We discuss  the existence of positive 2π-periodic solutions of the second-order ordinary differential equation with nonlinear derivative term by using positive operator perturbation method and fixed point index theory in cones. Under certain inequality conditions of the 
nonlinear term f(t,x,y), we obtain the existence of positive 2π-periodic solutions of the equation.

We used the Faedo-Galerkin method to investigate the three-dimensional magnetofluid equations with nonlinear damping terms and  delay terms on a bounded domain  and solved the problem of well-posedness of the solutions. Firstly, the existence of strong solutions was proven when α≥16/5. Secondly, the uniqueness of strong solutions was proven by using the Gagliardo-Niernberg inequality.

We considered the finite time blow-up to  a strongly damped  wave equation with variable-exponent nonlinear term. With the help of concave method and appropriately selected  parameters, we gave a new  blow-up criterion  for this problem and estimated the upper and lower bounds on  the blow-up time. The results show that the blow-up criterion contains special  implications for any high initial energy, and  the problem has a finite  time blow-up solutions.

By using Leray-Schauder degree theory and Banach contraction mapping principle, we studied the existence and uniqueness of solutions for boundary value problems of conformable fractional delay differential equations when the nonlinear term satisfied the growth condition and the Lipschitz condition, we obtained the results of existence and uniqueness of solution for the boundary value problem respectively, and gave an example to illustrate the applicability of the obtained results.

By using the fixed point theorem on expansion and  compression of cones, we study the existence of positive solutions for a class of periodic boundary value problems of third-order ordinary differential equations. The  results show that there exists  at least one positive solution to the above problem when the nonlinear term f satisfies appropriate conditions.

By applying mathematics induction and combinatorial analysis, we gave D(2)-vertex-distinguishing total colorings of cactus graphs G_T with maximum degree of 3, and then obtained χ_2vt(G_T)≤6. The result shows that D(β)-VDTC conjecture holds for cactus graphs with maximum degree of 3.

Aiming at  the blurring problem of remote sensing images, we designed an image restoration algorithm based on non-subsampled shearlet  transformation and sparse prior. Firstly, the image recovery model was created by setting the sparse a priori condition of remote sensing image under non-subsampled shearlet decomposition of the high-frequency image. Secondly, the model was solved by using the alternating direction multiplier method. Thirdly, the high-frequency image was restricted by the soft thresholding method, and the guided filtering was conducted in the low-frequency image to maintain the detailed information of the image as much as possible. Finally, the high-frequency image and the low-frequency image were reconstructed, the  
 reconstructed image was subjected to deep denoising by  using  convolutional neural networks, ultimately restoring a clear image. The deblurring algorithm was compared with H-PNP, GSR, and L2TV algorithms through experiments. The experimental results show that the algorithm can effectively remove  blurring and noise in remote sensing images, preserve the edge details of the image, and  the objective evaluation indexes are higher than the other three comparative experimental algorithms.

Firstly, by using the fractional order concentration-compactness principle, we proved the compactness conditions for a class of nonlinear Kirchhoff equations with electromagnetic fields and critical Hardy-Littlewood-Sobolev term in the whole space to overcome the problem of lack of compactness conditions caused by unbounded regions and critical term in this equation. Secondly, combined with the symmetric mountain path theorem, we proved that the equation satisfied the mountain path structure, and proved the multiplicity of the solution to the equation by using genus theory.

By using T-weak continuous operator method and classical Galerkin technique, we discussed the initial boundary value problem of a class of incompressible magneto-micropolar fluid equations in a two-dimensional bounded smooth region, and obtained the existence and uniqueness theorems of the global weak solutions for the problem,   further improving the regularity of the weak solutions.

By using the compactness method, we gaved the existence of solutions to the Cauchy problem of the energy-critical fractional  nonlinear Schrodinger equation and proved the existence of global solution to the Cauchy problem. By constructing the approximation equation and taking the limit of the solution sequence satisfying the approximation equation, the obtained limit function was the global weak solution of the energy-critical fractional nonlinear Schrodinger equation, and it was proved that the weak solution satisfied the energy inequality and mass conservation property.

We considered the characterization problem of weighted right core inverses and weighted right pseudo core inverses over *-rings by 
using algebraic methods. Firstly, the concept of weighted right core inverse was introduced. Secondly,  three equations and right invertible elements were used to give its characterization. Finally, the definition and characterization of weighted right pseudo core inverse were given.

