Search Results (19)

In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition for the angle measurement between two spacelike vectors whose vector product is lightlike. Later, we generalize the Lorentzian rotation about a lightlike axis, and determine matrices of these transformations using the Cartan frame and the well-known Rodrigues formula, then using the Cayley map, and finally using the generalized split quaternions. We see that such transformations give parabolic rotational motions on general cones or general hyperboloids of one or two sheets, while they also give linear rotational motions on general hyperboloids of one sheet. Full article

The system of extended ordered XOR-inclusion problems (in short, SEOXORIP) involving generalized Cayley and Yosida operators is introduced and studied in this paper. The solution is obtained in a real ordered Banach space using a fixed-point approach. First, we develop the fixed-point lemma for the solution of SEOXORIP. By using the fixed-point lemma, we develop a three-step iterative scheme for obtaining the approximate solution of SEOXORIP. Under the Lipschitz continuous assumptions of the cost mappings, the strong convergence of the scheme is demonstrated. Lastly, we provide a numerical example with a convergence graph generated using MATLAB 2018a to verify the convergence of the sequence generated by the proposed scheme. Full article

In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks. Full article

This paper introduces a novel closed-form coordinate-free expression for the higher-order Cayley transform, a concept that has not been explored in depth before. The transform is defined by the Lie algebra of three-dimensional vectors into the Lie group of proper orthogonal Euclidean tensors. The approach uses only elementary algebraic calculations with Euclidean vectors and tensors. The analytical expressions are given by rational functions by the Euclidean norm of vector parameterization. The inverse of the higher-order Cayley map is a multi-valued function that recovers the higher-order Rodrigues vectors (the principal parameterization and their shadows). Using vector parameterizations of the Euler and higher-order Rodrigues vectors, we determine the instantaneous angular velocity (in space and body frame), kinematics equations, and tangent operator. The analytical expressions of the parameterized quantities are identical for both the principal vector and shadows parameterization, showcasing the novelty and potential of our research. Full article

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-${\displaystyle \frac{1}{2}}$ systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems. Full article

In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using the well-known Rodrigues, Cayley, and Householder maps, we produce the generalized Galilean rotation matrices. Finally, we show that these rotation matrices can also be used to determine parabolic rotational motion. Full article

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. 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

The twisted hypersurfaces $\mathfrak{x}$ with the $(0,0,0,0,1)$ rotating axis in five-dimensional Euclidean space ${\mathbb{E}}^5$ is considered. The fundamental forms, the Gauss map, and the shape operator of $\mathfrak{x}$ are calculated. In ${\mathbb{E}}^5$, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces $\mathfrak{x}$ are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of $\mathfrak{x}$ are determined. Additionally, the Laplace–Beltrami operator relation of $\mathfrak{x}$ is given. Full article

The rigid body displacement mathematical model is a Lie group of the special Euclidean group SE (3). This article is about the Lie algebra se (3) group. The standard exponential map from se (3) onto SE (3) is a natural parameterization of these displacements. In technical applications, a crucial problem is the vector minimal parameterization of manifold SE (3). This paper presents a unitary variant of a general class of such vector parameterizations. In recent years, dual algebra has become a comprehensive framework for analyzing and computing the characteristics of rigid-body movements and displacements. Based on higher-order fractional Cayley transforms for dual quaternions, higher-order Rodrigues dual vectors and multiple vectorial parameters (extended by rotational cases) were computed. For the rigid body movement description, a dual tangent operator (for any vectorial minimal parameterization) was computed. This paper presents a unitary method for the initial value problem of the dual kinematic equation. Full article

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space ${\mathbb{E}}_2^5$. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a $5\times 5$ matrix. Full article

We investigate the class of helical hypersurfaces parametrized by $\mathfrak{x}=\mathfrak{x}(u,v,w)$, characterized by a light-like axis in Minkowski spacetime ${\mathbb{L}}^4$. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of $\mathfrak{x}$. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with $\mathfrak{x}$. We explore the conditions under which the curvatures of $\mathfrak{x}$ possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in ${\mathbb{L}}^4$. Full article

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. We compute the first and second fundamental form matrices, Gauss map, and shape operator matrix of the hypersurface. Additionally, we compute the curvatures of the hypersurface by using the Cayley–Hamilton theorem. Moreover, we give some relations for the mean and the Gauss–Kronecker curvatures of the hypersurface. Finally, we obtain the Laplace–Beltrami operator of the hypersurface. Full article

In this paper, the generalized helical hypersurfaces $\mathbf{x}=\mathbf{x}(u,v,w)$ with a time-like axis in Minkowski spacetime ${\mathbb{E}}_1^4$ are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of $\mathbf{x}$ are calculated. Moreover, the curvatures of the generalized helical hypersurface $\mathbf{x}$ are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of $\mathbf{x}$ are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in ${\mathbb{E}}_1^4$. Full article

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems. Full article