Hypercomplex Numbers Research Articles

Orbit propagation in Minkowskian geometry

The geometry of hyperbolic orbits suggests that Minkowskian geometry, and not Euclidean, may provide the most adequate description of the motion. This idea is explored in order to derive a new regularized formulation for propagating arbitrarily perturbed hyperbolic orbits. The mathematical foundations underlying Minkowski space–time $$\mathbb {M}$$ are exploited to describe hyperbolic orbits. Hypercomplex numbers are introduced to define the rotations, vectors, and metrics in the problem: the evolution of the eccentricity vector is described on the Minkowski plane $$\mathbb {R}_1^2 $$ in terms of hyperbolic numbers, and the orbital plane is described on the inertial reference using quaternions. A set of eight orbital elements is introduced, namely a time-element, the components of the eccentricity vector in $$\mathbb {R}_1^2 $$ , the semimajor axis, and the components of the quaternion defining the orbital plane. The resulting formulation provides a deep insight into the geometry of hyperbolic orbits. The performance of the formulation in long-term propagations is studied. The orbits of four hyperbolic comets are integrated and the accuracy of the solution is compared to other regularized formulations. The resulting formulation improves the stability of the integration process and it is not affected by the perihelion passage. It provides a level of accuracy that may not be reached by the compared formulations, at the cost of increasing the computational time.

Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies

Abstract. In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, neutrosophic intervals, neutrosophic hypercomplex numbers of dimension n, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures. The aim of this paper is to construct examples of splitting the literal indeterminacy ሺࡵሻ into literal sub-indeterminacies ሺࡵ ,૚ࡵ,...,૛࢘ࡵሻ, and to define a multiplication law of these literal sub-indeterminacies in order to be able to build refined ࡵ െ neutrosophic algebraic structures. Also, examples of splitting the numerical indeterminacy ሺ࢏ሻ into numerical sub-indeterminacies, and examples of splitting neutrosophic numerical components into neutrosophic numerical sub-components are given.

ФОРМУЛЫ ФИЗИЧЕСКИХ ЗАКОНОВ В ГИПЕРКОМПЛЕКСНОЙ СРЕДЕ: ТЕОРИИ ОТНОСИТЕЛЬНОСТИ

ФОРМУЛЫ ФИЗИЧЕСКИХ ЗАКОНОВ В ГИПЕРКОМПЛЕКСНОЙ СРЕДЕ: ТЕОРИИ ОТНОСИТЕЛЬНОСТИ

CORDIC-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

Quaternions have offered a new paradigm to the signal processing community: to operate directly in a multidimensional domain. We have recently introduced the quaternionic approach to the design and implementation of paraunitary filter banks: four- and eight-channel linear-phase paraunitary filter banks, including those with pairwise-mirror-image symmetric frequency responses. The hypercomplex number theory is utilized to derive novel lattice structures in which quaternion multipliers replace Givens (planar) rotations. Unlike the conventional algorithms, the proposed computational schemes maintain losslessness regardless of their coefficient quantization. Moreover, the one regularity conditions can be expressed directly in terms of the quaternion lattice coefficients and thus easily satisfied even in finite-precision arithmetic. In this paper, a novel approach to realizing CORDIC-lifting factorization of paraunitary filter banks is presented, which is based on the embedding of the CORDIC algorithm inside the lifting scheme. Lifting allows for making multiplications invertible. The 2D CORDIC engine using sparse iterations and asynchronous pipeline processor architecture based on the embedded CORDIC engine as stage of processor is reported. Also it is necessary to notice, that the quaternion multiplier lifting scheme based on the 2D CORDIC algorithm is the structural decision for the lossless digital signal processing. This approach applies to very practical filter banks, which are essential for image processing, and addresses interesting theoretical questions.

On a visualization of the convergence of the boundary of generalized Mandelbrot set to (n-1)-sphere

In this article we analyze the generalized Mandelbrot set in higher-order hyper- complex number spaces following both the Cayley-Dickson construction algebraic spaces and the spaces defined by Clifford algebras. The particular case of the generalized 3D quasi-Mandelbrot set was also considered. In particular, we investigated the increase of power of the iterated variable and proved that when this power tends to infinity, the Man- delbrot set is convergent to the unit circle. The same is true for the generalized Mandelbrot sets in higher-dimensional hypercomplex number spaces, i.e. when the power of iterated variable tends to infinity, the generalized Mandelbrot set is convergent to the unit ( n-1)- sphere. The results of our investigation were visualized for the generalized Mandelbrot set in a complex number space and the generalized quasi-Mandelbrot set in a 3D Euclidean space.

