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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]

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 • Classic QNN backpropagation techniques are outdated and should be avoided.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]

• 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]

• 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]

A free-form motion can be considered as a smoothly varying rigid-body transformation. Motions can be created by establishing functions in an appropriate space of matrices. While a smooth motion is created, the geometry of the motion itself is not always immediately clear. In a geometric algebra environment, motions can be created using extensions of the ideas of Bézier and B-spline curves and the geometric significance of the construction is clearer. A motion passing through given precision poses can be obtained by direct analogy with the curve approach. This paper considers the more difficult problem of dealing additionally with velocity constraints at the ends of the motion: here the analogy is less obvious. A geometric construction for the end pairs of control poses is established and is demonstrated by creating motions satisfying given pose and velocity constraints. [ABSTRACT FROM AUTHOR]

• The proposed method discovers the repeated motion parts in given historical motion sequences. • The proposed method alleviates the over-smoothing problems in deep GCNs structure. • The proposed method analyzes and predicts human motions with reasonable accuracy. 3D skeleton-based human motion prediction is an essential and challenging task for human-machine interactions, which aims to forecasts future poses given a history of their previous motions. Recent works based on Graph Neural Networks (GCNs) show promising performance for motion prediction due to the powerful ability of feature aggregation of GCNs. However, with the deep and multi-stage GCN model deployment, its feature extraction mechanism tends to result in feature similarity over all joints, and degrade the prediction performance. In addition, such a graph structure in recent works was still insufficient to process the high dimensional structural data in Euclidean space when inference through multi-layer networks. To solve the problem, we propose a novel Geometric Algebra-based Multi-view Interaction network (GA-MIN), which captures and aggregates motion features from two interactions: 1) global-interaction, which refactors various spectrum dependencies using geometric algebra-based structure, and 2) self-interaction, which leverage self-attention mechanism to capture compact representations. Extensive experiments are conducted on three public datasets: Human3.6M, CMU Mocap, and 3DPW, which prove that the proposed GA-MIN outperforms state-of-the-art methods on 3D Mean Per Joint Position Error (MPJPE) and Mean Angle Error (MAE) on average. [ABSTRACT FROM AUTHOR]

In this work, by reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the interpolation, virtual reality, graphics engineering, haptics. We derive intuitive geometric equations to handle surface operations like in kidney surgery. The interpolation can handle the interpolation and dilation in 3D of points, lines, planes, circles and spheres. With this procedure, we interpolate trajectories of surgical instrument. Using quaternions, we formulate the quaternion spike neural network for control. This new neural network structure is based on Spike Neural Networks and developed using the quaternion algebra. The real valued training algorithm was extended so that it could make adjustments of the weights according to the properties and product of the quaternion algebra. In this spike neural network, we are taking into account two relevant ideas the use of Spike neural network which is the best model for oculo-motor control and the role of geometric computing. As illustration. the quaternion spike neural network is applied for control of robot manipulator. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in human–machine interaction and robotics. [ABSTRACT FROM AUTHOR]

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

The crucial procedure of mobility and singularity identification of parallel mechanisms is widely recognized as how to determine their motions (constraints) concisely and visually. In this paper, we propose a geometric algebra (GA) based approach to determine the motions/constraints, mobility and singularity of parallel mechanisms mainly utilizing the geometric and algebraic relations. Firstly, the motions, constraints and their relations are represented by conformal geometric algebra (CGA) formulas in a concise form by employing the characterized geometric elements with G 4 , 1 . Secondly, the mobility of parallel mechanism, including its number and property and the axes of motions, not only at origin configuration but also in the prescribed workspace, is obtained by the procedure proposed in this paper. Thirdly, the singularity of parallel mechanism is identified by the two indices proposed in this paper with shuffle and outer products. Finally, a typical example is given to illustrate the motions/constraints, mobility and singularity analysis. This approach is beneficial to kinematic analysis and optimal design of parallel mechanisms, especially for which would be carried out in automatic and visual manner using computer programming languages. [ABSTRACT FROM AUTHOR]

