We established a class of measles epidemic model based on secondary vaccine failure induced by vaccination. By using the next generation matrix method, we first obtained the basic reproduction number R₀ of the model and gave its biological meaning, and then proved that R₀ was a sharp threshold parameter by constructing an appropriate Laypunov function. When R₀≤1, the disea
se-free equilibrium is globally asymptotically stable; when R₀>1, the diseasefree equilibrium is unstable and the endemic equilibrium is globally asymptotically stable.

By using Schauder fixed-point theorem, we discuss the existence and uniqueness of radial solutions to elliptic boundary value problem with gradient terms, where Ω={x∈R^N: |x|<1}， N≥2, f: [0,1]×R×R⁺→R is continuous. The existence results of radial solutions and positive radial solutions to the problem are obtained under the condition that the nonlinear term f(r,ξ,η) is allowed to grow superlinearly with respect to ξ and η. In addition, we also discuss the uniqueness of radial solutions of this problem.

By using the principle of time maps, we study the exact multiplicity and bifurcation diagrams of positive solutions of Dirichlet problem for a prescried mean curvature equation in the onedimensional Minkowski space, where λ>0 and L>0 are parameters. We obtain the results that the above problem has no positive solution, exactly one positive solution and exactly two positive solutions when the nonlinear terms f(u)=u(e^u-1) and f(u)=e^u-1, respectively.

By using the Krasnoselskii fixed-point theorem, the author discuss existence of infinite positive periodic solutions for nonlinear systems with parameters and obtain infinite positive periodic solutions, where λ is a positive parameter, a,b: ［0,1］→［0,∞) are continuous function and are not always 0 on any subinterval of ［0,1］,  f,g: ［0,1］×［0,∞)×［0,∞)→［0,∞) are continuous function.

The existence and non-existence of generalized traveling waves for two species competitive lattice system in general time inhomogeneous media were considered. By establishing comparison principle for sub-solutions and super-solutions of the related cooperative system and constructing appropriate sub-solutions and super-solutions of the cooperative system, the author proved that the generalized traveling waves of the system  exist when their least mean speed was greater than an explicit threshold, and there was no generalized traveling wave whose least mean speed was less than this threshold.

By using the properties of differential operators, covariant derivative operator, Lie-derivative operator and the expressions of curvature tensor fields on Riemannian manifold, we discussed the characteristics of n(n>3)-dimensional conformally flat Yamabe soliton with the semi-symmetric metric ρ-connection under compact condition, and gave a necessary and sufficient condition that sectional curvature of Yamabe sotion with this structure was constant. The results show that the sectional curvature of Yamabe soliton with this structure is constant -1, the soliton constant is -n(n-1), and the soliton field is a Killing vector field.

By using the fixed point theorem in cones, the author study the existence of multiple positive solutions for nonlinear third-order three-point boundary value problems where ρ>0 is a constant, 0<η<1, α>0, f: ［0,1］×［0,∞)→［0,∞) is a continuous function. The results show that when f satisfies certain conditions, there are countable multiple positive solutions for the problem.

We considered the dynamical properties of a class of plant-herbivore diffusion system. Firstly, we proved the global existence, dissipation and persistence of the solutions of system by using the method of upper and lower solutions and the comparison theorem of parabolic equations. Secondly, based on the principle eigenvalue theory of elliptic operator and Lyapunov function method, we gave the existence, the local and global asymptotic stability of constant steady state solutions, and established the criterion of the Turing instability of the system. Finally, the effectiveness of results was verified by some numerical simulations.

The author considered a parabolic partial differential equation with weight function. By using Giorgi’s iteration technique, the maximal norm estimate of weak solution sequence of the perturbation problem of the equation was given. By means of the energy estimate and Simon’s compactness theorem, the existence of weak solutions of the equation was proved by the limit process.

