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Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA). [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 study of signal processing has recently devoted significantly more attention to adaptive filtering techniques. By addressing the shortcoming of the conventional geometric algebra-based fixed step-size least mean square algorithm that is unable to satisfy in terms of both reducing steady-state error and a faster convergence speed, simultaneously, this study presents an improved logarithmic function-based variable step-size least mean square geometric algebra adaptive filtering algorithm by establishing the step-size factor μ and error signal e(n) nonlinear function relationship. The instantaneous values of a current error estimate e(n) and the previous error estimate e (n - 1) are used to determine the step size of the defined algorithm. Besides, an extensive discussion is given on the performance of algorithm under influence of parameters γ and T as well as comparative analysis with other existing geometric algebra-based adaptive filters. Computer simulation reveals that the proposed approach not only has a low steady-state error, robustness against impulsive noise, and fast convergence speed, but it also overcomes some existing algorithm's instability under steady-state phase. [ABSTRACT FROM AUTHOR]

With regards to the problem of multidimensional signal processing in the field of adaptive filtering, geometric algebra based higher-order statistics algorithms were proposed. For instance, to express a multidimensional signal as a multi-vector, the adaptive filtering algorithms described in this work leverage all of the benefits of GA theory in multidimensional signal processing. GA space is employed to extend the traditional least-mean absolute third (LMAT) and newly deduced least-mean mixed third-fourth (LMMTF) adaptive filtering methods for multi-dimensional signal processing. The objective of the presented GA-based least-mean absolute third (GA-LMAT) and GA-based least-mean mixed third-fourth (GA-LMMTF) algorithms is to minimize the cost functions by using higher-order statistics of the error signal e(n) in GA space. The simulation's results revealed that at significantly smaller step size, the given GA-LMAT algorithm is better than the others in terms of steady-state error and convergence rate. Besides, the defined GA-LMMTF algorithm mitigates for the instability of GA-LMAT as the step size increases and illustrates an improved performance relative to mean absolute error and convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. [ABSTRACT FROM AUTHOR]

From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time. [ABSTRACT FROM AUTHOR]

Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure. [ABSTRACT FROM AUTHOR]

Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist spaces are constructed by the join operator. For coupled limbs coupled with closed loops in the MCM, the equivalent limbs can be found by solving the analytical expressions of the twist space on each closed loop’s output link. Then, the twist spaces of the coupled limbs can be easily obtained. The twist space of the MCM’s output link is the intersection of all the limb twist spaces, which can be calculated by the meet operator. The proposed method provides a simplified way of analyzing the mobility of MCMs, and three typical MCMs are chosen to validate this method. The analytical mobility of the MCM’s output link can be obtained, and it naturally indicates both the number and the property of the degrees of freedom (DOFs). [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]

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]

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]

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]

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]

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]

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]

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]

Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s, but recursive in n. Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation. [ABSTRACT FROM AUTHOR]

The paper presents an initial step towards employing the advantages of educational theatre and implementing them into social robotics in order to enhance the emotional skills of a child and at the same time to augment robots with actors' emotional talent. Emotional child-robot interaction helps to catch quickly a child's attention and enhance information perception during learning and verbalization in children with communication disorders. An innovative approach for learning through art by transferring actors' emotional and social talents to socially assistive robots is presented and the technical and artistic challenges of tracking and translating movements expressing emotions from an actor to a robot are considered. The goal is to augment the robot intervention in order to enhance a child's learning skills by stimulating attention, improving timing of understanding emotions, establishing emotional contact and teamwork. The paper introduces a novel approach to capture movements and expressions of a human head, to process data from brain and inertial tracking devices and to transfer them into a socially assistive robot. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) show powerful processing ability on graph structure data for nodes and graph classification. However, existing GNN models may cause information loss with the increasing number of the network layer. To improve the graph-structured data features representation quality, we introduce geometric algebra into graph neural networks. In this paper, we construct a high-dimensional geometric algebra (GA) space in the Non-Euclidean domain to better learn vector embedding for graph nodes. We focus our study on few-shot learning and propose a geometric algebra graph neural network (GA-GNN) as the metric network for cross-domain few-shot classification tasks. In the geometric algebra space, the feature nodes are mapped into hyper-complex vector, which helps reduce the distortion of feature information with the increased hidden layers. The experimental results demonstrate that the approach we proposed achieves the state-of-the-art few-shot cross-domain classification accuracy in five public datasets. [ABSTRACT FROM AUTHOR]

This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis. We go on to show how operations and structure such as involutions, versors, and the Z 2 -grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and N -grading, and some of the gaps in mathlib that currently make this challenging. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic modeling and generalized force analysis of three-(rotation pair)-(prismatic pair)-spherical pair (3-RPS) parallel mechanism were carried out for the first time based on the five-dimensional geometric algebra space—(4,0,1) conformal geometric. Compared with the traditional homogeneous matrix method, the maximum error values of generalized force of the three branch chains are 1.90 × 10 - 4 N , 1.39 × 10 - 4 N and 6.0 × 10 - 5 N ,respectively. The results are basically consistent with the homogeneous matrix method. For composite rigid body transformation of two rotational motions, the rotation matrix method needs 27 times of multiplication and 18 times of addition, while the conformal geometric method only needs 16 times of multiplication and 15 times of addition. The computational efficiency of this method can be improved. In five-dimensional space, derivative operation can be linearly mapped to multiplication operation in three-dimensional space, so that dynamic equation has no derivative term. The dynamic model can separate variables of known and unknown, and realize parallel computation. This method provides a new idea for dynamic modeling and solving of parallel mechanism. [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]

Lyapunov stability theory for smooth nonlinear autonomous dynamical systems is presented in terms of Geometric Algebra. The system is described by a smooth nonlinear state vector differential equation, driven by a vector field in R n . The level sets of the scalar Lyapunov function candidate are assumed to be compact smooth vector manifolds in R n . The level sets induce an associated global foliation of the state space. On any leaf of this foliation, a geometric subalgebra is naturally attached to the corresponding tangent vector space of the smooth vector manifold. The pseudoscalar (field) of this subalgebra completely characterizes the tangent space. Asymptotic stability of the system equilibria is described in terms of equilibria of, easily computable, rejection vector fields with respect to the pseudoscalar field. Nonexistence of invariant sets of the Lyapunov function directional derivative, along the defining vector field, are also tested using a simple tangency condition. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We describe a possibility for geometric calculation of specific conics' intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position. Then we show how symbolic operations may be calculated directly in GAALOPWeb software, that the basis coefficients may be read off in the appropriate basis and, moreover, the result may be immediately and truly visualized. We compare the functionality with Maple package Clifford. [ABSTRACT FROM AUTHOR]

In this paper, we present Geometric Algebra as a powerful language to describe quantum operations using its geometric intuitiveness. Using the web-based GAALOPWeb, an online geometric algebra algorithm optimizer for computing with qubits, we describe new formulations for the NOT operation, as well as a strategy to describe the Z gate and especially the Hadamard operation both for one and multiple qubits. [ABSTRACT FROM AUTHOR]

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case n = 4 , the proofs for the case n = 5 and the case of arbitrary dimension n. The results can be used in symbolic computation. [ABSTRACT FROM AUTHOR]

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties. [ABSTRACT FROM AUTHOR]

Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals. [ABSTRACT FROM AUTHOR]