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In this paper, we find that the integrated form $d_F$ of a complex Finsler metric $F$ is inner. The distance $d_F$ is complete if and only if every closed bounded subset of a complex manifold $M$ is compact. We prove a version of theorem for normal families of holomorphic mappings between two complex Finsler manifolds, i.e, the theorem of Montel in complex Finsler manifolds. Our results extend the basic theorem of strongly negatively curved families for a Hermitian manifold [Wu, Acta Math. 119(1967), 193-233] or [Grauert, Reckziegel, Math. Z. 89(1965), 108-125]. As applications, we obtain a complex Finsler version of theorems $A$-$F$ in [Wu, Acta Math. 119(1967), 193-233], including the Cartan-Carath\'eodory-Kaup-Wu theorem, the theorem of the automorphism group on a complex Finsler manifold and some rigid results.

In this paper, first of all, according to Lu's and Zhang's works about the curvature of the Bergman metric on a bounded domain and the properties of the squeezing functions, we obtain that Bergman curvature of the Bergman metric on a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain is bounded. Secondly, by the property of curvature symmetry on a K\"ahler manifold, we have the property: if holomorphic sectional curvature of a K\"ahler manifold is bounded, we can deduce that its sectional curvature is bounded. After that, applying to the Schwarz lemma from a complete K\"ahler manifold into a complex Finsler manifold, we get that a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain admit complete strongly pseudoconvex complex Finsler metrics such that their holomorphic sectional curvature is bounded from above by a negative constant. Finally, by the Schwarz lemma from a complete K\"ahler manifold into a complex Finsler manifold, we prove the uniform equivalences of the Kobayashi metric and Carath\'eodory metric on a bounded strongly convex domain with smoothly boundary.

Data heterogeneity across clients is one of the key challenges in Federated Learning (FL), which may slow down the global model convergence and even weaken global model performance. Most existing approaches tackle the heterogeneity by constraining local model updates through reference to global information provided by the server. This can alleviate the performance degradation on the aggregated global model. Different from existing methods, we focus the information exchange between clients, which could also enhance the effectiveness of local training and lead to generate a high-performance global model. Concretely, we propose a novel FL framework named FedCME by client matching and classifier exchanging. In FedCME, clients with large differences in data distribution will be matched in pairs, and then the corresponding pair of clients will exchange their classifiers at the stage of local training in an intermediate moment. Since the local data determines the local model training direction, our method can correct update direction of classifiers and effectively alleviate local update divergence. Besides, we propose feature alignment to enhance the training of the feature extractor. Experimental results demonstrate that FedCME performs better than FedAvg, FedProx, MOON and FedRS on popular federated learning benchmarks including FMNIST and CIFAR10, in the case where data are heterogeneous.

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly K\"ahler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly K\"ahler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant., Comment: to appear in Annali di Matematica Puraed Applicata (1923-) (2022) 201: 1935-1964

Suppose that $M$ is a K\"ahler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold., Comment: 27 pages, has been accepted for publication in SCIENCE CHINA Mathematics. arXiv admin note: substantial text overlap with arXiv:2105.08284