Let L=(-Δ)²+V² be a high order Schrodinger type operator in Rⁿ (n≥5), where a nonnegative potential V belongs to the reverse Holder class RH_q(q>n/2). Let V_ρ(e^-tL) be the variation operator associated with the high order Schrodinger type operator L. Based on the theory of atomic decompesitions of Herz type Hardy spaces, by using the properties of the Schrodinger type operators, the boundedness of the commutators composed by such variation operators and Lipschitz fuctions on L^q is proved. Furthermore, it is proved that the commutators of this kind of variation operators are bounded from Herz type Hardy spaces 
to the Herz spaces and are bounded on Morrey-Herz spaces.

By using operator decomposition technique and priori estimation of the solution of the equation, we gave the uniform asymptotic compactness of the random dynamical system, and proved the existence of the random attractor for the stochastic suspension bridge equation with additive noise.

We considered the American put option pricing problem under Black-Scholes model. Firstly, based on the Black-Scholes model, we designed a neural network algorithm for the model, and gave the numerical approximation of the American option price. Secondly, the effectiveness of the algorithm was proved by comparing with the traditional binomial tree method.

We considered variable selection problem in one-way classified ANOVA under order constraints. We proposed two Bayesian variable selection method based on the stochastic search variable selection (SSVS), and designed a simple and easy to operate Gibbs sampling algorithm for a posteriors sampling. The results of numerical simulation and application examples show that the proposed method is effective.

By utilizing the Rosenthal moment inequality of negatively superadditive dependent (NSD) random arrays and truncated method, 
we discussed  the weak convergence, the L^r convergence and the complete convergence for maximum weighted sums of NSD random arrays under the condition of {X_nk, 1≤k≤k_n, n≥1} on {a_nk, 1≤k≤k_n, n≥1} uniform integrability.

In order to improve computational efficiency for unconstrained optimization problems, we proposed a new class of modified Hager-Zhang conjugate method, which possessed sufficient descent feature and trust region  trait without line search. The theoretical research results show that under some proper assumptions, the new algorithm not only converges globally for general function under weak Wolfe-Powell line search, but also has R-|linear convergence rate for uniformly convex functions. Numerical results show that the new algorithm performs better than the classical Hager-Zhang algorithm and its two classical modified algorithms.

We studied the discretization problems of a special class of  multivariate Hermite-type interpolation by using  the theory of discrete approximation algorithm. Namely,  a given Hermite-type interpolation problem was discretized into the limit of a series of Lagrange interpolation problems. When the interpolation condition of  Hermite-type interpolation  problem corresponded to a second-order D-invariant subspace,  a necessary and sufficient  condition for the problem to be discretized under the idea of discrete approximation algorithm was given by using the structural property of the space. The nonlinear  equations corresponding to the  condition were smaller in scale and more efficient in computation.

The problem of  calculating the eigensensitivity  of damped structures. Aiming at the problem that the traditional Nelson’s method could not  calculate the eigensensitivity with repeated frequency, we proposed a new extended Nelson’s method. The non-singularity of its coefficient matrix was proved by constructing a new particular solution control equation of  sensitivity. It could be used to calculate the sensitivity of repeated frequency complex mode shapes in structures, and the Nelson’s method was extended. The method kept the advantages of the traditional Nelson’s method and only used the mode shape corresponding to the repeated frequencies to solve the equation, and there was no need to expand the scale  of solving the equation  and rearrange the matrix. Numerical examples verify  the effectiveness of the method.

Through the analysis of the research characteristics of  Hilbert-type series inequalities in each stage, we analyzed and summarized  
the development process and the current status of research of Hilbert-type series inequalities from simple to deep, from special to general, from homogeneous kernel to non-homogeneous kernel and from low  dimension to high dimension,  and gave some of the latest research results.

Firstly, we prove that a unital associative ring R is a left generalized weak zero insertive (GWZI) ring if and only if ab=0 implies aⁿRb=0 for some positive integer n. Secondly, we prove that a ring R is a left GWZI ring if and only if so is S_n(R) for any integer n≥2 by using the method of matrix partition.

