Hypercomplex Numbers Research Articles

HYPERCOMPLEX NEUTROSOPHIC SIMILARITY MEASURE & ITS APPLICATION IN MULTI-CRITERIA DICISION MAKING PROBLEM

Abstract. Neutrosophic set is very useful to express uncertainty, impreciseness, incompleteness and inconsistency in a more general way. It is prevalent in real life application problems to express both indeterminate and inconsistent information. This paper focuses on introducing a new similarity measure in the neutrosophic environment. Similarity measure approach can be used in ranking the alternatives and determining the best among them. It is useful to find the optimum alternative for multi criteria decision making (MCDM) problems from similar alternatives in neutrosophic form. We define a function based on hypercomplex number system in this paper to determine the degree of similarity between single valued neutrosophic sets and thus a new approach to rank the alternatives in MCDM problems has been introduced. The approach of using hypercomplex number system in formulating the similarity measure in neutrosophic set is new and is not available in literature so far. Finally, a numerical example demonstrates how this function determines the degree of similarity between single valued neutrosohic sets and thereby solves the MCDM problem.

A broad class of discrete-time hypercomplex-valued Hopfield neural networks

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley–Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley–Dickson algebras.

Метод генерації гіперкомплексних числових систем для моделювання цифрових реверсивних фільтрів 4-го порядку

A digital filter is a digital system used to filter a discrete signal. It can be implemented either by software or by specialized equipment. The use of hypercomplex numbers for building digital filter structures can provide significant benefits. Digital filters with hypercomplex parameters have higher speed and better performance in terms of total parametric sensitivity. Although in previous works software filter of 3 th order structure synthesis algorithms were proposed, in this paper only software filter of 4 th order structure synthesis algorithms are considered. Studies have shown that the expression analysis method for the transfer function denominator norms of the hypercomplex filter allows you to effectively select the hypercomplex number system, which has all the necessary properties. It is shown that only quadriplex number system K and hypercomplex number system obtained by auto doubling of a double-number system W are suitable for the structures synthesis of fourth-order hypercomplex digital filters. These hypercomplex number systems allow to get a complete set of filter transfer function shift operator steps. On the other hand, they have isomorphic weakly filled hypercomplex number systems. Transition to such systems can significantly reduce the number of real operations during the filter operation time. In the work, all transformations and calculations in the hypercomplex domain were performed using the software module for hypercomplex calculations in the Maple computer algebra framework. This fact confirms the efficiency of this software module. Tabl.: 2. Refs: 17 titles.

A Dual Quaternion-Based Method for Estimating PTV Margins in Intensity-Modulated Radiotherapy

While a general pose or spatial displacement in 3D space is often decomposed into a translation and a rotation, they are not independent of each other, as the composition of translation and rotation does not commute. ICRU 83 notes that most planning systems allow margins along the three Cartesian dimensions only, although inter- and intra- fraction rotations are routinely imaged and reported. Efforts to construct PTV margins that account for systematic errors from rotations typically sample imaged anatomic rotations and translations in a population, and then use Monte Carlo or other random sampling techniques to construct a space of displacements for a structure by treating the rotations and translations as independent. Significant errors can occur for off-axis rotations as the translational shift of the reference frame origin depends on the axis and magnitude of rotation. Because most rigid rotations in the body occur about the spine, resulting in roll, pitch and yaw, a process to produce a PTV for a distant target that accounts for these shifts is needed. Given two spatial positions of a tumor, the relative spatial displacement between them can be reduced to a screw displacement, which is a combined rotation about and translation along the same axis, called screw axis. Everything about the screw displacement, i.e., the location and orientation of the screw axis as well as the angle of rotation about and distance of translation along the screw axis, can be represented holistically by an 8-dimensional hypercomplex number called dual quaternion. In this way, a set of spatial displacements is represented by a set of points in the space of dual quaternions (S) and the problem of finding the “workspace”, i.e., the enveloping space of all attainable spatial displacements from the sampled 6D population data can be converted into that of fitting the closest fit volume in S that contains all dual-quaternion points. Given the shape and size of a tumor, the resulting PTV that accounts for kinematic errors can then be obtained as swept volume of the kinematic workspace. For a given 3D model of a tumor as well as a set of dual quaternions that represent errors in both rotation and translation, an algorithm has been developed to compute the convex hull of the given displacements and then the swept volume generated by the tumor model. The swept volume represents the expanded PTV that accounts for position and orientation errors. Dual quaternion provides a compact presentation for translational and rotational errors that not only circumvents the non-commutative issue in combining rotation and translation in different order but also are independent of the coordinate systems. In addition, this representation captures naturally the effect of off-axis rotation on the swept volume.

