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Abstract In this paper, we introduce a bilevel optimization framework for addressing inverse mean-field games, alongside an exploration of numerical methods tailored for this bilevel problem. The primary benefit of our bilevel formulation lies in maintaining the convexity of the objective function and the linearity of constraints in the forward problem. Our paper focuses on inverse mean-field games characterized by unknown obstacles and metrics. We show numerical stability for these two types of inverse problems. More importantly, we, for the first time, establish the identifiability of the inverse mean-field game with unknown obstacles via the solution of the resultant bilevel problem. The bilevel approach enables us to employ an alternating gradient-based optimization algorithm with a provable convergence guarantee. To validate the effectiveness of our methods in solving the inverse problems, we have designed comprehensive numerical experiments, providing empirical evidence of its efficacy. 

Abstract Cycads represent one of the most ancient lineages of living seed plants. Identifying genomic features uniquely shared by cycads and other extant seed plants, but not non-seed-producing plants, may shed light on the origin of key innovations, as well as the early diversification of seed plants. Here, we report the 10.5-Gb reference genome ofCycas panzhihuaensis, complemented by the transcriptomes of 339 cycad species. Nuclear and plastid phylogenomic analyses strongly suggest that cycads andGinkgoform a clade sister to all other living gymnosperms, in contrast to mitochondrial data, which place cycads alone in this position. We found evidence for an ancient whole-genome duplication in the common ancestor of extant gymnosperms. TheCycasgenome contains four homologues of thefitDgene family that were likely acquired via horizontal gene transfer from fungi, and these genes confer herbivore resistance in cycads. The male-specific region of the Y chromosome ofC. panzhihuaensiscontains a MADS-box transcription factor expressed exclusively in male cones that is similar to a system reported inGinkgo, suggesting that a sex determination mechanism controlled by MADS-box genes may have originated in the common ancestor of cycads andGinkgo. TheC. panzhihuaensisgenome provides an important new resource of broad utility for biologists.