The shortest path problem between two points in time-dependent network was solved by using artificial bee colony algorithm. According to the characteristics of the first in first out network in time-dependent network, the path selection strategy in the original algorithm was improved to optimize the quality of generated individuals. The strategy of the algorithm added a Hash table for each individual (i.e. each path) to record the segments encountered in searching for paths. By looking up the table, we could find the better solution of the current individual. The experimental results show that the improved method can effectively improve the quality of the final solution and greatly shorten the running time.

Aiming at the problem that the traditional matrix-completion unconstrained optimization model had poor robustness in dealing with missing matrices damaged by singular noise, we proposed an adaptive robust matrix completion method. In this method, truncated kernel norm was used as the low-rank approximation of the rank function in the objective function, and the F-norm robust to singular noise was used as the loss term to recover the missing values in the matrix, so as to reduce the influence of outliers on the algorithm and improve the recovery accuracy. In the process of solving this model, a dynamic weight parameter was introduced by using convex optimization technique, which could be used to adjust the next update adaptively according to the current recovery error when updating the recovery value, and then an effective iteration method was established to solve the optimization problem. The experimental results show that the algorithm has better robustness and accuracy when dealing with matrices damaged by singular noise, so that better image restoration effects can be obtained.

Aiming at the problem of frequent identity exchange in multi-target tracking due to target occlusion, we proposed a pedestrian multi-target tracking algorithm. Firtsly, the algorithm used YOLOv4 as the detector to detect the target and determine the coordinates of the detection frame, the extended Kalman filter was used to predict the trajectory. Secondly, the Hungarian algorithm was used as the data association module, the detection frame predicted by extended Kalman filter was matched with the detection frame of target detection by cascade matching method, and the trajectory anomaly correction algorithm was added for the occluded target. Finally, the experiments were carried out on the test set of the MOT16 data set. The experimental results show that the algorithm achieves 56.5% tracking accuracy, and has a good effect on the occlusion phenomenon, which effectively improves the problem of frequent identity switching and target loss caused by occlusion.

Aiming at the problem of high location precision and real-time in robot servo grasping, we proposed a fast algorithm for feature matching and target location. Firstly, Shi-Tomasi detection algorithm was adopted to extract the feature points. Secondly, a new definition method of feature descriptors was proposed, the sub-image was intercepted with the feature point as a center, and the feature direction was determined by the partial derivative of two-dimensional Gaussian function, the local image was rotated according to the feature direction, and the local binary pattern of the standard local image after rotation was extracted as the feature descriptor, which had good locality as well as invariance of translation and rotation. Finally, the feature matching was realized by calculating the Hamming distance between the feature descriptors, the homography matrix was estimated to locate the position and direction of the target in the scene. The experimental results show that the algorithm has the advantages of fast 
matching speed, high location precision and good stability, and can meet the requirements of location precision and real-time in robot servo grasping.

Aiming at the problems of chromatic aberration of dust images and the tendency of halo of dust image in bright areas after image enhancement, we proposed a new dust image enhancement algorithm based on dark channel. Firstly, by using grayscale world algorithm to adjust chromatic aberratin in Lab color space, the phenomenon of color distortion could be effectively avoided. Secondly, gamma correction function and dark channel defogging algorithm were used to avoid image noise, excessive color enhancement and halo. Finally, the brightness compensated image and the contrast enhanced image were weighted and fused to further improve the visibility of the image and make image details more visible. The experimental results show that the method improves the clarity and brightness of dust image, and can get higher quality dust image.

Aiming at the problem that small targets in remote sensing images had some defects, such as less information, easy to be interfered by the background, weak feature expression and so on, which led to the current general target detection algorithm was not ideal in the detection of such small targets. In order to improve the detection ability of small targets in remote sensing images, we proposed an improved algorithm based on RFBNet. The algorithm was based on the framework of RFBNet. Firstly, the self-correcting convolution was used to replace the conventional convolution in the feature extraction network to expand the receptive field and enrich the output, so as to enhance the ability of weak feature extraction. Secondly, a multi-scale feature fusion module was designed to enrich the abstract information of the shallow feature map. Finally, a dense prediction module was designed, and contextual information was integrated in the shallow layer in advance, so that the output of each layer in the final stage contained rich and closely related multi-scale feature information. The proposed algorithm was tested on UCAS_AOD and NWPU VHR-10 datasets, and the average detection accuracy reached 83.4% and 94.8%, respectively. The experimental results show that the proposed algorithm can effectively improve the accuracy of target detection in remote sensing images, and has a significant improvement for the problem of small-scale target detection in remote sensing images.

