Hypercomplex Numbers Research Articles

On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication.

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.

Scaled Global Operators and Fueter Variables on Non-zero Scaled Hypercomplex Numbers

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers Ht,t∈R∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t,\\, t\\in \\mathbb {R}^*$$\\end{document}, of which the H-1=H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{-1}=\\mathbb {H}$$\\end{document} is the space of quaternions and H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{1}$$\\end{document} is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on Ht\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t$$\\end{document}. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function.

Free-Product Banach Algebras Induced by Scaled Hypercomplex Numbers

Free-Product Banach Algebras Induced by Scaled Hypercomplex Numbers

Quaternion Convolutional Neural Networks: Current Advances and Future Directions

Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.

Hyper-complex number quaternion system-based method for image representation

Hyper-complex number quaternion system-based method for image representation

On the Scaled Hypercomplex Numbers: Quaternions through Split Quaternions

In this paper, we analyze a certain algebraic structure ℋ[−1, 1] containing all t-scaled hypercomplex numbers of ℍt where the scales t are from −1 to 1, i.e., −1 ≤ t ≤ 1 in ℝ. The algebraic, operator-theoretic, and operator-algebraic properties of ℋ[−1, 1] are studied under the local dynamics on the closed interval [−1, 1] inherited from the dynamics on the continuum ℝ. Also, some analytic properties of an interesting type of operators switching scales of hypercomplex numbers acting on ℋ[−1, 1] are considered, and we investigate how they affect the analysis on ℋ[−1, 1].

Schur analysis over the unit spectral ball

We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers.

Regular Functions on the Scaled Hypercomplex Numbers

In this paper, we study the regularity of R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}-differentiable functions on open connected subsets of the scaled hypercomplex numbers Htt∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \\mathbb {H}_{t}\\right\\} _{t\\in \\mathbb {R}}$$\\end{document} by studying the kernels of suitable differential operators ∇tt∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \ abla _{t}\\right\\} _{t\\in \\mathbb {R}}$$\\end{document}, up to scales in the real field R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}.

On Higher Order Jacobsthal Hyper Complex Numbers

In this work, we define a new class of hyper complex numbers whose components are higher order Jacobsthal numbers, and call such numbers as the higher order Jacobsthal $ 2^{s} $-ions. We obtain some algebraic properties of the higher order Jacobsthal $ 2^{s} $-ions such as recurrence relation, Binet-like formula, generating function, exponential generating function, Vajda's identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Morever we derive the matrix representation of the higher order Jacobsthal $ 2^{s} $-ions, and so prove Cassini's identity as a further type.

Free Probability on Banach Algebras Induced by Scaled Hypercomplex Numbers

Free Probability on Banach Algebras Induced by Scaled Hypercomplex Numbers

Матрична алгебра В як евклідовий простір

Representation of information by means of hypercomplex numerical systems is used in various problems of science and technology: in classical mechanics, solid body mechanics, electrodynamics, radio electronics, computer animation, and others [1]. Often a hypercomplex system (that is, a system whose elements are considered to be hypercomplex numbers) is understood as any finite-dimensional algebra over a field. An important place among such algebraic structures is occupied by matrix algebras. The impossibility of constructing algebras with division does not at all mean the impossibility of constructing algebras without division, but their properties are close to the first ones (use of defined division). Since each algebra of finite rank can be monomorphically immersed in some complete matrix algebra, this caused, so to speak, an inverse approach to the construction of new algebras. A certain subalgebra stands out from a complete matrix algebra, which is a matrix representation of an algebra of finite rank. It is the implementation of such an approach that makes it possible to endow elements of algebra of finite rank with matrix characteristics, in particular, a canonical representation of algebra elements is constructed through the spectral representation of a matrix, and the algebra itself is endowed with a topological structure through one of the matrix norms. At the same time, an additional condition is often imposed, that it be an algebra over the field of real or complex numbers. The article constructs a real algebra of finite rank, the elements of which are matrices of the second order with the same sum of rows and columns. We endowed it with a norm and a scalar product, demonstrating that it is a Euclidean space. This algebra is a matrix representation of the algebra of hypercomplex numbers, which we called binary in our research [4].

Certain invertible operator-block matrices induced by C^{*}-algebras and scaled hypercomplex numbers

The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a C^{*}-subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex numbers, and that of operator-valued cases of (ii), and (iv) to confirm our invertibility of (ii) and (iii) are equivalent to the general invertibility of left( 2times 2right) -block operator matrices.

