Mathematics

Nonmonotone transformations of fuzzy numbers arise in loss functions, dispersion measures, and higher-order risk criteria, but inverse-credibility laws apply directly only to monotone maps. Existing methods typically either compute transformed alpha-cuts without a credibilistic scalar functional or use endpoint formulas whose validity depends on monotonicity. We develop a calculus that preserves interior extrema before integration. For any continuous transformation of a fuzzy number whose transformed output is integrable, the expected value admits an extrema-based alpha-cut representation that requires neither regularity nor an inverse credibility distribution. For continuous nondecreasing radial losses, we derive the exact risk, prove a sharp factor-two bound for the conservative endpoint proxy, characterize both equality regimes, and localize the proxy excess to center-containing cuts. We also establish Lipschitz perturbation and certified midpoint-quadrature bounds. At the decision level, proxy constraints are conservative and proxy minimization has a factor-two guarantee. A reproducible experiment further shows that endpoint substitution can reverse the optimizer and contract the feasible set. The framework is further specialized to higher-order credibilistic moments: for regular fuzzy numbers, signed odd moments retain inverse-distribution formulas, whereas absolute and even moments are obtained from the extrema-based formulation without requiring regularity. Triangular and Gaussian closed forms, standardized coefficients, a scale-free bias function, and deterministic audits demonstrate the resulting error, stability, computation, and decision guarantees.

Effective control of prosthetic legs is essential for restoring natural mobility and enhancing the quality of life for amputees. This paper presents a Direct Neural Dynamic Programming (DNDP) algorithm that leverages coupled action and critic neural networks to solve the optimal control problem for prosthetic gait trajectory tracking. By continuously learning and adapting to individual gait patterns, the DNDP action networks generate the control for the knee and ankle joints to reduce the gait tracking error. The proposed approach enables real-time optimization of control strategies without relying on precise system models, offering adaptability to subject-specific dynamics and varying conditions.

Graph theory and network theory have become critical tools for modeling, analyzing, and assessing complex systems, such as communication networks, transportation systems, power grids, wireless sensor networks, and supply chain infrastructures [...]