Aiming at the problems of overfitting and low classification accuracy in brain tumor magnetic resonance image classification, we proposed a brain tumor classification method based on an improved EfficientNetV2 network. The method  introduced the coordinate attention mechanism in the EfficientNetV2 network, which simultaneously obtained the feature information of brain tumor from both vertical and horizontal directions and accurately identified the lesion features of brain tumor. It helped the model to locate and identify the lesion area information more comprehensively and accurately, and effectively suppressed the influence of background information on the detection results, so that the model had higher classification accuracy. The problem of low classification accuracy caused by  insufficient acquisition of feature information was solved. In order to further improve the classification accuracy, the Hard-Swish activation function was introduced, which could not only improve the computational speed of the brain tumor classification network model, but also effectively improve the classification accuracy. Meanwhile, the improved model was equipped with Dropout layer and normalization layer, which could better suppress the occurrence of overfitting, accelerate the convergence speed of the model, improve the robustness of the model, and significantly improve the classification accuracy. The experimental results show that the improved model obtains classification accuracy of 98.4% in the validation set, and the effectiveness of the improved model in brain tumor classification task is verified by comparison experiments and ablation experiments.

Aming at  the problem that traditional clothing classification algorithms based on convolutional neural networks could not meet the needs of massive and diverse clothing classification, we  proposed a clothing classification network based on convolutional attention fusion.  The network adopted a parallel structure, including a ResNet branch and a Transformer branch, and  fullly utilizing  the local features extracted by the convolution operation and the global features extracted by the self-attention mechanism to enhance the representation learning ability of the network, thereby improving the performance and generalization ability of the clothing classification algorithm.  In order to verify the effectiveness of the method, we conducted comparative experiments on the Fashion-MNIST and DeepFashion datasets.   The results show that on the Fashion-MNIST dataset, the method achieves an accuracy rate of 93.58%, and on the DeepFashion dataset, the method  achieves an accuracy rate of 71.1%, which is superior to the  experimental results of other methods.

By using the method that the determinant of the adjacency matrix of the singular graph was equal to zero, we discussed the singularity of the generalized θ-graphs and the generalized plum blossom φ-graphs, and gave the necessary and sufficient conditions for the generalized θ-graph θ(a₁,a₂,…,a_k) and the generalized plum blossom graph φ(a₁,a₂,…,a_k) to be a singular graph, respectively. The probability values of singular graph occurring in these two types of graphs were calculated.

Firstly, based on the Black-Scholes stock price model,  the neural stochastic differential equation (NSDE) model was established by parameterizing the asset return rate and volatility as a drift network and a diffusion network, respectively. Secondly, in the empirical analysis, the underlying asset as a single stock option was used as the research object, and real stock data was used for  the network training  and testing. The experimental results show that the NSDE model can overcome the defects of the constant assumption of the Black-Scholes model. Finally, for the case where the price of the underlying asset of the option was unobservable, we  proposed that the price of any target option and the price of a known option could be constrained within the Wasserstein distance of their risk-neutral equivalent martingale measure, and theoretically  proved the method.

We used an ideal convergence structure to solve the characterization problem of directed topology, and provided necessary and  sufficient conditions for the topological transformation of ideal S limits and ideal generalized S limits. The results show that the directed topology, the ideal S limit topology and the ideal generalized S limit topology are the same  in T₀ topological spaces.  The ideal S convergence in a directed space is topological if and only if it is a c-space. The ideal generalized S convergence in a directed space is topological if and only if it is a locally strongly compact space.

We used  finite element method to numerically solve the viscous Cahn-Hilliard equation. Firstly, the equivalent form of the viscous Cahn-Hilliard equation was obtained by introducing the Lagrange multiplier r of the auxiliary variable. Secondly, the second order linear finite element numerical scheme for the viscous Cahn-Hilliard equation was given by using the mixed finite element approximation  in space and the implicit backward differentiation formula (BDF)  for discretization in time, and the unconditional stability in energy and error estimation of the given scheme were analyzed in detail. Finally, a series of numerical examples were used to verify the accuracy and effectiveness of the given scheme. The results show that the proposed numerical scheme is ideal and has the characteristics of simultaneously satisfying linear, unconditional stability in energy and second order accuracy.