Monogenic Functions in Commutative Algebras Associated with Classical Equations of Mathematical Physics

The methods involving the functions analytic in a complex plane for plane potential fields inspire the search for the analogous efficient methods for solving the spatial and multidimensional problems of mathematical physics. Many such methods are based on the mappings of hypercomplex algebras. The essence of the algebraic-analytic approach to elliptic equations of mathematical physics consists in the finding of a commutative Banach algebra such that the differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. The use of differentiable functions given in commutative Banach algebras combines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs. The paper contains the review of results reflecting the formation and the development of the mentioned approach.

Методика выбора гиперкомплексных числовых систем для моделирования цифровых реверсивных фильтров третьего и четвертого порядков

Методика выбора гиперкомплексных числовых систем для моделирования цифровых реверсивных фильтров третьего и четвертого порядков

Hypercomplex Tensor Completion via Convex Optimization

Expressing multidimensional information as a value in hypercomplex number systems (e.g., quaternion, octonion, etc.) has great potential, in data sciences, e.g., signal processing, to enjoy their nontrivial algebraic benefits which are not available in standard real or complex vector systems. Strategic utilizations of such benefits would include, e.g., hypercomplex singular value decomposition (SVD) and low-rank approximation of matrices. In this paper, as powerful mathematical tools for wider signal processing applications of hypercomplex number systems, we first propose novel definitions of SVD and best low-rank approximation of matrices based on algebraic translations of Cayley–Dickson (C–D) number systems. We then derive an algorithmic solution to hypercomplex tensor completion problem based on a convex optimization technique. Numerical experiments in a scenario of color tensor completion problem show that the proposed algorithm recovers much more faithfully the original color information, masked randomly by noise, than part-wise tensor completion algorithms.

Complex and Hypercomplex-Valued Support Vector Machines: A Survey

In recent years, the field of complex, hypercomplex-valued and geometric Support Vector Machines (SVM) has undergone immense progress due to the compatibility of complex and hypercomplex number representations with analytic signals, as well as the power of description that geometric entities provide to object descriptors. Thus, several interesting applications can be developed using these types of data and algorithms, such as signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, time series prediction, and visual servoing, among others. This paper presents and discusses the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric Support Vector Machines.

Методика вибору гіперкомплексних числових систем для моделювання цифрових реверсивних фільтрів

Digital filter is a digital system, which is used for filtering a discrete signal. It can be implemented as a software method, and with the help of specialized equipment. In both cases, the digital filter can be used for filtering, both in real time and for a pre-stored signal value. In this paper, we consider only the algorithms for the synthesis of the software filter structure. However, it should be noted that on the basis of these structures, implementations and hardware solutions of digital filters can beat. The use of HNS for the synthesis of digital filter structures can give a tangible advantage. Digital filters with hyper complex parameters have greater momentum and better characteristics in integrated parametric sensitivity.The paper discusses the definition of hyper complex digital filter and the conversion of its transfer function to the form of hyper complex function. A technique for selecting a hyper complex number system has been developed for the equivalent of a hyper complex filter to a real analogue. It is shown that the main method for determining the expediency of using one or another HNS for the synthesis of the structure of a hyper complex digital filter is the general form of the normic denominator of the hyper complex transfer function of the filter. This turn should be a complete polynomial of the shift operator, and also contain all the components of the multiplier with the shift operator in the denominator.However, these conditions are only necessary in the synthesis of the hyper complex filter, since the system of equivalence equations may not have real solutions when fulfilling these conditions. In this case, it is necessary to switch to another HNS, if there is such an opportunity, or to switch to HNS of higher orders. The latter path needs further research.

Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

Comparison of Neural Network–Based Adaptive Controllers Using Hypercomplex Numbers for Controlling Robot Manipulator

This study investigates an adaptive controller by applying a neural network, in which all the network parameters, states, signals and functions are expressed using hypercomplex numbers and algebras; its application to dynamics control of a robot manipulator. To design hypercomplex–valued neural networks where each neural network is a multilayer feedforward network with a split–type activation function of neurons using a tapped–delay–line input, we consider the following four types of hypercomplex numbers: complex, hyperbolic, bicomplex and quaternion numbers. In the control system, we utilise a feedback error–learning scheme to conduct the training of the network through a back–propagation algorithm. In the computational experiments, we explore a hypercomplex–valued neural network–based controller as a trajectory control problem of a three–link robot manipulator, in which the position of the end–effector follows to the desired trajectory in a 3–dimensional space. The simulation results validate the feasibility and effectiveness of the quaternion neural network–based controller for this task.

Explanation of Dark Matter, Dark Energy and Dark Space: Discovery of Invisible Universes

Theoretical and experimental studies of special processes in linear electric circuits have proved the principle of physical reality of imaginary numbers discovered 500 years ago. This principle of physical reality of imaginary numbers has allowed astrophysics to prove the existence of invisible parallel universes and explain the phenomena of dark matter, dark energy and dark space associated therewith. Physical entities corresponding to imaginary, complex and hypercomplex numbers in the other exact sciences together form an invisible and still unknown world that is to be learned by science of the future.

Image hash authentication algorithm for orthogonal moments of fractional order chaotic scrambling coupling hyper-complex number

In this paper, the image Hash authentication algorithm for orthogonal moments of fractional order chaotic scrambling coupling hyper-complex number Tchebichef is proposed. The improved PDE function and Ring segmentation mechanism are used to process the initial image and output the secondary image to improve the robustness of Hash’s scaling and rotation; and by taking full account into the color information of the image, the image R, G and B position is regarded as its imaginary part. In addition, the invariant moment of the hyper-complex number Tchebichef is desgined, and the robust characteristics of the secondary image is fully extracted to greatly improve its ability to recognize the rotation tampering. At the same time, in order to improve the security of the Hash sequence, the fractional order Logistic mapping and chaotic scrambling technique is constructed to encrypt the eigenvector; and the image information is authenticated by the Hamming distance. Finally, the robustness and tampering recognition accuracy of the proposed Hash algorithm are tested.

Product of Three Octonions

This paper considers octonions that are the eight-dimensional hypercomplex numbers characterized by multiplicative non-associativity. The decomposition of the product of three octonions with conjugate central factor into the sum of mutually orthogonal triple anticommutator, triple commutator and associator, is introduced in an obvious way by commuting the factors and alternating the multiplication order. The commutator is regarded as a generalization of the cross product to the case of three arguments both for quaternions and for octonions. It is shown that the decomposition found coincides with the decomposition of the triple octonion product into symmetric-antisymmetric or, in other words, symmetric-skew-symmetric parts. It is verified that the resulting additive decomposition is equivalent to the known solution derived and presented by Okubo in insufficiently perfect form. Based on the results of Okubo, a slight correction of modern definitions of the triple cross product is proposed.

The Schwarz Type Lemma in Upper Half Space in Clifford Analysis

In this paper, we firstly construct the Euler’s expression of hyper-complex numbers in $$H_{n+1}$$ , as an application of the Euler’s expression, we prove that the well-known homogeneous monogenic polynomials $$V_{l_1\dots l_p}(\mathbf{x})$$ are functions with values in $$H_{n+1}$$ . Secondly we construct a type of general Mobius transformation in Clifford analysis, the mapping properties are given; The Jacobi determinant and the monogenic property under these Mobius transformations are shown. Finally, by using the above Mobius transformation and modifying the Schwarz lemma in Zhang (J Math Anal Appl 443:1130–1141, 2016), we establish the Schwarz–Pick type lemmas for monogenic functions in the upper half space $$H^+_{n+1}$$ . A new generalization of the Schwarz lemma for harmonic functions with values in $$H_{n+1}$$ is also given.