Aiming at the shortcomings of traditional term frequency-inverse document frequency (TF-IDF) algorithm for text classification with specific attributes, especially the low accuracy of words with specific meaning under specific classification, we proposed an improved TF-IDF text clustering algorithm. Comparative experiments were carried out through the papers published by scientific research institutions in Jilin Province from 2015 to 2019. The improved TF-IDF algorithm and the traditional TF-IDF algorithm were used to calculate the frequency of keywords in the papers, then K-means++ method was used to cluster. Finally, random forest algorithm was used to evaluate the accuracy of clustering. The experimental results show that the improved TF-IDF algorithm improves the accuracy of classification.

Aiming at the problem that MinMax k-means algorithm was easy to generate empty solutions, slow convergence speed and low computational efficiency, we proposed an incremental MinMax k-means clustering algorithm. The algorithm started from a given initial clustering number, and generated new cluster centers by increasing a fixed step length. The fast dividing method based on data balance was used to generate incremental cluster centers, so as to avoid the large amount of calculation problem caused by traversing data and too many running times of k-means clustering algorithm in traditional incremental clustering center selection. Compared with MinMax k-means and related algorithms, the experimental results show that the algorithm is superior to the comparison algorithm in calculation efficiency and accuracy, and effectively improves the sensitivity of MinMax k-means clustering to initialization center and easy to generate empty solutions.

Aiming at the problem of the training data set with large samples, high dimension, and complex features, a self-adaptive multiple kernel learning algorithm was proposed by integrating support vector machine with ant colony optimization algorithm. The affinity propagation clustering algorithm was used to find the similar features adaptively, and then the parameters of the kernel function were selected adaptively by ant colony algorithm, so as to select the optimal kernel function quikly. Experimental results of five groups of UCI data sets show that the proposed algorithm has higher classification accuracy and F₁ value than the traditional support vector machine, which verifies the effectiveness and feasibility of the proposed algorithm.

Aiming at the problem of noise in non-continuous and non-stationary speech signals, we proposed a variational mode decomposition (VMD) denoising algorithm based on parameter optimization. Firstly, the grey wolf optimization algorithm was used to search the optimal decomposition parameter combination of the VMD algorithm: decomposition mode number K and penalty factor α. By using the combination of the obtained parameter combination to decompose the speech signal, K characteristic mode function components IMF were obtained. Secondly, the effective modal components were selected by the correlation coefficient, and the invalid modal components were processed by the wavelet threshold. Finally, the wavelet threshold processed modal component and effective modal component were reconstructed to denoise the speech signal. Experimental results show that compared with other classical algorithms, the proposed algorithm can effectively improve the signal-to-noise ratio (SNR), reduce the mean square error, and improve the quality of speech signals.

For the ultra-dense network communication scenario, we proposed an interference management and resource allocation algorithm based on cluster allocation to eliminate the impact of reduced throughput due to the large number of low-power base stations deployed in the ultra-dense network. Firstly, the femtocell base station was assigned a cluster based on three conditions: distance, inter-cell interference, and available resource. Secondly, the interference relationship was modeled as an interference-weighted undirected graph according to the allocation result, and the femtocell base station was classified according to appropriate standards, and different coloring algorithms were used for different types of femtocell base stations. Finally, spectrum resource block allocation was completed according to the overall coloring result graph. The simulation results show that the proposed algorithm can select the most suitable cluster for the femtocell base station to join, reduce inter-cluster interference, improve the utilization of spectrum resources, and improve the throughput of the network.