The operable representation of robot mechanisms is the key to the computerization of type synthesis of parallel mechanisms (PMs). Geometric algebra, an effective mathematical tool for geometric representation and computation, is introduced to describe the position and orientation characteristics (POC) of robot mechanisms, including serial and parallel mechanisms, in this paper. According to the correspondence between the POC of the joint axis and the motion output characteristics of the moving link, the POC union is defined by the outer product operation for the serial kinematic chain and the POC intersection is defined by the shuffle product operation for the parallel kinematic chain. In this paper, a moving coordinate system independent of the position of a mechanism is established for the expression of the joint axes. The direct symbolic algorithm for the motion output characteristics of robot mechanisms, including union and intersection, is proposed and validated via several case studies of 3R open chain, Sarrus linkage, 3-RCC PM and 3-RPS PM. The analysis procedure shows intuition and explicitness that is suitable for computer-aided derivation of the POC set of a mechanism. [ABSTRACT FROM AUTHOR]

Integrating dynamic analysis models into geographic information system (GIS)-based evacuation simulations is important yet complex. Different models must be smoothly assembled according to the data processing flow to obtain a dynamic, data-forced evacuation simulation. However, because of the diversity of data types and dynamic data updating among different models, closely integrated evacuation simulations are complex and inefficient. In this study, geometric algebra (GA) is introduced to develop a dynamic evacuation simulation framework for a hazardous gas diffusion scheme. In the framework, geospatial data are first integrated into a unified virtual scene with different forms of multivector representation. The major simulation models of gas diffusion, risk assessment, and dynamic evacuation routing compose the major steps of the evacuation simulation. On the basis of the generalized multivector structure, dynamic exchange and updating geospatial data at different evacuation steps can be performed seamlessly with the multivector structure and GA operators. The framework is tested with a case study of a three-dimensional residential area, which shows that our framework can support the integration of dynamic evacuation processes and the model integration is direct and smooth. This framework may also provide a new solution for the integration and dynamic data updating in spatiotemporal GIS. [ABSTRACT FROM AUTHOR]

• A new overconstrained 2R1T parallel manipulator with fixed actuators. • Only 13 single-DOF joints, simple inverse kinematics and few singularities. • Great performances in the levels of kinematics, stiffness and dynamics. • Suitable parallel module of five-axis hybrid machines through machining experiment. • Addition and multiplication in the geometric algebra avoid complex calculations. Five-axis hybrid machine tools are important equipment for highly efficient precision machining of complex workpieces, in which a key part is parallel manipulators (PMs) with two rotations and one translation (2R1T). In view of many single-degree-of-freedom (DOF) joints and virtual output rotational axes in most existing 2R1T PMs, a new 2R1T 2PRU-PSR PM with two certain rotational axes is proposed, which is composed of 13 single-DOF joints and only one S joint and is fully actuated with stationary actuators. Using geometric algebra as a mathematical tool, the mobility and kinematic performance evaluation considering the motion/force transmissibility are presented. With the motion/force transmissibility taken as the optimal index, the dimensional parameters of the 2PRU-PSR PM are optimized to obtain an improved transmission workspace without singular configurations. Based on the optimized parameters, stiffness and dynamic analyses of the 2PRU-PSR PM are presented in the framework of geometric algebra. The results reveal that the 2PRU-PSR PM exhibits excellent performance in the levels of kinematics, stiffness, and dynamics. Finally, a five-axis hybrid prototype is built based on the optimized 2PRU-PSR PM and two serial guides. Machining experiments show that the 2PRU-PSR PM is suitable as a parallel module for five-axis hybrid machines. [ABSTRACT FROM AUTHOR]

Traditional rational motion design describes separately the translation of a reference point in a body and the rotation of the body about it. This means that there is dependence upon the choice of reference point. When considering the derivative of a motion, some approaches require the transform to be unitary. This paper resolves these issues by establishing means for constructing free-form motions from specified control poses using multiplicative and additive approaches. It also establishes the derivative of a motion in the more general non-unitary case. This leads to a characterization of the motion at the end of a motion segment in terms of the end pose and the linear and angular velocity and this, in turn, leads to the ability to join motion segments together with either C 1 - or G 1 -continuity. [ABSTRACT FROM AUTHOR]

