quaternion and geometric algebra formalisms. It is shown that although these formalisms are fundamentally similar, the latter one seems to be clearer both for computations and interpretation of results. [ABSTRACT FROM AUTHOR]

For a long time, non-sinusoidal, non-linear electric circuit analysis has been a prevalent topic. The supplementary analysis tool and domain are subjects of debate in many scientific groups, leading to a range of norms and definitions. Since its beginnings, the electric power system has advanced thanks to the development of Power Electronic devices, converters, and Renewable Energy sources. Electronic equipment has transformed the electrical system and brought industrial applications a plethora of benefits. Unfortunately, this results in distortions in the power system (voltage and current). For this, it is necessary to comprehend power flow in nonsinusoidal linear and non-linear circuit situations. Therefore, it is always necessary to use a distinct mathematical framework to examine the circuit in such a setting. Finally, an agreement on norms that adhere to well-known, established standards can be obtained under non-sinusoidal circumstances. To show the precision of geometric algebra in power flow calculations, the work presented here combines the usage of the harmonic domain and geometric algebra in circuits with disturbances for sinusoidal and non-sinusoidal excitation. [ABSTRACT FROM AUTHOR]

The geometric product represents a core concept for establishing geometric algebras and in case of vectors matches the formal sum of their inner product and their wedge product. The geometric product is reconsidered for the case of 3-vectors by means of usual matrix algebra in this article. Therefore, a symmetric matrix product and an antisymmetric matrix product are introduced, whose matrix sum yields a third product that renders the information of a vector pair's geometric product in terms of a matrix associated to the vectors. The three matrices - that correspond to inner product, wedge product, and geometric product, respectively - are named wheel product, curl product, and full product. The observation about the structural correspondence of the geometric product with matrix theory may be used for future practical computations and unveils connections of geometric algebra with related disciplines. [ABSTRACT FROM AUTHOR]

Arm-and-hand tracking by technological means allows gathering data that can be elaborated for determining gesture meaning. To this aim, machine learning (ML) algorithms have been mostly investigated looking for a balance between the highest recognition rate and the lowest recognition time. However, this balance comes mainly from statistical models, which are challenging to interpret. In contrast, we present $\mu C^{1}$ and $\mu C^{2}$ , two geometric model-based approaches to gesture recognition which support the visualization and geometrical interpretation of the recognition process. We compare $\mu C^{1}$ and $\mu C^{2}$ with respect to two classical ML algorithms, k-nearest neighbor (k-NN) and support vector machine (SVM), and two state-of-the-art (SotA) deep learning (DL) models, bidirectional long short-term memory (BiLSTM) and gated recurrent unit (GRU), on an experimental dataset of ten gesture classes from the Italian Sign Language (LIS), each repeated 100 times by five inexperienced non-native signers, and gathered with wearable technology (a sensory glove and inertial measurement units). As a result, we achieve a compromise between high recognition rates ($>90\%$) and low recognition times ($ < 0.1 {\mathrm{ s}}$) that is adequate for human–computer interaction. Moreover, we elaborate on the algorithms’ geometric interpretation based on geometric algebra, which supports some understanding of the recognition process. [ABSTRACT FROM AUTHOR]

This paper proposes a systematic method for the identification of the load circuit parameters (say the $\boldsymbol{R}$ , $\boldsymbol{L}$ , and $\boldsymbol{C}$ elements) based only on the information of the instantaneous voltage and current measured at the point of common coupling (pcc). Geometric Algebra (GA) and concepts of differential geometry are used to produce a rigorous mathematical framework. The identification is formulated as a multidimensional geometrical problem that is solved conveniently by means of GA. Once the passive elements of the load have been identified, the active and reactive powers can be computed from first electromagnetic principles (Maxwell Equations). The theory is general and is verified with linear and nonlinear circuits. The paper shows single-phase circuits but the theory can be extended to three-phase circuits. The method is easy to program and has shown to be very robust for all tested cases. Because of its generality, the method presented will find applications beyond electric circuits. [ABSTRACT FROM AUTHOR]

A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange's identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange's identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices. [ABSTRACT FROM AUTHOR]

In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology. [ABSTRACT FROM AUTHOR]

