Hypercomplex Numbers Research Articles

On hyper complex numbers with higher order Pell numbers components

On hyper complex numbers with higher order Pell numbers components

MATRIX THEORY OVER DGC NUMBERS

Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given.

Multi-dimensional hypercomplex continuous orthogonal moments for light-field images

Image moments have always been topical in the domain of image processing, and the proposed hypercomplex moments and multi-channel moments have greatly broadened the application of image moments in recent years. However, the hypercomplex moments are limited by the specific number of imaginary parts of the hypercomplex number, and the redundancy of imaginary parts or the overflow of image channels will occur when the number of imaginary parts does not fit the image structure. The multi-channel moments allow flexible adjustment of the number of channels, but their loose structure will result in the loss of correlation between multiple channels of images. Moreover, the hypercomplex moments and multi-channel moments cannot handle images with multi-dimensional structures such as light-field (LF) images. For the purpose of addressing the above problems, this paper proposes generalized hypercomplex continuous orthogonal moments (HCOMs) and extends them to multi-dimensional hypercomplex continuous orthogonal moments (MHCOMs) in multi-dimensional space. MHCOMs can be used to process images with arbitrary multi-dimensional structures, and when they are used for LF image processing, there is no need to map the LF images to two-dimensional space to change the multi-dimensional structural properties of LF images. By calculating MHCOMs of LF images, a set of multi-dimensional hypercomplex tensors containing robust feature information of LF images can be obtained, and the original LF images can be reconstructed using this information. Furthermore, the MHCOMs-based zero-watermarking scheme is designed for LF images, which has excellent robustness and can achieve lossless copyright protection for LF images.

The Construction of Armendariz Ring using Formal Triangle Matrix Ring

Trinion and Quaternion numbers are one of the hypercomplex numbers which is an extensions of the complex number. From Trinion and Quaternion numbers, a bimodule can be formed which is an ordered pair of Trinion and Quaternion. Furthermore, Trinion number, Quaternion number, and their bimodule can be formed into a Formal Triangle Matrix. The Formal Triangle Matrix is better known as the Upper Triangle Matrix. Since Trinion number, Quaternion number and their bimodule are rings, then the Formal Triangle Matrix can be called as the Formal Triangular Matrix Ring. The purpose of this study is to construct the Armendariz Ring using the Formal Triangular Matrix Ring. The obtained results will show that the Formal Triangular Matrix Rings are the -Skew Armendariz Ring and the -Skew -Armendariz Ring, where is a Ring Endomorphism and is -derivation.

Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics

Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $\A^2=n\A$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.

Operators induced by certain hypercomplex systems

In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

Differentiability condition of function with hypercomplex number as variable of commutative ring

Differentiability condition of function with hypercomplex number as variable of commutative ring

Unrestricted Pell and Pell – Lucas 2N-ons

In this study, we define unrestricted Pell and Pell – Lucas hyper-complex numbers. We choose arbitrary Pell and Pell – Lucas numbers for the coefficients of the ordered basis 〖{e〗_0,e_1,⋯,e_(N-1)} of hyper-complex 2^N-ons where N∈{0,1,2,3,4} and call these hyper-complex numbers unrestricted Pell and Pell-Lucas 2N-ons. We give generating functions and Binet formulas for these type of hyper-complex numbers. We also obtain some generalization of well – known identities such as Catalan’s, Cassini’s and d’Ocagne’s identities.

Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.

Spacecraft-Mounted Robotics

Space-mounted robotics is becoming increasingly mainstream for many space missions. The aim of this article is threefold: first, to give a broad and quick overview of the importance of spacecraft-mounted robotics for future in-orbit servicing missions; second, to review the basic current approaches for modeling and control of spacecraft-mounted robotic systems; and third, to introduce some new developments in terms of modeling and control of spacecraft-mounted robotic manipulators using the language of hypercomplex numbers (dual quaternions). Some outstanding research questions and potential future directions in the field are also discussed.

Extendable orthogonal sets of integral vectors

Motivated by a model in quantum computation, we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the quaternions and other hypercomplex numbers.

Face Recognition by Principal Component Regression using Hypercomplex Numbers

Face Recognition by Principal Component Regression using Hypercomplex Numbers

Halos and undecidability of tensor stable positive maps

A map is tensor stable positive (tsp) if is positive for all n, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

Light-field image watermarking based on geranion polar harmonic Fourier moments

Light-field images provide a spatial and angular description of the light, and they can capture rich visual information in the natural world. In recent years, the research on light-field imaging has achieved many fruitful results. As more and more people are now focusing their attention to light-field imaging, the copyright protection of light-field images has become an urgent problem that needs to be addressed. Digital watermarking can effectively protect the copyright ownership of light-field images. Currently, there exists no light-field image watermarking scheme that can resist various geometric attacks. In this work, relying on geranion theory and polar harmonic Fourier moments (PHFMs), geranion polar harmonic Fourier moments (GPHFMs) are constructed and utilized for light-field image watermarking. The geranion represents a hypercomplex number containing one real part and thirty-one imaginary parts, and the imaginary parts of geranion can be used to encode multiple image color components while maintaining the correlation between each component. GPHFMs are stable image features, with good image reconstruction ability and stability. Essentially, the light-field image watermarking based on GPHFMs offers good imperceptibility and robustness, can resist various attacks, and effectively solves the problem of current watermarking schemes related to their inability of resisting geometric attacks. Furthermore, it is verified through experiments that the proposed scheme is more robust than the previously reported schemes.