This work addresses the inverse kinematics of serial robots using conformal geometric algebra. Classical approaches include either the use of homogeneous matrices, which entails high computational cost and execution time, or the development of particular geometric strategies that cannot be generalized to arbitrary serial robots. In this work, we present a compact, elegant and intuitive formulation of robot kinematics based on conformal geometric algebra that provides a suitable framework for the closed-form resolution of the inverse kinematic problem for manipulators with a spherical wrist. For serial robots of this kind, the inverse kinematics problem can be split in two subproblems: the position and orientation problems. The latter is solved by appropriately splitting the rotor that defines the target orientation in three simpler rotors, while the former is solved by developing a geometric strategy for each combination of prismatic and revolute joints that forms the position part of the robot. Finally, the inverse kinematics of 7 DoF redundant manipulators with a spherical wrist is solved by extending the geometric solutions obtained in the non-redundant case. • The IK of serial robots with a spherical wrist is solved in closed-form. • The position problem is solved by the manipulation of different geometric entities. • The orientation problem is solved by the algebraic manipulation of a rotor. • The developed method is extended to 7 DoF serial robots with a spherical wrist. [ABSTRACT FROM AUTHOR]

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p -form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl 1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl 1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium. [ABSTRACT FROM AUTHOR]

Abstract: Experimental data is subject to uncertainty as every measurement apparatus is inaccurate at some level. However, the design of most computer vision and pattern recognition techniques (e.g., Hough transform) overlooks this fact and treats intensities, locations and directions as precise values. In order to take imprecisions into account, entries are often resampled to create input datasets where the uncertainty of each original entry is characterized by as many exact elements as necessary. Clear disadvantages of the sampling-based approach are the natural processing penalty imposed by a larger dataset and the difficulty of estimating the minimum number of required samples. We present an improved voting scheme for the General Framework for Subspace Detection (hence to its particular case: the Hough transform) that allows processing both exact and uncertain data. Our approach is based on an analytical derivation of the propagation of Gaussian uncertainty from the input data into the distribution of votes in an auxiliary parameter space. In this parameter space, the uncertainty is also described by Gaussian distributions. In turn, the votes are mapped to the actual parameter space as non-Gaussian distributions. Our results show that resulting accumulators have smoother distributions of votes and are in accordance with the ones obtained using the conventional sampling process, thus safely replacing them with significant performance gains. [Copyright &y& Elsevier]

• To overcome the tradeoff between the low steady state error and the fast convergence speed of the GA-LMS adaptive filter, we propose the CGA-LMS adaptive filter. • To verify the advantage of the proposed CGA-LMS adaptive filter, we give the steady-state performance of analysis the CGA-LMS adaptive filter. • To speed up the overall convergence rate of the GA-LMS adaptive filter, a novel geometric algebra combination algorithm with the geometric algebra instantaneous transfer strategy is proposed. • To process the noncircular 3D and 4D signals, the CWL-GA-LMS and CWL-GA-LMS algorithms are investigated. • We compare the learning performance of the proposed CGA-LMS and CGA-LMS-TS algorithms with that of the GA-LMS. Recently, the geometric-algebra theory and the geometric-algebra based adaptive filters have been applied to numerous applications, such as 3D wind speed, computer vision and fusion prediction of dynamic pressure. However, similar to the real-valued adaptive filter, the geometric-algebra based adaptive filters also have the tradeoff problem between the low steady state error and the fast convergence speed. To overcome this shortcoming, this paper proposes a novel geometric algebra adaptive algorithm by convexly combining two geometric algebra least mean square algorithms with two different step sizes. Afterwards, this paper gives a detail steady state performance analysis of the CGA-LMS algorithm by using the geometric algebra theory. Moreover, to address the phenomenon that the slow filter may lag considerably behind the fast filter, which slows down the overall convergence of the combined geometric-algebra filter, we proposed a novel instantaneous transfer strategy, further leading to the CGA-LMS algorithm with transfer strategy (CGA-LMS-TS). To process the noncircular 3D and 4D signals, we have proposed the convex combination of widely linear GA-LMS (CWL-GA-LMS) algorithm. The CWL-GA-LMS with transfer strategy (CWL-GA-LMS) is also investigated. Simulation results for multivector-valued input are presented to verify the performance of the proposed algorithms and the correctness of the performance analysis. [ABSTRACT FROM AUTHOR]