Aiming at the problems that the existing trust-aware secure routing protocols local decision-making led to the overall lack of security, energy consumption and load imbalance in wireless sensor networks (WSNs), we proposed a trust-aware secure routing method for WSNs based on improved genetic algorithm. In this method, the comprehensive trust value of every node was calculated according to direct trust value, indirect trust value, volatile factor and residual energy. Aiming at maximizing the  comprehensive trust value, minimizing the network energy consumption and load balancing, the corresponding fitness function was constructed. The cluster head selection and route search were encoded by a single chromosome, and the optimized next generation was formed through improved genetic operations, thus, the optimal cluster head set and the optimal route paths of each cluster head were found. The simulation test results show that the method can ensure the network security, improve energy efficiency, balance network load and prolong network lifetime.

In order to solve the problem of the reflection of corporate customer value, we proposed an improved model of TFA customer segmentation, which took customer development space T, purchase frequency F, and average purchase amount A as indicators to fully reflect the customer value and development space. Firstly, the K-means clustering algorithm was improved by introducing local density value ρ and information entropy H to optimize the traditional K-means clustering method in the initial clustering center selection problem. Secondly, by building a machine learning framework, clustering experiments were carried out on selected artificial data sets and real data sets to verify the effectiveness of the model. The experimental results show that the model can more effectively classify customers, fully reflect the customer value and its development space, and improve the efficiency of the algorithm by improving the clustering algorithm.

Aiming at the problem of low accuracy of the cell classification of single cell transcriptome data, we proposed a novel cell ensemble classification algorithm. The algorithm could make full use of advantages of different classification models and reduce the classification error of single cell data. The experimental results on a chronic myeloid leukemia data and a triple-negative breast cancer data show that the cell classification based on the ensemble algorithm is more clear and accurate, which verifies the effectiveness of the proposed algorithm.

Aiming at the noise problem of contrast-enhanced ultrasound (CEUS) image, we proposed a denoisng method of contrast-enhanced ultrasound image based on convolutional neural networks. Firstly, the number of rare contrast-enhanced ultrasound image samples was expanded by data enhancement methods such as image translation, image flipping and image rotation. Secondly, the number of samples was further expanded by overlapping small image blocks. Finally, image blocks and artificial noise were used as input training sets to train the denoising model based on convolutional network structure. The experimental results show that the proposed method can be effectively extended to different sizes of CEUS images, and the peak signal-to-noise ratio of CEUS images after denoising is higher than that of traditional image denoising methods.

A two-dimensional functional photonic crystal was designed. The photonic crystal could change the functional form of the dielectric constant of the dielectric column only by changing the distribution of the external light intensity or the distribution of the electric field, so as to obtain the required band gap structure. The band gap structures of two-dimensional functional photonic crystals with different dielectric column radii and different dielectric constant functions were selected, and the Dirac-like point and band gap structures were obtained by numerical calculation. The conclusions could provide theoretical basis for the design of optical devices.

The author studied the Noether symmetries and conserved quantities of non-conservative singular systems in phase space on time scales. Firstly, the internal constraints caused by singularity were treated as external non-holonomic constraints, and the canonical equations of constrained Hamiltonian systems were obtained by using Hamilton principle under Δ derivative on time scale. Secondly, the special infinitesimal transformation with time invariance was introduced, and the criterion and theorem of Noether symmetry were obtained according to the invariance of Hamilton action. Finally, the effectiveness of the method and results was
 illustrated by an example. The results show that the structural properties of canonical equations of constrained Hamiltonian systems on time scale are still maintained, the singularity of the system makes the Noether symmetry no longer directly lead to the Noether-type conserved quantity, and it is necessary to construct a certain gauge function to make the Noether symmetry satisfy the structural equation.

