The algebro-geometric solutions for Degasperis-Procesi hierarchy

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy.Comment: 65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004 by other author

The Reduced Order Method for Solving the Linear Complementarity Problem with an M-Matrix

In this paper, by seeking the zero and the positive entry positions of the solution, we provide a direct method, called the reduced order method, for solving the linear complementarity problem with an M-matrix. By this method, the linear complementarity problem is transformed into a low order linear complementarity problem with some low order linear equations and the solution is constructed by the solution of the low order linear complementarity problem and the solutions of these low order linear equations in the transformations. In order to show the accuracy and the effectiveness of the method, the corresponding numerical experiments are performed

Research and Application of PDM Borehole Technology for HDD

AbstractBy introducing the structure character and working principle of PDM, borehole technologies of PDM directional drilling were studied in this text, including borehole technology principle, directional method, technology flow, drilling technology parameters, branch borehole technology etc. Shaanxi Tingnan Coal .Ltd field application was taken as an example to do a simple introduction on PDM matching equipment and technology promotion. The borehole technology and related result can be used as reference for horizontal directional drilling research and construction

Vibration analysis of a cylinder with slight diameter and thickness variations

The cross section of circular cylinder in their dynamic model is always considered as a perfect circle, which means radii at every point on the circle are the same. In real engineering structure, there are slight fluctuations in shape of the circular cylinder which is different from those in ideal model. Meanwhile, effects of structural fluctuations on its dynamic characteristic are rarely analyzed before. To study problem mentioned above, the geometric shape of a typical, apparently symmetrical cylinder is examined experimentally to demonstrate that a small variation in diameter and thickness indeed exists in practice firstly. Because fluctuations in diameter and thickness of the cylinder are related to each other, we need to separate effects of a slight variation in its diameter and thickness on structural dynamic characteristics to search the key factor of influence. Then, two simplified modes, which are modeled by finite element method, are used to study the effects of diameter or thickness variation alone on the natural frequencies and modal shapes of the free cylinder. It is revealed that the diameter variation described by the simplified model captures the key influencing elements which affect the modal characteristics of the free cylinder. Finally, a free cylinder with both variations is analyzed numerically and the results are verified experimentally. This work illustrates that significant discrepancies inevitably exist between the measured results of an actual free cylinder and an assumed symmetrical model even if there is only a very slight variation in its geometric shape

Gradient-based compressive sensing for noise image and video reconstruction

In this study, a fast gradient-based compressive sensing (FGB-CS) for noise image and video is proposed. Given a noise image or video, the authors first make it sparse by orthogonal transformation, and then reconstruct it by solving a convex optimisation problem with a novel gradient-based method. The main contribution is twofold. Firstly, they deal with the noise signal reconstruction as a convex minimisation problem, and propose a new compressive sensing based on gradient-based method for noise image and video. Secondly, to improve the computational efficiency of gradient-based compressive sensing, they formulate the convex optimisation of noise signal reconstruction under Lipschitz gradient and replace the iteration parameter by the Lipschitz constant. With this strategy, the convergence of our FGB-CS is reduced from O(1/k) to O(1/k2 ). Experimental results indicate that their FGB-CS method is able to achieve better performance than several classical algorithms

Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy

Though the completely integrable Camassa--Holm (CH) equation and Degasperis--Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker--Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker--Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy

Feature Representation Learning with Adaptive Displacement Generation and Transformer Fusion for Micro-Expression Recognition

Micro-expressions are spontaneous, rapid and subtle facial movements that can neither be forged nor suppressed. They are very important nonverbal communication clues, but are transient and of low intensity thus difficult to recognize. Recently deep learning based methods have been developed for micro-expression (ME) recognition using feature extraction and fusion techniques, however, targeted feature learning and efficient feature fusion still lack further study according to the ME characteristics. To address these issues, we propose a novel framework Feature Representation Learning with adaptive Displacement Generation and Transformer fusion (FRL-DGT), in which a convolutional Displacement Generation Module (DGM) with self-supervised learning is used to extract dynamic features from onset/apex frames targeted to the subsequent ME recognition task, and a well-designed Transformer Fusion mechanism composed of three Transformer-based fusion modules (local, global fusions based on AU regions and full-face fusion) is applied to extract the multi-level informative features after DGM for the final ME prediction. The extensive experiments with solid leave-one-subject-out (LOSO) evaluation results have demonstrated the superiority of our proposed FRL-DGT to state-of-the-art methods

The 3-dimensional printing for dental tissue regeneration: the state of the art and future challenges

Tooth loss or damage poses great threaten to oral and general health. While contemporary clinical treatments have enabled tooth restoration to a certain extent, achieving functional tooth regeneration remains a challenging task due to the intricate and hierarchically organized architecture of teeth. The past few decades have seen a rapid development of three-dimensional (3D) printing technology, which has provided new breakthroughs in the field of tissue engineering and regenerative dentistry. This review outlined the bioactive materials and stem/progenitor cells used in dental regeneration, summarized recent advancements in the application of 3D printing technology for tooth and tooth-supporting tissue regeneration, including dental pulp, dentin, periodontal ligament, alveolar bone and so on. It also discussed current obstacles and potential future directions, aiming to inspire innovative ideas and encourage further development in regenerative medicine