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.
This study presents a new extension of the length-biased truncated Lomax distribution by incorporating the sine Topp-Leone family, resulting in a flexible model called the sine Topp-Leone length-biased truncated Lomax distribution. The proposed distribution demonstrates remarkable adaptability in modeling data with increasing hazard rates. Key statistical and reliability properties of the new model are thoroughly examined, including the survival function, hazard rate, reversed hazard rate, quantile function, moments, incomplete moments, and Renyi and Tsallis entropy measures. Parameter estimation is conducted using both classical and Bayesian methods, considering symmetric and asymmetric loss functions. Due to the computational challenges of Bayesian estimation, Markov Chain Monte Carlo techniques with independent gamma priors are employed. Simulation studies confirm the consistency of the proposed estimators, showing improved accuracy with increasing sample size, with Bayesan estimates provide lower absolute biases and mean squared errors. Finally, applications to two real datasets demonstrate the superior flexibility and effectiveness of the proposed distribution compared to existing alternatives.
This work investigates a porous elastic system under the Green–Naghdi heat conduction theory. We provide an explicit characterization of the decay rate, which may be exponential or polynomial, depending on the relationship between the wave propagation speeds. To verify the asymptotic behavior of the solution, we present an algorithm for obtaining a numerical solution using finite element methods and finite differences in time.
Dynamic boundary conditions provide a stringent verification setting for physics-informed neural networks (PINNs), because the approximation must satisfy an interior parabolic equation together with an evolution law posed on the boundary. This paper develops a transparent benchmark study for the one-dimensional heat equation with Wentzell-type dynamic boundary conditions. Two manufactured solutions are employed. Benchmark A is a cosine solution for which homogeneous boundary forcing is admissible only under the compatibility condition \(k=\alpha\pi^2\); it is intentionally retained as a reduced Wentzell test because the endpoint derivatives vanish and the boundary-gradient term is inactive. Benchmark B is a fully active test with nonzero endpoint derivatives at both boundaries, so that the \(\beta u_x\) contribution is explicitly exercised. The numerical evidence is based on directly computed norms, residuals, five independent random-feature seeds, an implemented finite-difference reference solver, and a targeted ablation study. The reported neural solver is a least-squares physics-informed random-feature network with hard initial-condition enforcement, \(250\) hidden features, \(3980\) interior collocation points, and \(401\) boundary times per endpoint. Over five seeds, the mean relative errors are \(E_{L^2}=(8.65\pm1.27)\times10^{-6}\) and \(E_{L^{\infty}}=(1.12\pm0.20)\times10^{-5}\) for Benchmark A, and \(E_{L^2}=(9.45\pm2.42)\times10^{-8}\) and \(E_{L^{\infty}}=(2.77\pm0.99)\times10^{-7}\) for Benchmark B. The finite-difference baseline displays approximately second-order convergence under the parabolic scaling \(\Delta t=2h^2\). These results show that the two-case manufactured suite provides a useful verification framework for smooth one-dimensional Wentzell-type dynamics, while the scope of the claims remains restricted to this benchmark setting rather than general neural-solver superiority.
In this paper, we present new sharp forms of Iyengar-type integral inequalities for differentiable and twice-differentiable functions. Using exact integral identities and the Cauchy-Schwarz inequality, we derive optimal \(L^2\)-bounds for the deviation between the midpoint value and the integral mean of a function. These results refine and extend several classical Hermite-Hadamard and Iyengar inequalities.
This study introduces the Secant–Weibull Autoregressive Conditional Duration (SW–ACD) model and its extension with exogenous calendar effects (SW–ACD–X) . The primary innovation is the integration of the Secant-Weibull distribution as the innovation law, which allows the framework to capture non-monotonic intensity shapes, such as unimodal and bathtub patterns, that are typically inaccessible to standard monotonic models. A significant methodological contribution is the SW–ACD–X model, which endogenously incorporates intraday seasonality into the conditional mean equation. This joint estimation strategy provides an integrated alternative to traditional two-step pre-filtering by simultaneously capturing the interaction between deterministic diurnal patterns and stochastic duration clustering. The numerical properties of the proposed models are assessed through Monte Carlo simulations, which demonstrate asymptotic consistency while highlighting inherent identification challenges in small-sample regimes. Model estimation is implemented using a dual approach: Frequentist Maximum Likelihood and Bayesian Hamiltonian Monte Carlo (HMC) via the No-U-Turn Sampler (NUTS) in RStan. Empirical application to high-frequency IBM transaction data shows that the SW–ACD–X exhibits promising fit advantages over established benchmarks, including the W–ACD–X, LW–ACD–X, G-ACD-X , and Lomax–ACD–X models. Comprehensive model selection based on AIC, BIC, WAIC, and LOOIC confirms that the proposed model is a robust tool for analyzing market microstructure, liquidity dynamics, and the complex patterns of high-frequency durations.
In this paper, we introduce and study the class of modified \((p, h)\)-convex stochastic processes, which unifies and extends several existing notions of convexity in the stochastic setting. We establish fundamental arithmetic properties of this class and derive Hermite–Hadamard-type inequalities using classical and fractional Katugampola-type operators. We also investigated Ostrowski-type and Jensen-type inequalities in a simple and unified framework. Our results generalize and unify many existing results in the literature, providing a comprehensive framework for the study of convex stochastic processes.
