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 • Classic QNN backpropagation techniques are outdated and should be avoided.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

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]

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]

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]

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: 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]

• 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]

• 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]