This article proposes an enhanced accelerated iterative method for solving systems of nonlinear equations based on a double-step-length strategy. To improve computational efficiency, the Jacobian matrix in Newton’s method is approximated using an acceleration mechanism, resulting in a fully derivative-free scheme. Moreover, an inexact line-search technique, inspired by Li and Fukushima [1], is employed to determine the two step lengths adaptively. Under mild assumptions, the global convergence of the proposed method is rigorously established. Numerical experiments confirm the efficiency and robustness of the method, demonstrating superior performance compared with approaches reported in the literature. In addition, the proposed technique is successfully applied to motion control problems involving two-joint planar robotic manipulators, highlighting its practical applicability and effectiveness.
This article develops an application of the generalized method of lines to a time-dependent Navier–Stokes system in fluid mechanics. In the last section, we present a numerical example and the corresponding software in order to illustrate the applicability of the results.
In this paper, we study pairwise asymptotic criteria for sequences of compact sets in the Hausdorff hyperspace of a metric space. We introduce the notions of \(\mathcal I\)-statistically pre-Cauchy and frequent Cauchy compact-set sequences and examine their relation to convergence under suitable additional assumptions. After establishing basic permanence properties, we show that \(\mathcal I\)-statistical convergence implies the corresponding pre-Cauchy condition. For bounded sequences, we obtain two explicit characterizations, one in terms of double means of Hausdorff distances and the other in terms of bounded moduli. We also investigate the relation between frequent Cauchy behavior and frequent convergence under compactness assumptions on the hyperspace or on the closure of the range of the sequence. Finally, we present criteria under which pairwise asymptotic conditions can be upgraded to actual \(\mathcal I\)-statistical convergence, and we include examples that illustrate both the scope and the limitations of the pairwise approach in the hyperspace setting.
The article continues the study of a class of fourth-order nonlinear differential equations. Earlier, using an analytical approximation method, a theorem of existence and uniqueness (analogous to the Cauchy–Kovalevskaya theorem) was formulated and proved in the vicinity of a movable singular point of algebraic type (hereafter referred to as the movable singular point) in the complex domain. This work addresses the problem of the influence of the perturbation (error) of the movable singular point on the applicability domain of the analytical approximate solution. Two variants of estimating the applicability domain of the analytical approximate solution are considered.
The relationship between mathematics and music is as ancient as it is fascinating. This reciprocal contribution creates a unique synergy: while music provides “color” to mathematical abstractions, mathematics offers structural support to the most elusive of the arts. Although many arguments regarding this connection have been proposed – some profound, others tenuous – one fact remains certain: the scales of every musical culture are fundamentally grounded in arithmetic. Our analysis first axiomatizes the equal-tempered system as a geometric partition of the frequency spectrum, exploring its algebraic properties and transpositional invariance. Subsequently, the Pythagorean system is formalized through the powers of the \(3:2\) ratio, highlighting the inherent conflict between rational purity and the necessity of a closed harmonic circle. Finally, we discuss the “bracketing” property for a twelve-tone system, where the equal-tempered notes of the chromatic scale are encompassed by Pythagorean pairs derived from upward and downward cycles of fifths.
The enhanced thermal performance of nanofluids is highly important for improving heat transfer performance in diversified engineering applications. Furthermore, the interaction of chemical reactions is important in biomedical applications such as targeted therapy, drug delivery, and related processes. The present investigation aims to study the time-dependent stagnation-point flow of a two-phase model nanofluid along a stretching surface, emphasizing the diversified roles of Brownian motion vis-\`a-vis thermophoresis in the presence of chemical species. Moreover, the conducting fluid flowing through a porous medium affects the flow phenomena with the simultaneous involvement of thermal radiation and a heat source. The proposed mathematical model, originally expressed in dimensional form, is transformed into a non-dimensional form through the introduction of suitable similarity rules, and the resulting transformed set of equations is handled numerically. In particular, a numerical technique based on the fourth-order Runge–Kutta method is employed for the solution of the transformed equations. The physical significance of several parameters associated with the flow phenomena is presented graphically and discussed briefly. The key findings include the roles of Brownian motion and thermophoresis in enhancing nanoparticle transport, which improves thermal efficiency. The magnetization effect, together with the porous medium, plays a critical role in controlling the flow and thermal characteristics.
