The alternating direction implicit (ADI) methods are computationally efficient and numerically effective tools for computing low-rank solutions of large-scale matrix equations. It is known in the literature that the low-rank ADI method for Lyapunov equations is a Petrov-Galerkin projection algorithm that implicitly performs model order reduction. It recursively enforces interpolation at the mirror images of the ADI shifts and places the poles of the reduced-order models at the ADI shifts. In this paper, we show that the low-rank ADI methods for Sylvester and Riccati equations are also Petrov-Galerkin projection algorithms that implicitly perform model order reduction. These methods likewise enforce interpolation at the mirror images of the ADI shifts; however, they do not place the poles at the mirror images of the interpolation points. Instead, their pole placement ensures that the projected Sylvester and Riccati equations they implicitly solve admit a unique solution.By observing that the ADI methods for Lyapunov, Sylvester, and Riccati equations differ only in pole placement and not in their interpolatory nature, we show that the shifted linear solves-which constitute the bulk of the computational cost-can be shared. The pole-placement step involves only small-scale operations and is therefore inexpensive. We propose a unified ADI framework that requires only two shifted linear solves per iteration to simultaneously solve six Lyapunov equations, one Sylvester equation, and ten Riccati equations, thus substantially increasing the return on investment for the computational cost spent on the linear solves. All operations needed to extract the individual solutions from these shared linear solves are small-scale and inexpensive.Since all ADI methods implicitly perform model order reduction when solving these matrix equations, we show that the resulting reduced-order models can be obtained as an additional byproduct. These models not only interpolate the original transfer function at the mirror images of the ADI shifts but also preserve important system properties such as stability, minimum-phase property, positive-realness, bounded-realness, and passivity. Consequently, the proposed unified ADI framework also serves as a recursive, interpolation-based model order reduction method, which can preserve several important properties of the original model in the reduced-order model.The accuracy and rapid convergence of ADI methods depend on the choice of ADI shifts. Two existing self-generating shift strategies, which do not require any precomputation of shifts, are also reviewed. We provide the theoretical justification for these two strategies, which was not given in the original work. In addition, we propose a computationally efficient, subspace-acceleration-based self-generating shift strategy that is theoretically well grounded and significantly outperforms the existing approaches. With this shift strategy, the unified ADI framework becomes a fully automatic toolbox capable of solving large-scale matrix equations, performing model order reduction, and designing controllers and observers for large-scale plant models without user intervention. Three numerical examples demonstrate the effectiveness of the proposed ADI framework and shift-generation strategy, confirming its advantages over existing methods.
Given a graph F, an r-uniform hypergraph H is a Berge-F if there is a bijection ϕ:E(F)→E(H) such that e⊆ϕ(e) for each e∈E(F). Given a family F of r-uniform hypergraphs, an r-uniform hypergraph is F-free if it does not contain any member of F as a subhypergraph. The Turán number of F is the maximum number of hyperedges in an F-free r-graph on n vertices. Let Ms+1 denote a matching of size s+1, i.e., the graph consisting of s+1 independent edges. Khormali and Palmer [European J. Combin. 102 (2022) 103506] completely determined the Turán number of Berge matchings for sufficiently large n. Subsequently, Kang, Ni, and Shan [Discrete Math. 345 (2022) 112901] determined the exact value of the Turán number of Berge-Ms+1 for all n when r≤s−1 or r≥2s+2. In this paper, we settle the final open case s≤r≤2s+1, thereby completing the determination of the Turán number of Berge matchings.
Conventional carbon/magnetic composites for electromagnetic wave absorption (EMA) often suffer from weak interfacial coupling and discontinuous conductive networks, which restrict the synergistic interaction of conductive, dielectric, and magnetic loss mechanisms. In this study, a multiscale 1D@2D@1D carbon heterostructure (CPPGCo) was constructed through an interface-catalysis synergy strategy by integrating polydopamine-directed graphene assembly with Co-catalyzed in situ CNT growth. This process generated a chemically welded and spatially continuous CF/G/CNT/Co network with abundant heterointerfaces and embedded magnetic nanoparticles. Scanning thermal microscopy and CST simulations provide spatially resolved evidence supporting localized energy dissipation at multiscale heterointerfaces. The CPPGCo heterostructure exhibited excellent EMA performance, achieving a minimum reflection loss of -62.75 dB and an effective absorption bandwidth of 5.62 GHz at an ultralow filler loading of 7wt.%. When embedded in a cyanate ester matrix, the composite delivered high EMI shielding effectiveness (96.84 dB) and enhanced thermal conductivity (1.012 W m-1 K-1 ). This work highlights multiscale heterointerface engineering as an effective strategy for high-performance multifunctional EMA materials. (c) 2026 Published by Elsevier Ltd on behalf of The editorial office of Journal of Materials Science & Technology.
Ground Delay Programs (GDPs) are widely used to mitigate demand-capacity imbalances in air traffic operations. However, existing studies typically address GDP planning from either the perspective of air traffic control or that of individual airlines, with limited attention to their coordination. This study develops a collaborative framework for the airline-driven GDP that integrates proactive slot reduction and airline precancellation. The problem is formulated as a two-stage stochastic program. The first determines slot reductions and flight precancellations in anticipation of various potential capacity scenarios, whereas the second adjusts slot allocations and executes airline recovery actions based on the realized slot capacity. To address the computational challenges of this large-scale problem, we propose an efficient Lagrangian dual decomposition algorithm, which reduces the computation time by 49.06% on average compared with the standard column generation approach. Numerical experiments based on real-world data show that the proposed framework consistently achieves lower expected total recovery costs than rule-based deterministic sequential strategies commonly used in current GDP practice. Moreover, when fairness constraints are imposed, allowing slot exchanges among airlines further reduces total costs by an average of 1.37%. These findings highlight the importance of collaborative decision-making between air traffic managers and airlines, providing a robust and practical tool to enhance system-wide efficiency under capacity disruptions.
Generative Engine Optimization (GEO) is emerging as a strategic issue in AI-mediated retail supply chains. As conversational shopping systems increasingly mediate product search, comparison, and recommendation, firms compete not only for ranked-list positions but also for being surfaced, cited, or highlighted within compressed answer interfaces. We develop a two-stage analytical Hotelling duopoly in which retailers first invest in machine-legible GEO signals and then compete in prices, while a fraction of consumers purchase through a GenAI answer engine. The model endogenizes how probabilistic answer inclusion shapes demand allocation, pricing, welfare, fulfillment-capacity payoffs, and platform governance. The analysis yields several major findings. A higher highlighting probability raises both demand share and pricing power. Yet when retailers are initially symmetric, GEO creates a defensive arms race: both firms spend to preserve answer visibility, but relative visibility, prices, allocations, and consumer surplus remain unchanged, while total welfare declines. When retailers differ in inherited algorithmic visibility capital, the content-rich retailer’s advantage may persist and be reinforced under the maintained equilibrium conditions. Extensions show that fulfillment congestion attenuates the profit value of AI-steered demand, and that quality-weighted inclusion rules can reduce low-signal escalation by tying marginal GEO effectiveness to verifiable retail and operational quality. The paper contributes to research on retail supply-chain competition by showing that GenAI answer interfaces are upstream demand-allocation mechanisms rather than merely digital-marketing channels; their economic value depends on logistics execution, service reliability, and platform scoring design.