By using  variational methods and corresponding critical points theorems, we investigated a class of nonlocal elliptic equations Navier boundary value problems with p-biharmonic operators. We obtained two existence theorems for nontrivial generalized solutions 
 when nonlinear terms satisfied super-linear conditions.

We designed and synthesized an inorganic-organic hybrid complex based on Keggin-type polyacids,  and obtained [SiW₁₂O₄₀][Ag(bpy)₂]₄·H₂O (1)(bpy=2,2′-bipyridine)  by using  hydrothermal in situ synthesis method. The bpy in complex 1 was obtained by in situ decarboxylation of 2,2′-bipyridine-6,6′-dicarboxylic acid. The structure of complex was characterized  by single crystal X-ray diffraction,  elemental analysis,  infrared spectrum and thermogravimetric analysis, and its   fluorescence,  photocatalytic degradation of organic dyes,  electrochemical and antibacterial properties were  studied.

We proposed an improved approximate optimal gradient method to solve the unconstrained objective function in the graph partition problem. We first used  the modified BFGS updating formula and selected the linear combination of BB class step sizes as scalar matrices to obtain  the approximate optimal step sizes, then we introduced parameters to improve the classical Zhang-Hager line search form, construced the algorithm framework  and gave the proof of R-linear convergence. The experimental results show that the improved algorithm improves the performance of the original algorithm.

Based on the polynomial feature generation method for one-dimensional feature data, we proposed a data augmentation algorithm that used the polynomial feature generation method to generate feature data for high-dimensional feature data. At the same time, we proposed an  algorithm  that combined the generated polynomial feature data with the neural network model during convolutional neural network training, which could organically combine the  generated polynomial feature data with the convolutional neural network model, and  improve the low recognition accuracy  and the limited generalization performance of model caused by data limitations such as limited data samples, fixed total number of data samples, and differences in available data samples  when modeling convolutional neural network models. Experimental results show that the accuracy of the convolutional neural network model using this method achieves significant improvement.

Based on the fact  that infected animals found in animal detection still had  infectious characteristics, we established a dynamic model to analyze the influence of detection behavior on the spread of animal brucellosis. Firstly, the basic reproduction number of the model was given, and the existence of the equilibrium point was analyzed. Secondly, through the discussion of the equilibrium point, it was found that the model occured backward bifurcation. Lyapunov function was used to prove that when R₀<1, the equilibrium point of disease-free was globally asymptotically stable under certain condition, when R₀>1, the model was uniformly persistent. Thirdly,  the optimal control strategy was formulated and solved according to Pontryagin maximum principle. Finally,  the theoretical analysis results were validated by numerical simulation, indicating that the control strategy can effectively control the spread of animal brucellosis.

Aiming at the problem that the attention mechanism-based model ignored the  part-of-speech information of words in the aspect-level sentiment classification task, we proposed  an aspect-level sentiment classification method that incorporated a part-of-speech self-attention  mechanism. Firstly, the method  was based on the natural language processing part-of-speech tagging tool to obtain part-of-speech tagging sequence, and randomly initialized a part-of-speech embedding matrix to obtain part-of-speech embedding vector. Secondly,  the self-attention mechanism was used to learn the syntactic dependence between words. Finally the sentiment score of each word was calculated, the combination of word sentiment was used to express the polarity of sentiment in specific aspects.  The experimental results show that compared with baseline model with the best performance in 5 public datasets, this method improves the accuracy and macro F₁ score by 2% and 4.83% respectively, indicating  that the attention mechanism model incorporating part-of-speech information has better performance in aspect-level sentiment classification task.

Aiming at the high-dimensional “big p small n” problem where the number of genes in biomics data (denoted as p) was far more than the number of samples (denoted as n), we proposd a graph attention network GATOr with local and global attention mechanisms. Firstly, the model used Pearson correlation coefficient to calculate the correlation between features on the omic data, and constructed a single sample network of the omic data. Secondly, we proposed a graph attention network which combined local and global attention mechanisms to learn graph-based omics feature representation from a single-sample network, thereby transforming the high-dimensional characteristics of the omics data into low-dimensional representations. The experimental results show that compared with other traditional classification algorithms, GATOr achieves better performance in classification task accuracy and other indexes.