THE SET OF SEMI OCTAVE. II

Today, the theory of hypercomplex numbers is a rapidly developing field of mathematical knowledge due to its numerous applications in various branches of physics. For example, dual numbers allow us to model the physical space-time quite accurately mathematically, quaternions are used in electrodynamics, in the study of vortex motions, octaves also represent a mathematical model of a possible description of our reality [1-6]. In the article [7], by analogy with the work [8] of the Iranian mathematicians X. Mortazashl and M. Jafari, who gave the concept of a semi-quaternion, the definition of semi-octaves and operations on them is introduced, as well as some properties of these operations are established. This work continues the research started in [7]. Definitions of the norm of semi-octaves and linear equations over semi-octaves are introduced here, formulas for solving such equations are found. Analogs of the Euler and Moivre formulas, which originally took place for complex numbers, are also established for semi-octaves.

A general framework for hypercomplex-valued extreme learning machines

This paper aims to establish a framework for extreme learning machines (ELMs) on general hypercomplex algebras. Hypercomplex neural networks are machine learning models that feature higher-dimension numbers as parameters, inputs, and outputs. Firstly, we review broad hypercomplex algebras and show a framework to operate in these algebras through real-valued linear algebra operations in a robust manner. We proceed to explore a handful of well-known four-dimensional examples. Then, we propose the hypercomplex-valued ELMs and derive their learning using a hypercomplex-valued least-squares problem. Finally, we compare real and hypercomplex-valued ELM models’ performance in an experiment on time-series prediction and another on color image auto-encoding. The computational experiments highlight the excellent performance of hypercomplex-valued ELMs to treat multi-dimensional data, including models based on unusual hypercomplex algebras.

COORDINATE TRANSFORMATIONS OF THREE-PHASE VARIABLES USING QUATERNIONS

Among the variety of modern approaches to the mathematical description of the power quality indicators during the processes of transmission, distribution, conversion and calculation of the ac electric power, the representation of three-phase models in the form of a purely imaginary quaternion located in a separate subspace of the four-dimensional hypercomplex space allows, in relation to the generally accepted method of analyzing linear circuits, for example, symmetrical components with the selection of a direct, reverse and zero phase sequence for the fundamental harmonic, to take into a more complete account the features of energy consumption, especially in the presence of distortion in the modified forms of harmonic signals. In addition, the division of the quaternion into scalar (real) and partial (imaginary) makes it possible to significantly simplify the subsequent analytical processing of synthesis of a power converters control signals for active filtering and power supply of autonomous loads of an arbitrary type, including a single-phase configuration, by extracting from its composition individual components responsible for both the amplitude-phase asymmetry and the nonlinearity of the characteristics.
 
 The main algorithmic principles of organizing control structures as part of three-phase systems of various functional purposes, as a rule, are based on the conversion of reference signals and current values ​​of measured currents and voltages into state coordinates obtained by rotating the three-dimensional space plane by a given angle. At the same time, the calculated ratios for the numerical determination of the initial variables transformed by rotation in the quaternion basis are a function of only four kinematic parameters, which, other things being equal, leads to a simplification of the control law in relation to the traditional vector-matrix approach using nine direction cosines with six connection equations. In this regard, this paper is devoted to the applied problems of implementing linear transformations by E. Clarke and R.H. Park in terms of four-dimensional hypercomplex numbers, in compliance with the additional requirement of the invariance of scalar quantities after the transition.

Proper ARMA Modeling and Forecasting in the Generalized Segre’s Quaternions Domain

The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts of properness are suggested and statistical tests to check if a GSQ random vector is proper or not are proposed. Further, a method to determine in which specific hypercomplex algebra is most likely to achieve, if possible, the properness properties is given. Next, both the linear estimation and prediction problems are studied in the GSQ domain. Finally, ARMA modeling and forecasting for proper GSQ time series are tackled. Experimental results show the superiority of the proposed approach over its counterpart in the Hamilton quaternion domain.

Reduced Biquaternion Convolutional Neural Network for Color Image Processing

Reduced biquaternion is a four dimensional hyper–complex number which is commutative algebra and an extension of complex. Due to this property, the corresponding fast algorithm is designed in time frequency analysis which can better fit the convolution theorem than the non-commutative quaternion. In addition, the reduced biquaternion can be interpreted as a single point in 2-dimensional hyperbolic space with its algebraic structure which has more capacity and variable ability than the Euclidean space. However, the algebra properties of the reduced biquaternion have not yet been well explored in the deep convolutional neural network. In this paper, we develop a new deep network structure, namely reduced biquaternion valued convolutional neural network (RQV-CNN). The proposed RQV-CNN includes basic modules of reduced biquaternion convolution layer and fully connection layer. Extensive experiments on color image classification and color image denoising are conducted to evaluate the promising performance of the RQV-CNN framework. The results show that RQV-CNN outperforms the real-valued CNN (RV-CNN), complex-valued CNN (CV-CNN), and quaternion-valued CNN (QV-CNN) with same structures. The source code can be found at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/tasteimage/RQVCNN</uri> .

On Complex Numbers in Higher Dimensions

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.