To achieve 3D skeleton-based human motion prediction, attention-based methods have encouraged performance due to the observation that attention parts in the past state influence future actions. The model can identify the most relevant information for motion prediction by introducing an attention mechanism. However, existing methods tend to address the general allocation of attention without further exploiting the precision to pinpoint the exact location of the most relevant information. This oversight subsequently curtails the potential of prediction performance. To solve this problem, we propose a novel Gradient Multi-Foci Network (GMFnet), which leverages two-stage foci: spectral focus and spatial focus, to find the most pertinent information in a manner that emulates natural cognitive processes. The core idea of the proposed GMFnet is based on two aspects: spectral focus to model the repeatability of the observation action sequence by deploying an attention-based Related Sequences Directing Block (RSDB), spatial focus to capture the most valuable parts between motion joints by using Attention Feature Computational Unit (AFCU). Extensive experiments are conducted to reveal that GMFnet can capture the precision to pinpoint the exact attention location, thus enhancing the prediction performance. The proposed GMFnet outperforms state-of-the-art methods by 10.7 and 7.4% of MPJPEs for short-term and long-term prediction in Human 3.6M and by 14.3 and 4.8% of MPJPEs for short-term and long-term forecast in CMU-Mocap. Moreover, GMFnet outperforms even more in short term by 24.9% in AMASS. The code is available at https://github.com/JunyuShi02/GMFNet. [ABSTRACT FROM AUTHOR]

The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6 × n or n × n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C , such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation. [ABSTRACT FROM AUTHOR]

Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and Depenbrock's FBD transformations can be derived by imposing orthogonality on the voltage and current vectors defined in a Euclidean space by using GA. This paper presents these transformations as spatial‐like rotations and projections by means of the use of special algebraic objects named GA rotors. We prove that there is no need to use complex numbers nor matrices to perform the mentioned transformations and to provide geometrical intuition for electrical quantities and their transformations. Furthermore, power properties can be described using GA terminology by means of the proposed geometric power. The manipulation of the geometric power allows a useful current decomposition for a variety of applications such active filtering, frequency estimation, or electrical machinery. We provide in this paper an alternative approach focused on power systems under the paradigm of GA. [ABSTRACT FROM AUTHOR]

Since the introduction of quaternion by Hamilton in 1843, quaternions have been used in a lot of applications. One of the most interesting qualities is that we can use quaternions to carry out rotations and operate on other quaternions; this characteristic of the quaternions inspired us to investigate how the quantum states and quantum operator work in the field of quaternions and how we can use it to construct a quantum neural network. This new type of quantum neural network (QNN) is developed in the quaternion algebra framework that is isomorphic to the rotor algebra G 3 + of the geometric algebra and is based on the so-called qubit neuron model. The quaternion quantum neural network (QQNN) is tested and shows robust performance. [ABSTRACT FROM AUTHOR]

In this paper, power flows in electrical circuits are modelled in a mixed time-frequency domain by using geometric algebra and the Hilbert transform for the first time. The use of this mathematical framework overcomes some of the limitations of some of the existing methodologies, in which the so-called "active current" may not lead to the lowest Root Mean Square (RMS) current under distorted supply or unbalanced load. Moreover, this current may contain higher levels of harmonic distortion compared to the supply voltage. The proposed method can be used for sinusoidal and non-sinusoidal power supplies, non-linear loads and single- and multi-phase electrical circuits, and it provides meaningful engineering results with a compact formulation. It can also serve as an advanced tool for developing algorithms in the power electronics field. Several examples have been included to verify the validity of the proposed theory. [ABSTRACT FROM AUTHOR]

Noise and interference are the two most common and basic problems in wireless communication systems. The noise in wireless communication channels has the characteristics of randomness and impulsivity, so the performance of adaptive filtering algorithms based on geometric algebra (GA) and second-order statistics is greatly reduced in the wireless communication systems. In order to improve the performance of adaptive filtering algorithms in wireless communication systems, this paper proposes two novel GA-based adaptive filtering algorithms, which are deduced from the robust algorithms based on the minimum error entropy (MEE) criterion and the joint criterion (MSEMEE) of the MEE and the mean square error (MSE) with the help of GA theory. The noise interference in wireless communication is modeled by α -stable distribution which is in good agreement with the actual data in this paper. Simulation results show that for the mean square deviation (MSD) learning curve, the GA-based MEE (GA-MEE) algorithm has faster convergence rate and better steady-state accuracy compared to the GA-based maximum correntropy criterion algorithm (GA-MCC) under the same generalized signal-to-noise ratio (GSNR). The GA-MEE algorithm reduces the convergence rate, but improves the steady-state accuracy by 10–15 dB compared to the adaptive filtering algorithms based on GA and second-order statistics. For GA-based MSEMEE (GA-MSEMEE) algorithm, when GA-MSEMEE and the adaptive filtering algorithms based on GA and second-order statistics keep the same convergence rate, its steady-state accuracy is improved by 10–15 dB, and when GA-MSEMEE and GA-MEE maintain approximately steady-state accuracy, its convergence rate is improved by nearly 100 iterations. In addition, when the algorithms are applied to noise cancellation, the average recovery error of the two proposed algorithms is 7 points lower than that of other GA-based adaptive filtering algorithms. The results validate the effectiveness and superiority of the GA-MEE and GA-MSEMEE algorithms in the α -stable noise environment, providing new methods to deal with multi-channel interference in wireless networks. [ABSTRACT FROM AUTHOR]

