Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth

In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems

Infinitely many homoclinic solutions for a class of damped vibration problems

In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems x¨(t) + Bx˙(t) − A(t)x(t) + Hx(t, x(t)) = 0, where A(t) ∈ (R, RN) is a symmetric matrix for all t ∈ R, B = [bij] is an antisymmetric N × N constant matrix, and H(t, x) ∈ C 1 (R × Bδ , R) is only locally defined near the origin in x for some δ > 0. With the nonlinearity H(t, x) being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark’s theorem

Fast kNN Graph Construction with Locality Sensitive Hashing

Abstract. The k nearest neighbors (kNN) graph, perhaps the most popular graph in machine learning, plays an essential role for graphbased learning methods. Despite its many elegant properties, the brute force kNN graph construction method has computational complexity of O(n 2 ), which is prohibitive for large scale data sets. In this paper, based on the divide-and-conquer strategy, we propose an efficient algorithm for approximating kNN graphs, which has the time complexity of O(l(d + log n)n) only (d is the dimensionality and l is usually a small number). This is much faster than most existing fast methods. Specifically, we engage the locality sensitive hashing technique to divide items into small subsets with equal size, and then build one kNN graph on each subset using the brute force method. To enhance the approximation quality, we repeat this procedure for several times to generate multiple basic approximate graphs, and combine them to yield a high quality graph. Compared with existing methods, the proposed approach has features that are: (1) much more efficient in speed (2) applicable to generic similarity measures; (3) easy to parallelize. Finally, on three benchmark large-scale data sets, our method beats existing fast methods with obvious advantages

The existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness

In this paper, we show the existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness, by using the technique of penalized functionals and an infinite dimensional Morse theory developed by Kryszewski and Szulkin. Two applications are given to Hamiltonian systems and elliptic systems

Semilinear elliptic equations with dependence on the gradient

In this article we consider elliptic equations whose nonlinear term depends on the gradient of the unknown. We assume that the nonlinearity has a asymptotically linear growth at zero and at infinity with respect to the second variable. By applying Morse theory and an iterative method, we prove the existence of nontrivial solutions

Research on insulation joint damage of the station track circuit in high speed and heavy load condition

In high-speed and heavy-load operation environment, the insulation joints of the station track circuit are damaged in many stations, which caused the carrier information of the adjacent sections to interfere with each other, endangered the safety of the train and reduced the reliability of the track circuit. In order to solve the incorrect code problem caused by the insulation joint breakage, we analysed the insulation damage situation for the station track circuit and the structural principle of BES choke transformer. Then the different structures of the insulation damage protection circuit are given and the relevant parameters are obtained, and verifying the anti-interference ability of the new BES choke transformer. Finally, a complete four-terminal network model of the new BES choke transformer is established. And its four-terminal network parameters are calculated by matlab simulation, which provides the theoretical basis for establishing the track circuit complete system

Research on insulation joint damage of the station track circuit in high speed and heavy load condition

In high-speed and heavy-load operation environment, the insulation joints of the station track circuit are damaged in many stations, which caused the carrier information of the adjacent sections to interfere with each other, endangered the safety of the train and reduced the reliability of the track circuit. In order to solve the incorrect code problem caused by the insulation joint breakage, we analysed the insulation damage situation for the station track circuit and the structural principle of BES choke transformer. Then the different structures of the insulation damage protection circuit are given and the relevant parameters are obtained, and verifying the anti-interference ability of the new BES choke transformer. Finally, a complete four-terminal network model of the new BES choke transformer is established. And its four-terminal network parameters are calculated by matlab simulation, which provides the theoretical basis for establishing the track circuit complete system

Tomato Leaf Disease Recognition via Optimizing Deep Learning Methods Considering Global Pixel Value Distribution

In image classification of tomato leaf diseases based on deep learning, models often focus on features such as edges, stems, backgrounds, and shadows of the experimental samples, while ignoring the features of the disease area, resulting in weak generalization ability. In this study, a self-attention mechanism called GD-Attention is proposed, which considers global pixel value distribution information and guide the deep learning model to give more concern on the leaf disease area. Based on data augmentation, the proposed method inputs both the image and its pixel value distribution information to the model. The GD-Attention mechanism guides the model to extract features related to pixel value distribution information, thereby increasing attention towards the disease area. The model is trained and tested on the Plant Village (PV) dataset, and by analyzing the generated attention heatmaps, it is observed that the disease area obtains greater weight. The results achieve an accuracy of 99.97% and 27 MB parameters only. Compared to classical and state-of-the-art models, our model showcases competitive performance. As a next step, we are committed to further research and application, aiming to address real-world, complex scenarios