Mathematics

Classical binomial interval methods often exhibit poor performance when applied to extreme conditions, such as rare-event scenarios or small-sample estimations. Recent frequentist and Bayesian approaches have improved coverage in small samples and rare events. However, they typically rely on fixed error margins that do not scale with the magnitude of the proportion. This distorts uncertainty quantification at the extremes. As an alternative method to reduce these boundary distortions, we propose a novel hybrid approach. It blends Bayesian, frequentist, and approximation-based techniques to estimate robust and adaptive intervals. The variance incorporates sampling variability, Wilson score margin of error, a tuned credible level, and a gamma regularization term that is inversely proportional to sample size. Extensive simulation studies and real-data applications demonstrate that the proposed method consistently achieves better coverage proportions at all sample sizes and proportions. It provides more conservative interval widths below a sample size of 50 and competitively narrower widths from moderate to large sample sizes, especially beyond 50, compared to the Jeffreys’ and Wilson score intervals. Geometric analysis of the tuning curves demonstrates how the blended method adaptively tunes credible levels across binomial extremes. It starts at higher values for small samples and gradually flattens into near-linear, symmetric trajectories as sample size increases. This ensures robust coverage and balanced sensitivity. Our method offers a theoretically grounded, computationally efficient, and practically robust estimation of rare-event intervals. These intervals have applications in safety-critical reliability, epidemiology, and early-phase clinical trials.

Small flexible-wing aircraft are vulnerable to gusts due to their low inertia and operating regime at low-Reynolds-number regimes, compromising flight stability and mission reliability. This paper introduces a novel active gust alleviation device (AGAD) installed at the wingtip, which works in concert with the conventional tail-plane to form a multi-surface control system. To coordinate these surfaces optimally, a quasi-static aeroelastic aircraft model is established, and a linear–quadratic regulator (LQR) controller is designed. A key innovation is the integration of an extended state observer (ESO) to estimate the unmeasurable, gust-induced angle of attack in real time, allowing the LQR to effectively counteract unsteady disturbances. Comparative simulations against a baseline (tail-plane-only control) demonstrate the superiority of the combined AGAD-tail strategy: the peak gust responses in pitch angle and normal acceleration are reduced by over 57% and 20%, respectively, while structural loads at the wing root are also significantly attenuated. Furthermore, the AGAD enhances maneuverability, reducing climb time by 20% during a specified maneuver. This study confirms that the integrated AGAD and LQR-ESO framework provides a practical and effective solution for enhancing both the stability and agility of small flexible aircraft in gusty environments, with direct benefits for applications like precision inspection and monitoring.

We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of $\phi$-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a single set of poles (a common denominator). The shared pole set is obtained by assembling a stacked Hankel matrix from Chebyshev boundary data for all target functions and computing a single SVD; the zeros of the associated singular-vector polynomial, mapped via the standard CF slit transform, yield the poles. With the poles fixed, per-function residues and constants are recovered by a robust least squares fit on a suitable grid of the negative real axis. For any linear combination of resolvent operators applied to right-hand sides, the evaluation reduces to one shifted linear solve per pole with a single combined right-hand side, so the dominant cost matches that of computing a single $\phi$-function action. Numerical experiments indicate geometric convergence at a rate consistent withHalphen’s constant, and for highly stiff problems our algorithm outperforms existing Taylor and Krylov polynomial-based algorithms.