Inspired by Laughlin's theory of the fractional quantum Hall effect, we explore the topological nature of the quark–gluon plasma (QGP) and the nucleons in the context of the Clifford algebra. In our model, each quark is transformed into a composite particle via the simultaneous attachment of a spin monopole and an isospin monopole. This is induced by a novel kind of meson endowed with both spin and isospin degrees of freedom. The interactions in the strongly coupled quark–gluon system are governed by the topological winding number of the monopoles, which is an odd integer to ensure that the overall wave function is antisymmetric. The states of the QGP and the nucleons are thus uniquely determined by the combination of the monopole winding number m and the total quark number N. The radius squared of the QGP droplet is expected to be proportional to m N. We anticipate the observation of such proportionality in the heavy ion collision experiments. [ABSTRACT FROM AUTHOR]

Geometric algebra is a useful tool to overcome some problems in kinematics. Thus, the geometric algebra has attracted the attention of many researchers. In this paper, quaternion operators on curves and surfaces in Euclidean 3-space are defined by using geometric algebra. These operators generate the curves or the surfaces from the points, curves or surfaces. Using quaternion operators, we obtain motions that have orbits along the generated curve or surface. Also, these motions are expressed as 1-parameter or 2-parameter homothetic motions. [ABSTRACT FROM AUTHOR]

Summary: This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non‐sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. The analysis and understanding of how AC circuits operate in steady state are of a paramount importance for all the electrical engineers and practitioners around the world. Typically, lecturers of most undergraduate courses teach circuit theory using complex phasors, vector calculus, or linear algebra. However, these approaches have some important limitations in practice, which requires the development of strategies to improve teaching‐learning process related to AC circuit analysis. By formulating a new mathematical framework, the paper presents and discusses how GA can be of help in this approach. It is also described how the results obtained by using GA are validated using computer‐based simulations. It is highlighted how the proposed teaching methodology based on GA theory can be effective in helping students learn AC circuit analysis since it has several potential benefits derived from its simplicity, compactness, and use of easily identifiable geometrical concepts. [ABSTRACT FROM AUTHOR]

A hidden-variable model for quantum–mechanical spin, as represented by the Pauli spin operators, is proposed for systems illustrating the well-known no-hidden-variables arguments by Peres (Phys Lett A 151:107–108, 1990) and Mermin (Phys Rev Lett 65:3373–3376, 1990) and by Greenberger et al. (Bell's theorem, quantum theory, and conceptions of the universe, Kluwer, Dordrecht, 1989). Both arguments rely on an assumption of non-contextuality; the latter argument can also be phrased as a non-locality argument, using a locality assumption. The model suggested here is compatible with both assumptions. This is possible because the scalar values of spin observables are replaced by vectors that are components of orientations. [ABSTRACT FROM AUTHOR]

We present an eight-dimensional even sub-algebra of the 2 4 = 16 -dimensional associative Clifford algebra C l 4 , 0 and show that its eight-dimensional elements denoted as X and Y respect the norm relation ‖ XY ‖ = ‖ X ‖ ‖ Y ‖ , thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere. [ABSTRACT FROM AUTHOR]