Miraculous hypercomplex numbers

<p>The development of the number system has been a long and difficult process, and many landmark concepts and theorems have been put forward. By briefly reviewing the development of hypercomplex systems, the rules for constructing the unit elements are discussed. As a vector space defining multiplication, division, and norm of vectors, hypercomplex numbers synthesize the advantages of mathematical tools such as algebra, geometry, and analysis, faithfully describe the intrinsic properties of space-time and physical systems, and provide a unified language and a powerful tool for basic theories and engineering technology. In the application of hypercomplex numbers, the group-like properties of the basis vectors are the most important, and the zero factor has little influence on the algebraic operation. The multiplication table of the basis vectors fully describes the intrinsic properties of the hypercomplex system, and the matrix <strong>A</strong> constructed from the multiplication table satisfies the structure equation <strong>A</strong><sup>2</sup> = <em>n</em><strong>A</strong> and thus obtains a set of faithful matrix representations of the basic elements. This paper also uses typical examples to show the simple and clear concepts and wide application of hypercomplex numbers. Therefore, hypercomplex numbers are worth learning in basic education and applying in scientific research and engineering technology.</p>

Generalization of Neural Networks on Second-Order Hypercomplex Numbers

The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the architecture and building blocks of complex-valued neural networks. In this paper, we generalize models by considering other types of hypercomplex numbers of the second order: dual and double numbers. We developed basic operators for these algebras, such as convolution, activation functions, and batch normalization, and rebuilt several real-valued networks to use them with these new algebras. We developed a general methodology for dual and double-valued gradient calculations based on Wirtinger derivatives for complex-valued functions. For classical computer vision (CIFAR-10, CIFAR-100, SVHN) and signal processing (G2Net, MusicNet) classification problems, our benchmarks show that the transition to the hypercomplex domain can be helpful in reaching higher values of metrics, compared to the original real-valued models.

Top-down metacomputing with algebraic dimensionality raising for automating theory-building to enable directly computable multiphysics models

The need to address the fundamental issues associated with the computational model generation process in the context of multiphysics theories that observe conservation laws and relevant assumptions motivates the present work. Accordingly, the goal of this paper is to describe, discuss, and demonstrate the benefits and implications of using metacomputing (i.e., the utilization of computational technologies for computing or generating programs that, in turn, compute the desired results) for theory or model generation, further assisted by metacomputing for raising the semantic and syntactic dimensionality of physics models for continua. The issues associated with the model generation process are identified at the outset. Then, a top-down approach that contains three distinct metacomputing activities is described. First, a meta-theoretical description of generating continuum multiphysics models defined in terms of quantities instantiated from the field of real numbers is described, along with its computational implementation. Then, a computational framework for raising the semantic dimensionality from the reals to the hypercomplex numbers is presented for both 2D and 3D continuum multiphysics problems. The semantic dimensionality is expressed in terms of the order of the algebra used to define the tensorial quantities of interest and their associated operations, as they participate in representing physics models expressed initially as systems of partial differential equations. The utilization of hypercomplex algebras, such as those for complex numbers and quaternions, is demonstrated for representative applications of models. Subsequently, syntactic dimensionality is defined in terms of the basis of the space utilized to write equational systems. Finally, a prototype of a computational system for converting equational systems to directed graphs termed “Algebraic Solution Graphs” is described. Examples of how to utilize these technologies are also presented to convey the efficacy and efficiency of the proposed approach.

Method for Constructing a Commutative Algebra of Hypercomplex Numbers

Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences.

Gauge theory on fiber bundle of hypercomplex algebras

I introduce a way of constructing a fiber bundle whose fibers are given by hypercomplex algebras and woven by appropriate structure group, and present that a novel gauge theory can be built on the hypercomplex fiber bundle. In this work, I aim to answer a question about how nature selects one preferred vacuum among degenerate physical vacua, called vacuum selection problem. In the end, I found presence of the impenetrable domain wall that prohibits phase transition between the two vacua. To be specific, I found that in this theory, one particular vacuum between two degenerate physical vacua for Higgs-like scalar potential can be dynamically chosen with priority due to intrinsic even parity of both a scalar field and its vacuum under a Z2 symmetry, even though its scalar potential is given to be Z2-symmetric under both odd- and even-parity transformations of the scalar field. This means that the vacuum selection problem can be resolved in this gauge theory. I suggest that this work may be a gateway to addressing the theoretical origin of the true physical vacuum that nature takes.

Proper adaptive filtering in four-dimensional Cayley–Dickson algebras

A family of hypercomplex algebras in four dimensions (4D) is proposed to devise adaptive filters. Such a family, called β-quaternions, has multiplication rules for the complex units that depend on a parameter β, and this family contains, as particular cases, both standard Hamilton quaternions and split quaternions. In this framework, two notions of properness for random vectors are introduced and their implications on the statistical processing involved are analyzed. Then, statistical tests to check properness in practice and a method to select the best algebra where the properness conditions could hold are provided. Also, proper adaptive filters are suggested and row and column updating problems are studied. The main advantage of the techniques proposed compared with the standard ones is that a notable reduction in the computational burden is achieved. Finally, simulation examples validate the proper adaptive filters and demonstrate that our scheme performs better than the traditional quaternion domain.

Special Issue: Real, Complex and Hypercomplex Number Systems in Data Processing and Representation

The evolution of human society is inevitably associated with the widespread development of computer technologies and methods, and the constant evolution of the theory and practice of data processing, as well as the need to solve increasingly complex problems in computational intelligence, have inspired the use of complex and advanced mathematical methods and formalisms for representing and processing big data sets [...]

On hyper complex numbers with higher order Pell numbers components

On hyper complex numbers with higher order Pell numbers components