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them with a small imaginary step next to the real number line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof. [ABSTRACT FROM AUTHOR]

• In this paper, we analyze the transient performance of the geometric algebra based least mean M-estimate (GA-LMM) filtering algorithm in detail under some simplifying assumptions and give the step size range that ensure the mean square stability of the GA-LMM. Further, to eliminate the constraint of the constant step size on the performance of the GA-LMM, a novel variable step-size algorithm called VSS-GA-LMM is designed and the optimal step size is obtained by maximizing the difference of mean square deviation (MSD) between successive iterations, which effectively balances the contradiction between convergence rate and steady-state error. • The validity of the transient performance analysis about GA-LMM and the advantages of the GA-LMM and VSS-GA-LMM algorithms over other existing GA based algorithms are confirmed through numerical simulations. In this paper, the transient performance of the geometric algebra based least mean M-estimate (GA-LMM) filtering algorithm is analyzed in detail under some simplifying assumptions. Further, the variable step-size variant VSS-GA-LMM is designed to eliminate the constraint of the constant step size on the performance of the GA-LMM and the optimal step size is obtained by maximizing the difference of mean square deviation (MSD) between successive iterations, which effectively balances the contradiction between convergence rate and steady-state error. Finally, numerical simulations are presented to verify the validity of the theoretical analysis of the GA-LMM and the advantages of the GA-LMM and VSS-GA-LMM algorithms. [ABSTRACT FROM AUTHOR]

Abstract: Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3,0 and even Cl 3,1 + subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl 3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented. [Copyright &y& Elsevier]

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

Abstract: A means for constructing a free-form motion in three dimensions is given. A sparse sequence of prescribed poses is used to guide the motion, and some of these can also be specified to be precision poses through which the motion must pass. The resultant motion is given in a sequence of refined poses. Geometric algebra is used to specify these in a way that is analogous to the use of complex numbers in an equivalent approach for free-form curves. [Copyright &y& Elsevier]

Abstract: This paper presents the application of 2D and 3D Hough Transforms together with conformal geometric algebra to build 3D geometric maps using the geometric entities of lines and planes. Among several existing techniques for robot self-localization, a new approach is proposed for map matching in the Hough domain. The geometric Hough representation is formulated in such a way that one can easily relate it to the conformal geometric algebra framework; thus, the detected lines and planes can be used for algebra-of-incidence computations to find geometric constraints, useful when perceiving special configurations in 3D visual space for exploration, navigation, relocation and obstacle avoidance. We believe that this work is very useful for 2D and 3D geometric pattern recognition in robot vision tasks. [Copyright &y& Elsevier]

Abstract: Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. We explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order k-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of is expressed in terms of the concept of an h-twistor, which is a generalization of a Penrose twistor. [Copyright &y& Elsevier]

Abstract: In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. As an example of application of our approach, we propose a causal algorithm based on an Extended Kalman Filter, for the estimation of 3D structure and motion from 2D observations of points, lines and coplanar features, and we evaluate its performance on both synthetic and real sequences. [Copyright &y& Elsevier]

Abstract: Spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. One can then establish the relation to other approaches that describe spin with fermionic variables. The fermionic star product formalism and the fermionic path integral formalism are related in analogy to their bosonic counterparts. [Copyright &y& Elsevier]