This paper investigates the emergence of chaotic behavior in generalized shift dynamical systems defined on functional Alexandroff spaces. We provide precise characterizations of Devaney chaos and Li–Yorke chaos in terms of the injectivity of the inducing map, the absence of periodic points, and the existence of infinite orbits. Our results extend prior work on topological dynamics and symbolic systems, with adapted proofs utilizing the chain decomposition of functional Alexandroff spaces. Additionally, we highlight analogies to fractal structures such as the devil’s staircase in routes to chaos, where finite chains correspond to non-chaotic regimes and infinite chains support chaotic dynamics, though this serves as intuitive motivation rather than formal proof.
In this paper, we introduce Fibonacci and Lucas quasi-quaternions by combining classical number sequences with the structure of quasi-quaternion algebra. We investigate their fundamental algebraic properties, including real and imaginary parts, conjugates, norms, and recurrence relations. We establish Binet-type formulas, generating functions, and sum formulas for these sequences in the quasi-quaternionic setting. In addition, we derive several classical identities, such as the Cassini, Catalan, d’Ocagne, Vajda, and Honsberger identities, adapted to Fibonacci and Lucas quasi-quaternions. Furthermore, we present matrix representations of these structures and obtain explicit expressions for the powers of the associated matrices. We also consider De Moivre-type formulas in the quasi-quaternion framework and analyze the behavior of these sequences under repeated operations. The graphical representations complement the theoretical results by illustrating the structural and asymptotic behavior of these quasi-quaternionic sequences.
For quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) with discriminant \(d\), \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\), and four simple \(3\)-principalization types \(\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace\), we establish necessary and sufficient conditions for the Galois group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) of the unramified Hilbert \(3\)-class field tower of \(k\) to coincide with the Galois group \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) of the maximal metabelian unramified \(3\)-extension of \(k\). In the case of non-coincidence, we study the path between \(M\) and \(S\) in the descendant tree of the elementary bicyclic \(3\)-group \((\mathbb{Z}/3\mathbb{Z})^2\). For two complex \(3\)-principalization types \(\varkappa(k)\in\lbrace (2122),(4231)\rbrace\), we show that infinitely many non-metabelian possible Galois groups \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) with presumably unbounded derived length \(\mathrm{dl}(S)\) share a common metabelianization \(M=S/S^{\prime\prime}\), whence only partial criteria can be stated. Minimal discriminants \(d>0\) with assigned simple \(3\)-principalization type \(\varkappa(k)\) and fixed length \(\ell_3(k)\in\lbrace 2,3\rbrace\) of the \(3\)-class field tower are determined experimentally for nilpotency class \(\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace\) under assumption of the generalized Riemann hypothesis.
In this paper, we investigate the dynamical behavior of a four-dimensional fractional-order chaotic system using a Caputo fractional derivative. The examination of local stability reveals that the system undergoes Hopf bifurcations, leading to oscillatory states. Numerical findings based on Lyapunov exponents, bifurcation diagrams, and phase portraits validate the existence of chaos and hyperchaos over extensive parameter ranges. Furthermore, the system exhibits diverse chaotic phenomena, including self-excited attractors produced from unstable equilibria and coexisting attractors that emerge under different initial conditions. For numerical simulations, the Adams-Bashforth-Moulton predictor-corrector method is employed. Also, analytical calculations are carried out using the Maple program, and the MATLAB software program is used to illustrate the results. The results demonstrate that the proposed fractional-order system accurately represents multistability, coexisting chaotic attractors, and complex dynamics depending on parameters and derivative order.