This research paper focuses on quaternion neural networks (QNNs) - a type of neural network wherein the weights, biases, and input values are all represented as quaternion numbers. Previous studies have shown that QNNs outperform real-valued neural networks in basic tasks and have potential in high-dimensional problem spaces. However, research on QNNs has been fragmented, with contributions from different mathematical and engineering domains leading to unintentional overlap in QNN literature. This work aims to unify existing research by evaluating four distinct QNN backpropagation algorithms, including the novel GHR-calculus backpropagation algorithm, and providing concise, scalable implementations of each algorithm using a modern compiled programming language. Additionally, the authors apply a robust Design of Experiments (DoE) methodology to compare the accuracy and runtime of each algorithm. The experiments demonstrate that the Clifford Multilayer Perceptron (CMLP) learning algorithm results in statistically significant improvements in network test set accuracy while maintaining comparable runtime performance to the other three algorithms in four distinct regression tasks. By unifying existing research and comparing different QNN training algorithms, this work develops a state-of-the-art baseline and provides important insights into the potential of QNNs for solving high-dimensional problems. • Clifford Multilayer Perceptrons outperform other quaternion neural network methods. • Multilayer GHR Calc. networks approximate real nonlinear functions reasonably well. • Classic QNN backpropagation techniques are outdated and should be avoided. [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]

Ray-tracing is an efficient asymptotic computational electromagnetic method for studying wave propagation. The major bottleneck of ray-tracing algorithms is the computational complexity of the geometric-processing steps. There is a clear need for designing efficient ray-tracing algorithms and implementations in computational electromagnetics applications. We use Geometric algebra for the formulation of the pre-processing of electromagnetic ray tracing. Geometric algebra provides unified mathematical representations of geometric primitives and transformations. Various kinds of geometric algebras are currently used in many practical applications. We develop a unified 2D/3D ray-tracing procedure based on the homogeneous geometric algebra. The procedure is implemented using generative programming for optimizing code compilation. Results show that combining the powerful mathematical modeling capabilities of geometric algebra with efficient automatic code generation resulted in better performance of the ray-tracing implementation compared to typical approaches. [ABSTRACT FROM AUTHOR]

Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E² or E³ space. The projection operation from the E³ to the E² space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E² space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. The algorithm can be efficiently modified for the SSE4 instruction use or the GPU application, too. [ABSTRACT FROM AUTHOR]

Geometric problems are usually solved in the Euclidean space by using the standard vector algebra techniques. In this study, principles of the projective geometry and geometric algebra will be introduced via a novel method that significantly simplifies the solution of geometrical problems. Also, it supports the GPU parallel computation application. Besides that, an application of the principle of duality leads to a simple solution of the dual problems. We show that, the equivalence of the extended cross-product (outer product) and the solution of the system of linear equations. This gives a direct impact to scientific computation, solution of geometrical problems, robotics, computer graphics algorithms and virtual reality via fast computation through GPU parallel systems. Some numerical and graphical results are presented. [ABSTRACT FROM AUTHOR]

Directional solvent extraction (DSE) has been gaining interest as a water treatment technology in recent years. DSE utilizes the process of micellization for the purposes of species separation between water and complex chemical systems. In this article, we develop a conformal geometric algebra‐based formulation that models surfactants, their solubilities, and critical micelle concentration (CMC), with relation to temperature and pressure. Molecules are represented as spatially distributed networks embedded in R4,1 space, and the mathematical characterizations of these molecules are shown to be effective in modelling CMC as a function of temperature and pressure. One of the contributions of this work is the utilization of this formulation to develop a governing expression, in the form of a three‐dimensional relationship, between CMC, pressure, and temperature for a general surfactant. In prior works, the CMC–temperature plane and CMC–pressure plane expressions have been extensively documented for sodium alkyl sulfates. In this work, we extend the formulation to model the CMC of decanoic acid, sodium octyl sulfate, sodium decyl sulfate, sodium dodecyl sulfate, and sodium tetradecyl sulfate. Using this theoretical model, a relationship between CMC and the directional solubility of water in a surfactant is determined. Directional solubility is related to temperature and pressure, and on this basis, we devise a directional solubility–pressure–temperature expression for an arbitrary surfactant to improve the state of the art for DSE. From this expression, we propose a novel isothermal DSE process for water treatment. [ABSTRACT FROM AUTHOR]

