This study investigates the algebraic and differential properties of balancing-like polynomials, which constitute a polynomial extension of balancing-like sequences. Several fundamental properties are established, including matrix representations, addition and subtraction formulas, and polynomial analogues of the Cassini, Catalan, D’Ocagne, and Vajda identities. A product factorization is obtained, which yields an explicit description of the zeros of these polynomials. Moreover, a direct connection with the Chebyshev polynomials of the second kind is established, and the ordinary generating function of the balancing-like polynomials is derived. The derivative sequences associated with these polynomials are then systematically investigated. Explicit formulas, recurrence relations, and a self-convolution identity are obtained, together with a relation recovering the original polynomials from their neighboring derivative terms. Finally, the analysis is extended to higher-order derivatives, and a generating-function representation and a general recurrence relation for the resulting differential hierarchy are derived. The differential framework is further extended to non-integer orders by means of the Caputo fractional derivative. An explicit fractional derivative formula and a generating-function representation are obtained.
We study the multifractal structure of a branching random walk on the boundary of a finite-type Galton–Watson tree. For α∈R, we consider the level set EX(α)=t∈∂T:limn→∞SnX(t)n=α. The multi-type structure leads naturally to the weighted mean offspring matrix M(q) and to the deterministic pressure P˜(q)=logρ(M(q)). Using the corresponding Perron–Frobenius eigenvectors, we construct a family of multi-type Mandelbrot martingales and prove their almost sure and Lp uniform convergence on compact subsets of the admissible parameter domain. This yields the associated Mandelbrot measures and the uniform estimates required for the multifractal analysis. As a consequence, for every α in a suitable admissible set I, we obtain almost surely dimHEX(α)=P˜*(α), where P˜* denotes the Legendre transform of P˜ and dimH stands for the Hausdorff dimension. Thus the classical multifractal formalism for one-type branching random walks extends to the genuinely multi-type setting, with the scalar normalization replaced by the Perron–Frobenius spectral structure of the weighted offspring matrices.
The exact traveling-wave solutions, equilibrium structures, stability characteristics, bifurcation phenomena, and complicated nonlinear dynamics of a space-time fractional modified Benjamin–Bona–Mahony (mBBM) equation are thoroughly discussed in this article. The governing fractional partial differential equation is converted to a nonlinear ordinary differential equation by implementing a suitable traveling-wave transformation, wherein the fractional-order parameter β serves as a spatial–temporal scaling factor in the physical coordinate frame. This ordinary differential equation is then solved using the improved modified extended tanh-function approach. Various explicit traveling-wave solutions are successfully constructed and rigorously classified into smooth localized solitary waves (such as bell-shaped and kink-type profiles), doubly periodic Jacobi elliptic functions, singular periodic waves, and rational solutions. Their explicit domains and singularities are clearly discussed. Eigenvalue investigation and linearization are employed to identify the equilibrium points and evaluate their local stability. Furthermore, changes between regular and complex patterns of behavior, such as periodic, quasi-periodic, and chaotic dynamics, are shown via bifurcation and numerical analysis by introducing an external periodic excitation into the reduced planar system. The findings reveal that the fractional parameter β modifies the wave profile geometry and propagation speed, while the chaos is mainly influenced by external forcing parameters and nonlinear system coefficients. The improved modified extended tanh-function approach, along with dynamical-systems analysis, offers a comprehensive foundation for expressing both explicit wave patterns and complicated nonlinear processes.
Disturbances from multiple sources propagate through coupled motor, hydraulic, and mechanical channels in electro-hydrostatic actuators, while increasing observer bandwidth introduces a trade-off with sensitivity to high-frequency components. To address these challenges, this study proposes a cascaded control method that integrates parallel disturbance estimation with fractional-order error damping. A detailed model incorporating the instantaneous flow rate of a piston pump, leakage, fluid compressibility, and dynamic friction is developed, and a control-oriented model is derived through time-scale separation. A three-channel parallel observer is constructed according to the locations at which disturbances enter the state equations, enabling corresponding compensation within a position–pressure difference–rotational speed backstepping control cascade. A Caputo fractional derivative is introduced into the motor-level error feedback to shape the magnitude and phase characteristics within the target frequency band by adjusting the fractional order and damping gain. This term is implemented using a rational approximation over a finite frequency band. The approximation filter states are incorporated into a composite Lyapunov function. Under prescribed filter dissipation conditions, positive feedback gains, and uniformly bounded equivalent residuals, uniform ultimate boundedness of the augmented continuous-time closed-loop error dynamics is established, together with residual-dependent error bounds. Co-simulation results show that the position root-mean-square error during multilevel step tracking is 5.68 mm, representing reductions of 45.8%, 24.6%, and 12.6% relative to proportional–integral–derivative control, fractional-order linear active disturbance rejection control, and cascaded sliding-mode control with multiple-disturbance observation, respectively. In semi-physical experiments, the steady-state position error during the plateau phases remains largely within ±2 mm under loads of 5 kN and 10 kN. During motion-reversal tracking under variable loads, the position root-mean-square error is reduced by 26.8% and 14.8% relative to the first two baseline methods, respectively. These results demonstrate that, under the tested conditions, channel-specific disturbance compensation combined with fractional-order dynamic shaping improves actuator tracking performance and provides a basis for fractional-order controller design and finite-dimensional implementation analysis in coupled electrohydraulic systems.
The reliability analysis of integrated circuit (IC) isolation structures (ISs) faces two challenges: how to accurately describe the distribution of time-to-failure (TTF) data, and how to obtain the TTF distribution under actual working conditions (AWCs). To address these issues, this paper introduces the flexible five-parameter Beta Generalized Weibull (BGW) distribution, which can represent the five characteristic failure-rate shapes (IFR, DFR, CFR, BFR and UBT). The sign of the hazard derivative is analyzed through Glaser’s eta-function theorem together with the asymptotic behavior of the hazard rate at the origin and at infinity; this yields exact, necessary-and-sufficient parameter conditions for each shape, with the origin controlled by the product aγω and the tail controlled by γ. A TTF model based on the BGW distribution is then proposed to predict the TTF distribution under AWCs, and it is proved analytically that a change of stress affects only the scale parameter Δ, leaving all shape parameters unchanged. Because this invariance was previously stated as an assumption, it is verified empirically on the time-dependent dielectric breakdown (TDDB) data using per-level Weibull shape estimates with bootstrap confidence intervals, a Kruskal–Wallis test on standardized failure times, and a parametric-bootstrap likelihood-ratio test within the BGW family; all three tests support stress-invariant shapes. The reliability function is recast as a convergent infinite series, with the correct beta-integral kernel and an explicit ratio-test bound. Finally, the effectiveness and superiority of the method are verified using TDDB TTF data, and the untested 3800 V condition is predicted from the fitted linear voltage-acceleration model.