By using Goswami’s method, we discussed the mean value problem of the composition of a class of arithmetic functions and the integral part function, and gave an asymptotic formula.

Firstly, based on techniques such as  Taylor expansion and Chapman-Enskog multi-scale expansion,  the lattice Boltzmann method was used to accurately recover the discussed macroscopic equations, and  the equilibrium distribution function expressions of D1Q3 and D2Q9 models were derived. Secondly,  two numerical examples were used to verify the effectiveness of the proposed method. The results show that the lattice Boltzmann method can be used to solve the numerical solution of the Caputo type modified time fractional equation.

Aiming at the problems of the complicated underwater image imaging environment resulted in the subsequent image analysis often being affected by color bias and other factors, we proposed a deep convolutional neural network image restoration method based on multi-scale features and triple attention multimodal fusion. Firstly, the deep convolutional neural network introduced the image multi-scale transformation feature on the basis of extracting the image spatial feature. Secondly, by using channel attention, supervised attention and non-local attention, the scale correlation and feature correlation of image features were mined. Finally, by designing a multimodal feature fusion mechanism, the above two types of features could be effectively fused. The proposed method was tested on the open underwater image test set and compared with the current mainstream methods. The results show that this method is superior to the comparison method in quantitative comparison such as peak signal-to-noise ratio and structural similarity, as well as qualitative comparison such as color and details.

Aiming at the problem that it was difficult to achieve high-precision and high frame rate tracking under limited computing resources, we proposed a lightweight  siamese network target  tracking algorithm based on ananchor free.   Firstly, the modified lightweight network MobileNetV3 was used as the backbone network to extract features, and reduced parameters and computation of the network  while maintaining deep feature expression capability. Secondly, for traditional cross-correlation operation, we proposed deep cross-correlation module for graph cascading optimization, which highlighted important information of target features through rich feature response graphs. Finally, feature sharing was used  to reduce parameters and computation to improve tracking speed in the anchor classification regression prediction network. Comparative experiments were conducted on two mainstream datasets OTB2015 and VOT2018, the experimental results show that the algorithm has a significant accuracy  advantages compared to  SiamFC tracker, and is more robust in complex tracking scenes. At the same time, the tracking frame rate can reach 175 frames/s.

We considered the lower bound of VC (Vapnik-Chervonenkis) dimension for concept classes induced by general Bayesian networks where each random variable took any finite values. By analyzing the relationship between the number of parameters that could be freely set in a network and the corresponding VC dimension, we proved that adding 1 to the number of parameters that could be freely set in any discrete non-full Bayesian network was a lower bound of corresponding VC dimension.

We used taking condition method, probability method and moment generating function method to research the lifetime problem of Yule-Furry classical δ shock model, and gave the explicit expressions of the lifetime properties such as reliability, moment generating function and moment of lifetime of the model. The mean lifetime of the model was applied to the problem of cell carcinogenesis and numerically verified.

 The studies of reactions involving water radical cations and their cluster ion ((H₂O)⁺_n)  can  better understand the mechanisms of certain natural processes,  such as proton transfer in aqueous solutions,  the formation of hydrogen bonds,  the destruction of bio-molecules such as DNA,  and the discovery of novel gas phase reactions and products. In addition,  the potential of water radical cations in  radio-biology have broad application prospects and its use as a primary reactive ion have attracted much attention for efficient selective  chemical ionization as well as improving the analytical sensitivity. At present, there are many studies on the bonding properties and  structure  of protonated water clusters and hydrated electrons, but there are few studies on the isolation and physicochemical properties of (H₂O)⁺_n due to  their ultra-high reaction activity and extremely short lifetime. Since significant progress has been made in the  technology  of mass spectroscopy,  molecular spectrometry  and high-precision  theoretical calculation of quantum chemistry, we review the current knowledge of    (H₂O)⁺_n, including  the formation  methods  and generation mechanisms,  structural theoretical simulation and experimental verification,   chemical property analysis,  and their application research.

The structural problem of simple weight modules with a one-dimensional weight space in the quantum Loop algebra U_q(L(sl₂)) was solved by using a construction method, and it was obtained that any simple weight module with a one-dimensional weight space must be  isomorphic to one of the four classes of simple weight modules of U_q(L(sl₂)). In addition, a class of simple weight modules of  the quantum Loop algebra U_q(L(sl₂)) with  weight space dimension of 2, which was neither the highest weight nor the lowest weight, was constructed.

