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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]

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]

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

The Geometric Algebra Fulcrum Library (GA-FuL) version 1.0 is introduced in this paper as a comprehensive computational library for geometric algebra (GA) and Clifford algebra (CA), in addition to other classical algebras. As a sophisticated software system, GA-FuL is useful for practical applications requiring numerical or symbolic prototyping, optimized code generation, and geometric visualization. A comprehensive overview of the GA-FuL design is provided, including its core design intentions, data-driven programming characteristics, and extensible layered design. The library is capable of representing and manipulating sparse multivectors of any dimension, scalar kind, or metric signature, including conformal and projective geometric algebras. Several practical and illustrative use cases of the library are provided to highlight its potential for mathematical, scientific, and engineering applications. The metaprogramming code optimization capabilities of GA-FuL are found to be unique among other software systems. This allows for the automated production of highly efficient code, based on powerful geometric modeling formulations provided by geometric algebra.

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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]

The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms.

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.

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]

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]

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]

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]

Within a Geometric Algebra (GA) framework, this article presents a general method for synthesis of sliding mode (SM) controllers in Single Input Single Output (SISO) switched nonlinear systems. The method, addressed as the invariance control method, rests on a reinterpretation of the necessary and sufficient conditions for the local existence of a sliding regime on a given smooth manifold. This consideration leads to a natural decomposition of the SM control scheme resulting in an invariance state feedback controller feeding a Delta–Sigma modulator that, ultimately, provides the required binary-valued switched input to the plant. As application examples, the obtained results are used to illustrate the design of an invariance controller for a switched power converter system. Using the invariance control design procedure, it is shown how well-known second order sliding regime algorithms can be obtained, via a limiting process, from traditional sliding regimes induced on linear sliding manifolds for certain nonlinear switched systems. [ABSTRACT FROM AUTHOR]

We introduce several modifications of conic fitting in Geometric algebra for conics by incorporating additional conditions into the optimisation problem. Each of these extra conditions ensure additional geometric properties of a fitted conic, in particular, centre point position at the origin of coordinate system, axial alignment with coordinate axes, or, eventually, combination of both. All derived algorithms are accompanied by a discussion of the underlying algebra and computational optimisation issues. Finally, we present examples of use on a sample dataset and offer possible applications of the algorithms. [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]

Abstract Compared with the traditional manipulator, the hyper‐redundant manipulator has the advantage of high flexibility, which is particularly suitable for all kinds of complex working environments. However, the complex space environment requires the hyper‐redundant manipulator to have stronger obstacle avoidance ability and adaptability. In order to solve the problems of a large amount of calculation and poor obstacle avoidance effects in the path planning of the hyper‐redundant manipulator, this paper introduces the ‘backbone curve’ approach, which transforms the problem of solving joint path points into the behaviour of determining the backbone curve. After the backbone curve approach is used to design the curve that meets the requirements of obstacle avoidance and the end pose, the least squares fitting and the improved space joint fitting are used to match the plane curve and the space curve respectively, and the angle value of each joint of the manipulator is limited by the algorithm. Furthermore, a fusion obstacle avoidance algorithm is proposed to obtain the joint path points of the hyper‐redundant manipulator. Compared with the classic Jacobian iteration method, this method can avoid obstacles better, has the advantages of simple calculation, high efficiency, and can fully reflect the geometric characteristics of the manipulator. Simulation experiments have proven the feasibility of the algorithm.

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]

Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer–Weyl (Am J Math 57: 425–449, 1935, ) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted. [ABSTRACT FROM AUTHOR]

As shared, collaborative, networked, virtual environments become increasingly popular, various challenges arise regarding the efficient transmission of model and scene transformation data over the network. As user immersion and real-time interactions heavily depend on VR stream synchronization, transmitting the entire data set does not seem a suitable approach, especially for sessions involving a large number of users. Session recording is another momentum-gaining feature of VR applications that also faces the same challenge. The selection of a suitable data format can reduce the occupied volume, while it may also allow effective replication of the VR session and optimized post-processing for analytics and deep-learning algorithms. In this work, we propose two algorithms that can be applied in the context of a networked multiplayer VR session, to efficiently transmit the displacement and orientation data from the users' hand-based VR HMDs. Moreover, we present a novel method for effective VR recording of the data exchanged in such a session. Our algorithms, based on the use of dual-quaternions and multivectors, impact the network consumption rate and are highly effective in scenarios involving multiple users. By sending less data over the network and interpolating the in-between frames locally, we manage to obtain better visual results than current state-of-the-art methods. Lastly, we prove that, for recording purposes, storing less data and interpolating them on-demand yields a data set quantitatively close to the original one. [ABSTRACT FROM AUTHOR]

This paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations. [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]

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]

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]

Geometric algebra (GA) is proposed as a mathematical framework for revisiting fundamental aspects of sliding mode control (SMC) in nonlinear, switch‐controlled, single input systems. Sliding mode existence conditions, the switching policy, the invariance conditions, the associated equivalent control, and the characterization of ideal sliding dynamics, are all re‐examined using a geometric algebra (GA) standpoint. Two illustrative examples, from switched power electronics, are presented using the GA language. [ABSTRACT FROM AUTHOR]

Abstract 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 $$\alpha$$ 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 $$\alpha$$ α -stable noise environment, providing new methods to deal with multi-channel interference in wireless networks.

Recent RNN models deal with various dimensions of MTS as independent channels, which may lead to the loss of dependencies between different dimensions or the loss of associated information between each dimension and the global. To process MTS in a holistic way without losing the inter-relationship among dimensions, this paper proposes a novel Long-and Short-term Time-series network based on geometric algebra (GA), dubbed GA-LSTNet. Specifically, taking advantage of GA, multi-dimensional data at each time point of MTS is represented as GA multi-vectors to capture the inherent structures and preserve the correlation of those dimensions. In particular, traditional real-valued RNN, real-valued LSTM, and the back-propagation through time are extended to the GA domain. We evaluate the performance of the proposed GA-LSTNet model in prediction tasks on four well-known MTS datasets, and compared the prediction performance with other six methods. The experimental results indicate that our GA-LSTNet model outperforms traditional real-valued LSTNet with higher prediction accuracy, providing a more accurate solution for the existing shortcomings of MTS prediction models. [ABSTRACT FROM AUTHOR]

In conventional machine learning applications, each data attribute is assumed to be orthogonal to others. Namely, every pair of dimension is orthogonal to each other and thus there is no distinction of in-between relations of dimensions. However, this is certainly not the case in real world signals which naturally originate from a spatio-temporal configuration. As a result, the conventional vectorization process disrupts all of the spatio-temporal information about the order/place of data whether it be 1D, 2D, 3D, or 4D. In this paper, the problem of orthogonality is first investigated through conventional k-means of images, where images are to be processed as vectors. As a solution, shift-invariant k-means is proposed in a novel framework with the help of sparse representations. A generalization of shift-invariant k-means, convolutional dictionary learning is then utilized as an unsupervised feature extraction method for classification. Experiments suggest that Gabor feature extraction as a simulation of shallow convolutional neural networks provides a little better performance compared to convolutional dictionary learning. Other alternatives of convolutional-logic are also discussed for spatio-temporal information preservation, including a spatio-temporal hypercomplex encoding scheme. [ABSTRACT FROM AUTHOR]

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras G p , q of vector space of dimension n = p + q . We present basis-free formulas for all characteristic polynomial coefficients in the cases n ≤ 6 , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case n = 4 , and one of the formulas in the case n = 5 . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases n ≤ 5 . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation. [ABSTRACT FROM AUTHOR]

There is increasing demand for multi-level declassification of geographic vector field data in the big data era. Different from traditional encryption, declassification does not aim at making the original data unavailable through perturbation and transformation. During declassification process, the general geospatial features are usually retained but the detailed information is hidden from the perspective of data security. Furthermore, when faced with different levels of confidentiality, different levels of declassification are needed. In this paper, A declassification and reversion method with multi-level schemes is realized under the geometric algebra (GA) framework. In our method, the geographic vector field data is uniformly expressed as a GA object. Then, the declassification methods are proposed for vector field data with the rotor operator and perturbation operator. The declassification methods can progressively hide the detailed information of the vector field by vector rotating and vector perturbating. To make our method more unified and adaptive, a GA declassification operator is also constructed to realize the declassification computing of geographic vector field data. Our method is evaluated quantitatively by comparing the numerical and structure characterization of the declassification results with the original data. Divergence and curl calculating results are also compared to evaluate the reanalysis ability of the declassification results. Experiments have shown that our method can perform effective multi-level controls and has good randomness and a high degree of freedom in numerical and structure characteristics of geophysical vector field data. The method can well capture the application needs of geographic vector field data in data disclosure, secure transmission, encapsulation storage, and other aspects. [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]

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]

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]