Использование методов генерации изоморфных гиперкомплексных числовых систем для повышения эффективности умножения гиперкомплексных чисел

Использование методов генерации изоморфных гиперкомплексных числовых систем для повышения эффективности умножения гиперкомплексных чисел

Структура алгоритму швидкої двомірної згортки за допомогою ізоморфних гіперкомплексних числових систем

In the mathematical modeling of linear systems, it is necessary to repeatedly perform a linear convolution of discrete signals. The complexity of calculating the linear convolution rapidly increases with the length of the convoluted arrays and their dimension, thus the methods of «fast» convolution calculations are used. One of the most common methods is convolution using Fast Fourier Transform (FFT) algorithms, which are based on decomposition of the original large-dimensional problem into a large number of low-dimensional problems. Thus, it is very important to develop such methods for solving problems for small dimension, which use, possibly, a smaller number of real operations. There are a number of methods for the rapid calculation of linear convolution: the methods of Cook-Toom, Vine, Fast Fourier Transform (FFT), Cooley-Tuke, Good-Thomas, and others. The algorithms for performing convolution based on the transition to hypercomplex spaces are considered. The basis of this approach has been developed by the authors. Convoluted numerical sequences are considered as components of hypercomplex numbers belonging to some HNS. The product of these numbers will contain paired products of components of convolutional numerical sequences. Nevertheless, they will be combined in amounts not in the same composition as necessary for organizing convolution components. In addition, the number of real multiplications with multiplication of hypercomplex numbers in the general case is the same as in the direct calculation of convolution, that’s why there is no profit. In this way there are two problems: the first one is a reduction in the number of real operations when multiplying hypercomplex numbers; the second one is the organization of the choice of paired products of convolution components. The solution of these two problems allows synthesize such convolution algorithms, which by the number of operations are more efficient than direct calculation algorithms for convolution.

Versatile Quaternion Multipliers Based on Distributed Arithmetic

This paper introduces the idea of versatile circuits for multiplying 4-dimensional hypercomplex numbers in hardware. Depending on the settings of such a device, a variable quaternion can be left- or right-multiplied by a constant coefficient or by its conjugate, as various operations are useful in transform-type algorithms. Multiplierless circuits based on distributed arithmetic (DA) are reviewed that compute quaternion products by additions and bit shifts. It is shown that they can be made versatile by extending memory of partial results, but a better solution is our method for preprocessing bits used to address this memory. The method allows for using the same partial results to compute different inner products that are related to a quaternion multiplication. So versatile multipliers can be implemented with memory optimized so as to save area or to speed up reprogramming compared to the basic DA-based circuit. This has been demonstrated by hardware design experiments, which show that 13–69% area can be saved in a case of ASIC implementation, while for FPGA implementation, spending only 11% more logic resources allows a multiplier to be reprogrammed 75% faster. Additionally, it has been explained how versatile multipliers can be used to realize low-area analysis/synthesis filter banks.

High-dimensional information processing through resilient propagation in quaternionic domain

Abstract This paper proposes a fast but effective machine learning technique for the high-dimensional information processing problems. A quaternion is the hyper-complex number in four dimensions which possesses four components in a single body with embedded phase information among their components. Thus, a neural network with quaternion as unit of information flow has ability to learn and generalize magnitude and argument of high-dimensional information simultaneously. The slow convergence and getting stuck into bad minima are the main drawbacks of the back-propagation learning algorithm; therefore learning technique with faster convergence is the demand of quaternionic neurocomputing. The basic issues of quaternionic back-propagation (ℍ-BP) algorithm have been combated by proposed resilient propagation algorithm in quaternionnic domain (ℍ-RPROP). The efficient and intelligent behavior of the proposed learning algorithm has been vindicated by the wide spectrum of benchmark problems. The simulations on different transformations demonstrate its ability to learn 3D motion, which is very desirable in viewing of objects through different orientations in many engineering applications. The critical thing is the training through mapping over straight line having a small number of points and eventually generalization over complicated geometrical structures (Sphere, Cylinder and Torus) possessing huge amount of point cloud data. The 3D and 4D time series prediction as benchmark applications are also taken into the account for the significant contribution of the proposed work. The substantial edge of the proposed algorithm is its quicker convergence with better approximation capability in high-dimensional problems.

Algebraic models and methods of computer image processing. Part 1. Multiplet models of multichannel images

We present a new theoretical framework for multichannel image processing using commutative hypercomplex algebras. Hypercomplex algebras generalize the algebras of complex numbers. The main goal of the work is to show that hypercomplex algebras can be used to solve problems of multichannel (color, multicolor, and hyperspectral) image processing in a natural and effective manner. In this work, we suppose that the animal brain operates with hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is considered not as an K–D vector, but as an K–D hypercomplex number, where K is the number of different optical channels. The aim of this part is to present algebraic models of subjective perceptual color, multicolor and multichannel spaces.