We studied the one-dimensional non-Newtonian fluid model with gravitational potential and damping terms. The singularity and strong nonlinearity were solved by using the method of regularization equations and constructing  approximate solutions. The existence of the positive density solution was obtained by assuming the compatibility condition. Furthermore, the existence and uniqueness of local solutions for coupled equations under vacuum conditions were obtained.

We used the methods of conservative energy estimation and characteristic line estimation to solve the Sobolev norm growth estimation problem of the solution of the (1+1)-dimensional Maxwell-Chern-Simons-Higgs model, gave the L^∞ estimation of the first derivative of the finite energy solution of the model, and obtained the polynomial growth of the H² norm of the solution by improving the exponential growth.

We considered a  class of tumor-immune model, discussed the existence  conditions  of their equilibrium points, and used characteristic equations to analyze the local kinetic stability of each equilibrium point,  proving that the model underwent Hopf bifurcation under the corresponding conditions. By calculating the first Lyapunov coefficient, it can be concluded that if the coefficient is not zero, the model undergoes Hopf bifurcation,  the bifurcation is supercritical if the coefficient is less than zero, and the bifurcation is subcritical if the coefficient is greater than zero. Finally, numerical simulations are used to validate the theoretical analysis results.

Aiming at the problem of the difficulty in collecting noiseless point clouds and the low generalisation performance of  training on synthetic datasets using simulated noise,  we proposed a self-supervised denoising method that only required  noisy point clouds to complete  training in order  to achieve point cloud denoising in different environments. The method first performed downsampling on  the noisy point cloud by designing and implementing a special sampler to obtain the paired point cloud required for training the network, and then the problem of noise perturbation in network training was solved by designing a lightweight multi-scale denoising network. The experimental results on multiple datasets show that the method is effective and can obtain the same effect as supervised training in different scenarios.

Aiming at  the problem of few-shot relation extraction,  we proposed an induction network based on attention mechanism. Firstly, we used  dynamic routing algorithm in induction network to learn the class representation. Secondly, we proposed instance-level attention mechanism to  adjust support set and obtain high-level information between support set and query set samples, thereby obtaining  the support set samples that were more relevant to the query instances. The proposed  model effectively solved  the problem of how to extract relationships when the training data was insufficient. The experiment was conducted  on the few-shot relation extraction FewRel dataset, and the experimental results showed an  accuracy rate of (88.38±0.27)% in the 5-way 5-shot case,  (89.91±0.33)% in the 5-way 10-shot case, (77.92±0.44)% in the  10-way 5-shot case,  (81.21±0.39)% in the  10-way 10-shot case. The  experimental  results show that the model can adapt to tasks and outperforms other comparative  models, achieving better results than comparative  models in few-shot relation extraction.

We considered  a class of Kirchhoff type viscoelastic wave equation with logarithmic nonlinear term of variable exponents. Firstly, the energy identity for the problem was given. Secondly, by constructing auxiliary functions and using Holder inequality and Gagliardo-Nirenberg inequality, we obtained the result of blow up of equation solutions in finite time in the case where  the logarithmic nonlinear term  contained variable exponents.

The Hopf bifurcation of predator-prey symbiotic model with double time delay was discussed  by using the normal form theory and the central manifold theorem. By analyzing the characteristic equation and taking the development time of the prey and the development time of the co-genitor as the parameters, the stability of equilibrium point, the existence of Hopf bifurcation, the stability of bifurcation direction and bifurcating periodic solution were given. The result that  time delay  affected the stability of the system was obtained. The correctness of obtained conclusions was verified through numerical simulation.

Aiming at the redundant relations and entity overpalling problems in  the task of relational triple extraction,  we proposd a 2D entity pair tagging scheme based on the relation filter (RF2DTagging).  RF2DTagging model consisted of two parts: 1) A relation filter for filtering redundant relations, and 2) a 2D entity pair tagging scheme that could effectively solve various entity overlapping problems. To further validate the RF2DTagging model, we conducted experiments on three public Chinese relation extraction datasets CCKS2019-Task3, CMeIE and DuIE2.0. The experimental results show that the  model can effectively solve the above two problems,  and the overall performance is better than the comparison model.