Abstract: We describe a geometric structure optimizer that rapidly establishes whether or not SiO4 units in a hypothetical zeolite framework can exist as minimally-deformed regular tetrahedra. The optimizer, SiGH (Silica General Handler), enables an order of magnitude computational speed gain when processing large databases of zeolite graphs through the early rejection of infeasible graphs. [Copyright &y& Elsevier]

Abstract: We use a combination of real-space geometric algebra and reciprocal space dynamical matrix analyses to study the effect of cation substitution on the framework geometries of β-quartz, cordierite and leucite. We show that the geometric stress associated with the substitution in these framework silicates is absorbed by rigid-unit type motion of those coordination polyhedra near the substitution site. We find that the inherent flexibility of these structures enables screening of geometric stress, such that the associated energy cost is minimal and unlikely to influence substitution patterns. [Copyright &y& Elsevier]

The language of geometric algebra can be used in the development of computer graphics applications. This paper proposes a method to describe a 3D polygonal mesh model using a representation technique based on geometric algebra and the conformal model of the 3D Euclidean space. It describes also the stages necessary to develop an application that uses this formalism. The current application was used to validate the implementation of the main abstract operations characteristic to a geometric algebra computational environment (programming module GAP). The data structures that characterize this geometric algebra based modeling approach as well as the implementation of geometric algebra based methods for model visualization/transformation are developed in detail. The paper emphasizes the elegance and generality of the geometric algebra approach referring also to the necessary computational resources. [Copyright &y& Elsevier]

By means of a novel classical limiting method we derive classical Liouville equations for particles with spin 0 and 1 from the Klein–Gordon and the Duffin–Kemmer–Petiau equations in relativistic quantum phase space within a geometric algebra structure. [Copyright &y& Elsevier]

A new method is proposed for estimating the parameters of ball joints, also known as spherical or revolute joints and hinge joints with a fixed axis of rotation. The method does not require manual adjustment of any optimisation parameters and produces closed form solutions. It is a least squares solution using the whole 3D motion data set. We do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation. This method is compared with other methods that use similar assumptions in the cases of random measurement errors, systematic skin movements and skin movements with random measurement noise. Simulation results indicate that the new method is superior in terms of the algorithm used, the closure of the solution, consistency and minimal manual parameter adjustment. The method can also be adapted to joints with translational movements. [Copyright &y& Elsevier]

• Upgraded geometric algebra power theory is presented for single phase circuits. • Geometric Power is redefined as the product of voltage and current reverse. • Several additions and modifications are introduced to reflect interharmonics and active current based on Fryze proposal. • This new method is validated through some real examples with linear and non-linear loads. The aim of this work is to present major upgrades to existing power theories based on geometric algebra for single-phase circuits in the frequency domain. It also embodies an interesting new approach with respect to traditionally accepted power theories, revisiting power concepts in both sinusoidal and non-sinusoidal systems with linear and nonlinear loads for a proper identification of its components to achieve passive compensation of true non-active current. Moreover, it outlines traditional power theories based on the apparent power S and confirms that these should definitively be reconsidered. It is evidenced that traditional proposals based on the concepts of Budeanu, Fryze and others fail to identify the interactions between voltage and current harmonics. Based on the initial work of Castro-Núñez and others, new aspects not previously included are detailed, modified and reformulated. As a result, it is now possible to analyze non sinusoidal electrical circuits, establishing power balances that comply with the principle of energy conservation, and achieving optimal compensation scenarios with both passive and active elements in linear and non-linear loads. [ABSTRACT FROM AUTHOR]

A design for a truss mechanism of a secondary mirror based on robotics is proposed. This design would allow for the construction of larger vehicle-mobile telescopes. As the new truss mechanism combines the original support structure and adjustment mechanism, the problems in designing new structures needs to be overcome. In this paper, the basic form of the truss mechanism is determined by finite element method, and the number of limbs meeting the requirements of resonance frequencies and stiffness is obtained. Degrees-of-freedom of the new truss mechanism is calculated by motion space based on geometry algebra and screw theory, It can provide more accurate and specific results compared with the G-K formula. The optimal structure is calculated to meet the requirement in degrees-of-freedom with the minimum possible limbs and kinematic pairs. After the form and the value of joints are determined, the deformations are calculated by stiffness evaluation index. Wavefront aberrations simulated with Zernike polynomials are used to verify the structure. [ABSTRACT FROM AUTHOR]

