Search Results (14)

Mathematics is the wide-ranging solid foundation of the engineering sciences which ensures their progress by providing them with its unique toolkit of rules, methods, algorithms and numerical systems. In this paper, an overview of the numerical systems that have currently found an application in engineering science and practice is offered, while also mentioning those systems that still await full and comprehensive applicability, recognition, and acknowledgment. Two possible approaches for representing hypercomplex numbers are proposed—based on real numbers and based on complex numbers. This makes it possible to justify the creation and introduction of numerical systems specifically suited to digital signal processing (DSP), which is the basis of all modern technical sciences ensuring the technological progress of mankind. Understanding the specifics, peculiarities, and properties of the large and diverse family of hypercomplex numbers is the first step towards their more comprehensive and thorough study, and hence their use in a number of high-tech intelligent applications in various engineering and scientific fields, such as information and communication technologies (ICT), communication and neural networks, cybersecurity and national security, artificial intelligence (АI), space and military technologies, industrial engineering and machine learning, astronomy, applied mathematics, quantum physics, etc. The issues discussed in this paper are, however, far from exhausting the scientific topics related to both hypercomplex numbers in general and those relevant to DSP. This is a promising scientific area, the potential of which has not yet been fully explored, but research already shows the enhanced computational efficiency and intelligent performance of hypercomplex DSP. Full article

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative. Full article

In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over ${\mathbb{Z}}_p$. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over ${\mathbb{Z}}_3$ and we present a method to encrypt plain texts, by using invertible elements in some of these algebras. Full article

A generalized construction procedure for algebraic number systems is hereby presented. This procedure offers an efficient representation and computation method for complex numbers, quaternions, and other algebraic structures. The construction method is then illustrated across a range of examples. In particular, the novel developments reported herein provide a generalized form of the Cayley–Dickson construction through involutive dimagmas, thereby allowing for the treatment of more general spaces other than vector spaces, which underlie the associated algebra structure. Full article

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types. Full article

The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over the real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexified Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration. Full article

Due to the computational aspects which appear in the study of algebras obtained by the Cayley–Dickson process, it is difficult to obtain nice properties for these algebras. For this reason, finding some identities in such algebras plays an important role in obtaining new properties of these algebras and facilitates computations. In this regard, in the first part of this paper, we present some new identities and properties in algebras obtained by the Cayley–Dickson process. As another remark regarding the computational aspects in these algebras, in the last part of this paper, we solve some quadratic equations in the real division quaternion algebra when their coefficients are some special elements. These special coefficients allowed us to solve interesting quadratic equations, providing solutions directly, without using specialized softs. Full article

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state $\left|\psi \right.⟩$ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group ${\pi }_1\left(V\right)$ of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean ${\mathbb{R}}^4$ ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture $GQ(2,2)$ (at index 15), and the combinatorial Grassmannian Gr $(2,8)$ (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$ s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’. Full article

This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found under which such algebras are separable. Algebras satisfying these conditions are described. Full article

Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups are investigated. Necessary structural properties of metagroups are studied. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley–Dickson algebras. Full article

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation. Full article

Given a 2^N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units e_a , 1 ≤ a ≤ 2^N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {e_a, e_b , e_c} , 1 ≤ a < b < c ≤ 2^N - 1 and e_ae_b = ±e_c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C_N; in particular, C₃ (octonions) is isomorphic to the Pasch (6₂, 4₃) -configuration, C₄ (sedenions) is the famous Desargues (10₃) -configuration, C₅ (32-nions) coincides with the Cayley-Salmon (15₄, 20₃) -configuration found in the well-known Pascal mystic hexagram and C₆ (64-nions) is identical with a particular (21₅, 35₃) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C_N leads to a conjecture that C_N is isomorphic to a combinatorial Grassmannian of type G₂(N + 1). Full article

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888). Full article