In order to solve the problems of hazards caused by abnormal operation of large-scale rotating equipment such as wind turbines and power systems, we designed an optical fiber vibration real-time monitoring system based on Mach-Zehnder interferometer. Firstly, the interferometer was composed of two 3 dB couplers to increase the sensitivity and stability of the sensor. Secondly, the optical fiber vibration sensor composed of Mach-Zehnder interferometer was used to collect the real-time vibration data of the rotor, and the signal processing unit was used to analyze the collected vibration information. Finally, experiments were tested on a small-scale rotor will rated speed of 2 680 r/min and rated power of 2.8 kW to complete the vibration information of the rotor under normal and fault conditions respectively. The results show that the system has a good monitoring effect on the oprating state of the rotor.

The Sm-doped CeO₂-based ceramic was prepared by the self-propagating combustion method, we studied the effect of different low sintering temperature and pressure on the electrical properties of the Ce_0.9Sm_0.1O_2-δ (SDC) electrolyte. X-ray diffraction, scanning electron microscopy and AC impedance spectroscopy were used to study the phase composition, microstructure and electrical properties of the SDC electrolyte under different variables. The results show that the average grain size of the SDC sample increases significantly with the sintering temperature, the sintering temperature and pressure do not affect the the unit cell parameters of the samples. Compared with the grain conductivity, the effect of the sintering temperature and pressure on the grain boundary conductivity is more obvious.

Firstly,  the  metal-semiconductor heterogeneous core-shell  structure was constructed by the combination of anisotropic gold nanostars (AuNS)  and TiO₂. Secondly, the sulfonatothiacalix[4]arene (STC[4]A) was functionalized on the surface of TiO₂/AuNS to prepare AuNS/TiO₂/calixarenes (AuNS/TiO₂/STC[4]A) composite photocatalytic materials.  The materials were characterized by ultraviolet-visible spectrum (UV-Vis)  Fourier transform infrared spectrum (FT-IR), scanning electron microscope (SEM), and the photocatalytic degradation activity of  AuNS/TiO₂/STC[4]A for Rhodamin B (RhB) was investigated. The experimental results show that the degradation rate of Rhodamine B by  AuNS/TiO₂/STC\[4\]A  can reach  99%, after five cycles, the degradation rate only decreases by 1.8%.

The relationship between Fusarium oxysporum and mass ratio  of  ginsenoside in ginseng root was investigated. In order to determine the effect of ginsen root inoculated with Fusarium oxysporum on the mass ratio of ginsenoside, the  change of mass ratio of three ginsenosides Rb1,Re and Rg1 within 120 h after inoculation with Fusarium oxysporum was monitored  by high performance liquid chromatography (HPLC). The results show that after Fusarium oxysporum infection, the mass ratio  of Rb1 increases, and  the mass ratio of Re and Rg1 does not change significantly.  Rb1 has obviously inhibitory effect on conidia of Fusarium oxysporum, the higher the mass concentration of Rb1, the lower the germination rate  and the higher the inhibition rate.  Re and Rg1 have almost no inhibitory effect on conidia of Fusarium oxysporum. Therefore, Rb1 may be a disease-resistant compound produced by ginseng roots against Fusarium oxysporum.

We studied reaction mechanism and efficiency of in situ remediation of Cr(Ⅵ) contaminated aquifer by Fe(Ⅲ)  enhanced tannic acid (TA). The results show that Fe(Ⅲ) can effectively improve the remediation efficiency of tannic acid to Cr(Ⅵ) contaminated aquifer by complexing with tannic acid. The two adjacent phenolic hydroxyl groups of pyrogallol structure in tannic acid molecule combine with Fe(Ⅲ) to prvide electrons,  so at least 9 mol tannic acid is needed to repair 1 mol Cr(Ⅵ). For the best strengthening effect of Fe(Ⅲ),  n（TA）∶n（Fe(Ⅲ)）=3∶5. At this time,  the tannic acid/Fe(Ⅲ) structure is the most stable and the storage capacity reaches the maximum.  After Cr(Ⅵ) pollution is repaired,  Fe(Ⅲ)  forms stable FeOOH under the action of tannic acid to adsorb Cr(Ⅲ) and form coprecipitation,  so as to achieve the purpose of remving total chromium.