Denote byḂ α,φ (Ω) the Orlicz-Besov space, where α ∈ R, φ is a Young function and Ω ⊂ R n is a domain. For α ∈ (−n, 0) and optimal φ, in this paper we characterize domains supporting the imbeddingḂ α,φ (Ω) into L n/|α| (Ω) via globally n-regular domains. This extends the known characterizations for domains supporting the Besov imbeddingḂ s pp (Ω) into L np/(n−sp) (Ω) with s ∈ (0, 1) and 1 ≤ p < n/s. The proof of the imbeddingḂ α,φ (Ω) → L n/|α| (Ω) in globally n-regular domains Ω relies on a geometric inequality involving φ and Ω , which extends a known geometric inequality of Caffarelli et al.

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Cell segmentation and counting play a very important role in the medical field. The diagnosis of many diseases relies heavily on the kind and number of cells in the blood. convolution neural network achieves encouraging results on image segmentation. However, this data-driven method requires a large number of annotations and can be a time-consuming and expensive process, prone to human error. In this paper, we present a novel frame to segment and count cells without too many manually annotated cell images. Before training, we generated the cell image labels on single-kind cell images using traditional algorithms. These images were then used to form the train set with the label. Different train sets composed of different kinds of cell images are presented to the segmentation model to update its parameters. Finally, the pretrained U-Net model is transferred to segment the mixed cell images using a small dataset of manually labeled mixed cell images. To better evaluate the effectiveness of the proposed method, we design and train a new automatic cell segmentation and count framework. The test results and analyses show that the segmentation and count performance of the framework trained by the proposed method equal the model trained by large amounts of annotated mixed cell images.

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Based on Lyapunov nite-time stability theory and backstepping strategy, we put forward a novel xedtime bounded H in nity tracking control scheme for a single-joint manipulator system with input saturation. The main control objective is to maintain that the system output variable tracks the desired signal at xed time. The advantages of this paper are the settling time of the tracking error converging to the origin is independent of the initial conditions, and its convergence speed is more faster. Meanwhile, bounded H in nity control is adopted to suppress the in uence of the external disturbances on the controlled system. At the same time, the problem of input saturation control is considered, which effectively reduce the input energy consumption. Theoretical analysis shows that the tracking error of the closed-loop system converges to a small neighborhood of the origin within xed time. In the end, a simulation example is presented to demonstrate the effectiveness of the proposed scheme.