Search Results (13)

In this study, we introduce the generalized Repunit sequence and its hybrid quaternion extension derived from a parametric recurrence relation that preserves the base-10 structure of classical Repunit numbers. Fundamental properties of the proposed sequences, including the characteristic equation, generating function, and Binet-type formula, are systematically investigated. Several algebraic identities, such as bilinear index-reduction formulas, are established to demonstrate the internal structure and consistency of the construction. Numerical experiments and graphical analyses are conducted to examine the structural behavior of the generalized Repunit sequence and its hybrid quaternion counterpart. While the scalar Repunit sequence exhibits regular and predictable growth, the hybrid quaternion extension displays significantly higher structural complexity and variability. Density distributions, contour plots, histogram representations, and discrete variation measures confirm the presence of enhanced diffusion and local irregularity in the quaternion-based structure. These statistical, graphical, and numerical findings indicate that generalized Repunit hybrid quaternion sequences possess properties that are relevant to encoding, masking, and preprocessing mechanisms in applied mathematical and computational frameworks. However, this work does not propose a complete cryptographic algorithm, nor does it claim compliance with established cryptographic security standards such as NIST SP 800-22. The results should therefore be interpreted as pre-cryptographic indicators that motivate further research toward rigorous security evaluation, algorithmic development, and broader applications in areas such as coding theory, signal processing, and nonlinear dynamical systems. Full article

This paper aims to define the Obata-type connection on a $4n$-dimensional smooth manifold equipped with a 3-parameter generalized quaternion structure. The paper demonstrates some general algebraic properties of this connection. The integrability of endomorphisms of the tangent bundle is interpreted as homomorphisms of certain skew-symmetric algebroids. The results confirm the uniqueness of a torsion-free connection compatible with generalized quaternion structures, assuming their integrability for non-zero parameters. Full article

This work breaks a 180-year-old framework created by Hamilton both with regard to the use of imaginary quantities and the definition of a quaternion product. The general quaternionic algebraic structure we are considering was provided by the author in a previous work with a commutative product and will be provided here with a non-commutative product. We replace the imaginary units usually used in the theory of quaternions by linearly independent vectors and the usual Hamilton product rule by a Hamiltonian-adapted vector-valued vector product and prove both a new geometric property of this product and a vectorial adopted Euler type formula. Full article

This paper primarily investigates the solutions to the system of equations $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$. This system generalizes the classical equation $AXB=C$, as well as the system of equations $AX=B$ and $XC=D$, and finds broad applications in control theory, signal processing, networking, optimization, and other related fields. Various methods for solving this system are introduced, including the generalized inverse method, the $\mathrm{vec}$-operator method, matrix decomposition techniques, Cramer’s rule, and iterative algorithms. Based on these approaches, the paper discusses general solutions, symmetric solutions, Hermitian solutions, and other special types of solutions over different algebraic structures, such as number fields, the real field, the complex field, the quaternion division ring, principal ideal domains, regular rings, strongly *-reducible rings, and operators on Banach spaces. In addition, matrix systems related to the system $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$ are also explored. Full article

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article

This survey provides a review of the theoretical research on the classic system of matrix equations $AX=C$ and $XB=D$, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert $C^{\ast }$-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in color images. Full article

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, $Q_8$, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion. Full article

In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)$\otimes$SU(2)$\otimes$U(1) symmetry. 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

In this work, we began to take forward kinematics of the Gough–Stewart (G-S) platform as an unconstrained optimization problem on the Lie group-structured manifold SE(3) instead of simply relaxing its intrinsic orthogonal constraint when algorithms are updated on six-dimensional local flat Euclidean space or adding extra unit norm constraint when orientation parts are parametrized by a unit quaternion. With this thought in mind, we construct two kinds of iterative problem-solving algorithms (Gauss–Newton (G-N) and Levenberg–Marquardt (L-M)) with mathematical tools from the Lie group and Lie algebra. Finally, a case study for a general G-S platform was carried out to compare these two kinds of algorithms on SE(3) with corresponding algorithms that updated on six-dimensional flat Euclidean space or seven-dimensional quaternion-based parametrization Euclidean space. Experiment results demonstrate that those algorithms on SE(3) behave better than others in convergence performance especially when the initial guess selection is near to branch solutions. Full article

In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic $C^{*}$ -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions. Full article