• Geometric algebra framework is applied to solve non sinusoidal and nonlinear circuits. • Power flow direction and magnitude for interharmonics is solved. • The geometric algebra framework applied to power systems is improved with the inclusion of interharmonic representation. • This new method is validated through some examples with linear and non-linear loads. The calculation of power flow in power systems with the presence of harmonics has been properly studied in the scientific literature. However, power flow calculation considering interharmonic components is still an open question. Traditional methods based on the IEEE1459 standard have proven to be valid and accurate only for linear and sinusoidal systems, but have been criticized for non-linear and non-sinusoidal systems because they are not able to explain correctly the current and voltage interactions beyond the active power. This paper proposes the use of a novel mathematical framework called geometric algebra (GA) to study the power flow considering the interaction of current and voltage harmonics and interharmonics. The use of GA enables the precise determination of the direction and magnitude of the total and single active power flow for each component, as well as other power elements related to the non-active power due to cross interaction. Moreover, this paper makes a novel contribution to the definition of interharmonics in geometric algebra space that has not been done before. To test the validity of the method, both linear and non-linear circuits are proposed and solved by applying voltages and currents with harmonic and interharmonic components. The results obtained show that power flow can be analyzed under the prism of the principle of energy conservation (PoCoE) in a way that allows a better understanding of the power spectrum due to the interaction of harmonics and interharmonics of voltage and current. [ABSTRACT FROM AUTHOR]

We present a formulation of the Boolean Satisfiability Problem in spinor language that allows to give a necessary and sufficient condition for unsatisfiability. With this result we outline an algorithm to test for unsatisfiability with possibly interesting theoretical properties. [ABSTRACT FROM AUTHOR]

Due to the increasing complexity of GIS data and service modes, there is an urgent need for the next generation of GIS with new representation and computation methods. A number of spatiotemporal models, analytical and visualization methods, as well as system architectures have been proposed. However, previous studies failed to integrate basic geographical theories with latest computing technologies. Without a well-defined body of underlying theories, new models and methods are limited in scope and not able to meet the ultimate requirements of the next-generation GIS, which demands multidimensional, highly dynamic and semantic-rich representations and computational power. Geometric algebra (GA) provides an ideal tool for the expression and calculation of multidimensional geometric objects, and has proved to be effective for GIS representation and computation applications in our previous studies. We propose to use GA as the basic mathematical language for the establishment of the next-generation GIS. We present the framework of a GA-based next-generation GIS and describe the representation space, data structure, and computational models in this paper. A few issues that have not been sufficiently addressed by previous studies are discussed in detail with potential solutions proposed. These include multi-scale representations, modelling of geographic processes, simulation of geographic interactions, and multi-element modelling. The GA-based next-generation GIS uses an integrated structure consisting of a theoretical architecture, model for information expression, and computational methods. Implementation of the approach aims to improve GIS capacities in applications such as global spatiotemporal modelling and analysis, regional geographic modelling and simulation, smart city applications, and many others. [ABSTRACT FROM AUTHOR]

• Widely linear, semi-widely linear and strictly linear AFs are constructed in the GA domain. As different from the GA-LMS, the GA-LMS algorithm proposed in this paper can be directly derived from the quaternion LMS, complex LMS and real LMS by setting the orthonormal basis parameter n accordingly. • To lower the computational complexity of the GA-LMS algorithms, we propose partial-update variants of the SL-GA-LMS, SWL-GA-LMS and WL-GA-LMS algorithms based on sequential, stochastic and M-max partial updates. • A detailed performance analysis of the PU-GA-LMS algorithms is provided. • The partial-update quaternion LMS (PU-QLMS), which is an isomorphism to the PU-GA-LMS, is investigated in detail. Geometric-algebra based adaptive filters have been successfully employed in many applications such as computer vision, data fusion and linear prediction where the unknown parameters of interest are high-dimensional multivectors. However, conventional geometric-algebra adaptive filters, such as the strictly linear geometric-algebra least mean square (SL-GA-LMS) algorithm, are only applicable to circular multivector-valued inputs with rotation-invariant probability distribution functions. To remove this limitation, we propose new semi-widely linear and widely linear GA-LMS algorithms. As geometric-algebra adaptive filters can have extremely high computational complexity, partial-update variants of these algorithms with reduced complexity are also developed employing stochastic, sequential and M -max partial updating strategies. Steady-state and transient performances of the proposed partial-update algorithms are analysed. As an isomorphism to the partial-update GA-LMS algorithms, widely linear, semi-widely linear and strictly linear quaternion LMS algorithms with partial updates are proposed and analysed for noncircular quaternion inputs. Finally, numerical studies are carried out to confirm the advantages of the proposed methods and the convergence analysis results for multivector and quaternion-valued inputs. [ABSTRACT FROM AUTHOR]