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 • Classic QNN backpropagation techniques are outdated and should be avoided.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

• 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]

• 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]

• 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 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]