An unified approach for developing rationalized algorithms for hypercomplex number multiplication

In this article we present a common approach for the development of algorithms for calculating products of hypercomplex numbers. The main idea of the proposed approach is based on the representation of hypernumbers multiplying via the matrix-vector products and further creative decomposition of the matrix, leading to the reduction of arithmetical complexity of calculations. The proposed approach allows the construction of sufficiently well algorithms for hypernumbers multiplication with reduced computational complexity. If the schoolbook method requires N 2 real multiplications and N(N-1) real additions, the proposed approach allows to develop algorithms, which take only (N(N-1)/2)+2 real multiplications and 3Nlog2N+(N(N-3)+4)/2 real additions. Streszczenie. W artykule zostalo przedstawione uogolnione podejście do syntezy algorytmow wyznaczania iloczynow liczb hiperzespolonych. Glowna idea proponowanego podejścia polega na reprezentacji operacji mnozenia liczb hiperzespolonych w formie iloczynu wektorowo- macierzowego i dalszej mozliwości kreatywnej dekompozycji czynnika macierzowego prowadzącej do redukcji zlozoności obliczeniowej. Proponowane podejście pozwala zbudowac algorytmy wyrozniające sie w porownaniu do metody naiwnej zredukowaną zlozonością obliczeniową. Jeśli metoda naiwna wymaga wykonania N 2 mnozen oraz N(N-1) dodawan liczb rzeczywistych to proponowane podejście pozwala syntetyzowac algorytmy wymagające tylko (N(N-1)/2)+2 mnozen oraz 3Nlog2N+(N(N-3)+4)/2 dodawan. (Uogolnione podejście do konstruowania zracjonalizowanych algorytmow mnozenia liczb hiperzespolonych - tytul polski artykulu).

Дослідження зв’язків між узагальненими кватерніонами та процедурою подвоєння гіперкомплексних числових систем

The class of non-commutative hypercomplex number systems of 4-dimension, constructed by using of non-commutative procedure of Grassman-Clifford doubling of 2-dimensional systems is investigated and established their relations with the generalized quaternions. Algorithms of the operations and methods of algebraic characteristics calculation in them, such as conjugation, normalization and a type of zero dividers have been investigated. Tabl.: 3. Refs: 14 titles.

“General Theory of Particle Mechanics” arising from a fractal surface

The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical and relativistic mechanics can be regarded as mathematical consequences of a single condition of stability of exceptional algebras of real, complex and quaternion numbers under transformations of primitive constituents of their units and elements. In the course of the study a notion of basic fractal surface underlying the physical three-dimensional space is introduces, and an original geometric treatment (admitting visualization) of some formerly considered abstract functions (mechanical action, space-time interval) are suggested.

Construction of hyperdeterminants

Constructions of several multidimensional determinants for matrices in dimensions n = 2m are described. With one exception, the constructions are facilitated by isomorphisms with rings of commutative hypercomplex numbers. Properties shared by the Cayley hyperdeterminant, such as the multiplicative property |AB| = |A||B|, are established and the degrees and homogeneity of the determinants are discussed. These considerations emphasize the close relationship between order 2 multidimensional matrices in dimension m and hypercomplex systems in dimension n = 2m.

Застосування неканонічних гіперкомплексних числових систем для оптимізації сумарної параметричної чутливості реверсивних цифрових фільтрів

Constructing optimal parametric sensitivity method based on using of non-canonical hypercomplex number systems (HNS) is proposed and investigated. It has been shown that the use of non-canonical HNS with grater number of non-zero structure constants in multiplication table can significantly improve the sensitivity of the digital filter. Fig.: 6. Refs: 11 titles.

Обчислювальні властивості одного класу некомутативних гіперкомплексних числових систем четвертої вимірності

Досліджено клас некомутативних гіперкомплексних числових систем (ГЧС) четвертої вимірності, які побудовано за допомогою некомутативної процедури подвоєння Грасмана-Кліфорда систем другої вимірності. Побудовано всі ГЧС цього класу, досліджено алгоритми виконання операцій у них а також методи обчислення таких алгебраїчних характеристик, як спряження, нормування, вигляд дільників нуля. Виведено формули представлення експоненціальноїфункціїу цих системах.