This paper presents a geometric construction for fitting planar motions to three control poses within a particular geometric algebra. The immediate impact of the geometric construction along with the control pose representation is the provision of simple, usable tools for the design and manipulation of motions in a similar way to the highly successful B-spline approach for curves and surfaces in CAD. • Free-form motions using geometric algebra; • use of the slerp construction; • fitting of planar motion through three control poses. [ABSTRACT FROM AUTHOR]

• Skeleton sequence space as a subset of Geometric Algebra is constructed to represent each skeleton sequence along both spatial and temporal dimensions. • Rotor-based view transformation overcomes the view variation challenge and reserves relative motions among skeletons. • Spatio-temporal view invariant model is constructed to model spatial configuration and temporal dynamics of skeleton joints and bones. • Four skeleton sequence shape and motion representations are learned to comprehensively describe skeleton-based actions, which are fed to a selected multi-stream convolutional neural network for action recognition. Skeleton-based action recognition is an increasing attentioned task that analyses spatial configuration and temporal dynamics of a human action from skeleton data, which has been widely applied in intelligent video surveillance and human-computer interaction. How to design an effective framework to learn discriminative spatial and temporal characteristics for skeleton-based action recognition is still a challenging problem. The shape and motion representations of skeleton sequences are the direct embodiment of spatial and temporal characteristics respectively, which can well address for human action description. In this work, we propose an original unified framework to learn comprehensive shape and motion representations from skeleton sequences by using Geometric Algebra. We firstly construct skeleton sequence space as a subset of Geometric Algebra to represent each skeleton sequence along both the spatial and temporal dimensions. Then rotor-based view transformation method is proposed to eliminate the effect of viewpoint variation, which remains the relative spatio-temporal relations among skeleton frames in a sequence. We also construct spatio-temporal view invariant model (STVIM) to collectively integrate spatial configuration and temporal dynamics of skeleton joints and bones. In STVIM, skeleton sequence shape and motion representations which mutually compensate are jointly learned to describe skeleton-based actions comprehensively. Furthermore, a selected multi-stream Convolutional Neural Network is employed to extract and fuse deep features from mapping images of the learned representations for skeleton-based action recognition. Experimental results on NTU RGB+D, Northwestern-UCLA and UTD-MHAD datasets consistently verify the effectiveness of our proposed method and the superior performance over state-of-the-art competitors. [ABSTRACT FROM AUTHOR]

Abstract: Deforming the algebraic structure of geometric algebra on the phase space with a Moyal product leads naturally to supersymmetric quantum mechanics in the star product formalism. The supersymmetric Hamiltonian emerges then from the classical one by the transition from commutative to noncommutative geometry on the phase space. [Copyright &y& Elsevier]

• Geometric algebra framework is applied to solve non sinusoidal and nonlinear circuits. • A new definition of non-active power based on geometric algebra is presented. • New current decomposition is introduced to minimize line looses and supply active power. • This new method is validated through some examples with linear and non-linear loads. A new approach for the definition of non-active power in electrical systems is presented in this paper. Through the use of geometric algebra, it is possible to define a new term called geometric non-active power, which is applicable to both sinusoidal and non-sinusoidal systems, and to both linear and nonlinear loads. The classic definitions of distortion and reactive power are compared and discussed in our proposal. We verify how geometric non-active power can appear in both purely resistive and purely reactive systems. The superiority of geometric algebra is revealed through several examples of electrical circuits previously analysed in specialised literature. Furthermore, a new geometrical current decomposition is proposed, for the first time, to provide a greater physical sense to existing geometric power. The results obtained confirm that classic concepts based on apparent power S are based on a lack of physical meaning, which is why geometric algebra theory should be adopted instead. [ABSTRACT FROM AUTHOR]