Two families F, G of k-subsets of {1,2,..., n} are called non-trivial cross-intersecting if FnG not equal empty set for all F is an element of F, G is an element of G and n{F: F is an element of F}= empty set = n{G : G is an element of G}. In this note, we establish the product version of the Hilton-Milner Theorem for k >= 8 in the missing range. That is, if F, G subset of ([n] k). are non-trivial cross-intersecting, n >= 2k+ 1and k >= 8, then |F||G| <= (( n - 1 k - 1) - (n - k - 1 k - 1). + 1)(2). (C) 2026 The Authors. Published by Elsevier Inc.
In this paper, we study a bilinear saddle point problem of the form minxmaxyF(x)+〈Ax,y〉−G(y), where F and G are μF- and μG-strongly convex functions, respectively. By incorporating Nesterov acceleration for strongly convex optimization, we first propose a first-order primal–dual gradient algorithm. We show that it achieves the linear convergence rate O((1−min{μF/LF,μG/LG})k) for both the primal–dual gap and the iterative gap, where LF and LG denote the smoothness constants of F and G, respectively. We further develop a continuous-time accelerated primal–dual dynamical system with constant damping. Using the Lyapunov analysis method, we establish the linear convergence rate O(e−min{μF,μG}t). Notably, when A=0, our methods recover the classical Nesterov accelerated methods for strongly convex unconstrained problems in both discrete and continuous-time. Numerical experiments are presented to support the theoretical convergence results
The safety assessment of tunnels comprising both curved and straight segments connected by deformation joints remains a critical challenge in underground engineering, due to the complex interplay of geometric curvature, structural discontinuity, and arbitrary dynamic loading. This paper presents a novel analytical framework that delivers explicit closed-form solutions for the dynamic response of a hybrid curved-straight beam system on a Pasternak viscoelastic foundation under arbitrary dynamic loads. The model integrates curved segments, straight segments, and deformation joints into a unified piecewise beam formulation. A key contribution is the use of a Dirac-delta-based virtual-force representation to capture displacement and rotation jumps at deformation joints without homogenization, thus preserving physically meaningful risk indicators across transitions. Using modal superposition, we derive fully analytical expressions for displacement, velocity, and bending moment. The framework is validated through degenerative analysis, high-fidelity finite element simulations, and a real-world case study. Parametric studies reveal significant dynamic-response discontinuities along the tunnel, indicating that curved segments become more critical when the curvature radius is small relative to the characteristic deformation wavelength, with deformation joints acting as key amplification interfaces. The proposed solution offers not only computational efficiency but also a transparent, mechanics-based tool for performance-driven design and risk-informed monitoring of combined curved-straight tunnel systems, effectively bridging high-fidelity analysis with practical engineering decision-making.
The selective oxidation of H2S under ambient conditions remains a significant challenge due to the simultaneous requirements of efficient activation and controlled sulfur formation. Herein, we report a cooperative catalytic system comprising MgO-CuO nanoparticles supported on nitrogen-doped carbon aerogels (NBCAs) derived from chitosan. The catalyst was fabricated via a freeze-drying-assisted pyrolysis process, yielding uniformly dispersed dual metal oxides within a hierarchically porous carbon framework. Combined experimental characterization and density functional theory (DFT) calculations reveal a synergistic mechanism in which CuO primarily promotes H2S dissociation and O2 activation, and MgO provides a thermodynamically favorable pathway for deep oxidation of elemental sulfur to sulfates. The optimized Mg0.6Cu0.4-NBCA-600 catalyst delivers an exceptional sulfur breakthrough capacity of 5051.4 mg g-1 at room temperature and retains over 75% of its activity after five regeneration cycles. Moreover, the catalyst exhibits stable performance under varying oxygen concentrations and humidity levels, demonstrating high durability and adaptability under realistic operating conditions. These results elucidate the cooperative roles of dual metal oxides in ambient-temperature H2S oxidation and establish a structure-activity-mechanism relationship relevant to the rational design of regenerable desulfurization catalysts with integrated sulfur recovery.
This paper introduces a natural fuzzy order on fuzzy numbers that extends the natural orders on real numbers and interval numbers. We investigate its completeness properties and show that the space of uniformly bounded fuzzy numbers is conically complete and conically cocomplete. Moreover, it is proved that this space constitutes a [0, 1]-enriched domain if and only if the underlying continuous t-norm satisfies the (S) condition. These results provide a foundation for ordering fuzzy numbers.