Digital Fingerprinting Based on Quaternion Encryption Scheme for Gray-Tone Images

In this paper a new idea of digital images fingerprinting is proposed. The method is based on quaternion encryption in the Cipher Block Chaining (CBC) mode. Quaternions are hyper-complex numbers of rank 4 and thus often applied to mechanics in three-dimensional space. The encryption algorithm described in the paper is designed for graytone images but can easily be adopted for color ones. For the encryption purpose, the algorithm uses the rotation of data vectors presented as quaternions in a three-dimensional space around another quaternion (key). On the receiver’s side, a small amount of unnoticeable by human eye errors occurs in the decrypted images. These errors are used as a user’s digital fingerprint for the purpose of traitor tracing in case of copyright violation. A computer-based simulation was performed to scrutinize the potential presented quaternion encryption scheme for the implementation of digital fingerprinting. The obtained results are shown at the end of this paper.

Синтез матричних представлень узагальнених кватерніонів

It is synthesized matrix representations of a class of non-commutative hypercomplex number sys-tems of the fourth dimension built with the help of non-commutative procedures of doubling of the Grassmann-Clifford systems of the second dimension. Refs: 12 titles.

Алгоритми швидкого обчислення циклічної згортки з поданням дискретних сигналів гіперкомплексними числами

Представлено методи синтезу алгоритмів швидкого обчислення циклічної згортки числових масивів довжиною 2п. Ці методи базуються на поданні їх у спеціальних гіперкомплексних числових системах, які мають такі ізоморфні їм системи, що виконання гіперкомплексних операцій в них потребує меншої кількості дійсних операцій. Бібліогр.: 9 найм.

Rotacija z enotskim kvaternionom

A quaternion is a hyper-complex number. A rule for quaternion multiplications allows us to use it as a rotation in three-dimensional space. The aim of this article is to present quaternion rotations to the Slovene professional geodetic public. Quaternions are described in the article along with the manner to use them for rotations. Two experiments were performed to compare the rotations using quaternions versus rotations with Euler angles. The experiments revealed that transformations using quaternion are more efficient than transformations using Euler angles. An adjustment for the determination of transformation parameters is also more efficient if the mathematical model is based on quaternion rotation.

Hyperholomorphic Function Theory and Clifford Analyticity

As a special issue of this highly esteemed journal, we were pleased to invite the interested authors to contribute their original research papers as well as good expository papers to this special issue that will make better improvement on the theory of Clifford analysis and its application tomathematical physics, providing new approaches to differential geometry using Clifford’s geometric analysis. We suggested the following topics: theory of hyperholomorphic functions, regular functions,monogenic functions, hypercomplex number, dual number systems, spilt number systems, bicomplex numbers, and split biquaternions and pseudoquaternions; analytic extensions and applications; general theory of complex analytic spaces; complex partial differential operators, quaternion matrix equations, and generalized Cauchy-Riemann systems; domains of hyperholomorphy; and complex function spaces and hyperconjugate harmonic function. Certain papers that have significant results on complex analysis in the widest sense were intended to be welcome. Besides some papers belonging to the above-intended topics, we are also happy to publish, in this special issue, several other papers regarding analytic number theory, qseries and combinatorics, and special functions and the theory of group representations.

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).

SQUARE OF THE ERROR OCTONIONIC THEOREM AND HYPERCOMPLEX FOURIER SERIES

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions. </span></p></div></div></div>

A MULTIPLICATIVE DETERMINANT FOR 2m-DIMENSIONAL MATRICES

A multiplication for a specific nested collection of multidimensional matrices is defined by association with a system of n = 2m-dimensional hypercomplex numbers. A totally symmetric and multiplicative determinant is then derived from the system which extends the Cayley hyperdeterminant to these higher dimensions. The determinant is related to the zero divisors of the system of hypercomplex numbers. Properties of the determinant are then discussed.

OCTONIC GRAVITATIONAL FIELD EQUATIONS

Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell–Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein–Gordon equation for linear gravity are also developed.