We considered the boundary value problem of a class of piecewise-smooth singularly perturbed ordinary differential equation with critical case. Firstly, we constructed a asymptotical approximation of solution with internal and boundary layers by using the boundary function method and smooth matching method. Then we used the intermediate value theorem to prove the existence of solution of the problem, and gave the accuracy of the constructed asymptotical approximation.

The author considered the solvability of a class of Riemann-Liouville fractional boundary value resonance problems with infinite multipoint boundary conditions. Firstly, by using the fixed point theorem of cone extension and compression, the author obtained the existence of a positive solution when the nonlinear term f satisfied a certain condition. Secondly, under the condition that the nonlinear term f satisfied stronger conditions, the results of three positive solutions of the problem were obtained by using Leggett-Williams fixed point theorem.

We considered the asymptotic behavior of the heat equation on a semi infinite cylinder under nonstandard boundary conditions, where the solution satisfied the homogeneous Neumann condition in the local region on the side and the non-homogeneous Neumann condition in the other regions. Using the energy analysis method,  we obtained the spatial asymptotic theorem of the solution of the equation, and extended  results to the heat equation in binary mixtures.

By using inequality technique, we discussed the dynamic behavior of a class of Hopfield neural network systems with time-varying delays, and proved the existence and uniqueness of pullback attractor of the system.

The boundary entropy inequality problem of simplified chromatography equation was discussed by  using entropy flux pair and 
the viscosity vanishing method. Firstly, by judging whether the solutions of the initial value problem of the chromatography equation satisfied the boundary entropy inequality, we gave the interaction of the elementary waves on the boundary, and then gave the global solutions for the initial boundary value problem of the chromatography equation. Secondly,  the correctness of the theoretical analysis for the initial boundary value problem was verified by numerical simulation.

By using the Ito formula and the law of iterated logarithm, we studied the dynamic behavior of a class of flocking models disturbed by white noise. Firstly, we proved that the flocking model of two individuals presented unconditional flocking phenomenon. Secondly, for the flocking model of N individuals, we proved that the system had strong random flocking phenomenon under certain conditions.

By using the monotone iterative method of upper and lower solution and establishing a new maximum principle, we constructed 
a monotone iterative procedure for the ω-periodic solutions of nth order delay differential equation -u⁽ⁿ⁾(t)=f(t,u(t),u(t-τ₁),u(t-τ₂),…,u(t-τ_n)),t∈R, and proved existence and uniqueness of its ω-periodic solutions, where f: R×Rⁿ⁺¹→R was a continuous function which was ω-periodic on t, τ₁,τ₂,…,τ_n were positive constants.

Let W be a class of weak tilting modules and C be a class of cotilting modules，we prove  that (W,C) is duality pair on an arbitrary ring R, and discuss the properties of module class and cotorsion pairs related to W based on this result.

We consider Gorenstein AC-projective modules over triangular matrix rings. Let T be a triangular matrix rings, where A and B are rings and U is a (B,A)-bimodule. We prove that if _BU is a flat module and U_A is a finitely generated projective module, then a le
ft T-module  is a Gorenstein AC-projective module if and only if M₁ is a Gorenstein AC-projective left A-module, φM is a monomorphism, and Coker φ^M is a Gorenstein AC-projective left B-module.

We considered representation problems of Z-Quantale. Firstly, we proved that any unital Z-Quantale was isomorphic to the Z-Quantale formed by its strong self-continuous maps. Secondly, we  proved that any  unital Z-Quantale was isomorphic to a relation Z-Quantale. Finally, we discussed the relationship between the category of unital Z-Quantale and the category of relation Z-Quantale, and  proved that the category of unital Z-Quantale was equivalent to the category of relation Z-Quantale.

By calculation, we gave some basic properties of the generalized derivation algebra GDer(L), quasiderivation algebra QDer(L), 
centroid C(L), quasicentroid QC(L) and central derivation algebra ZDer(L) of the Poisson 3-Lie algebras, and gave a necessary and sufficient condition that quasicentroid was a Lie algebra.

Using the weight function method, the idea of parameterization and the technique of real analysis, we established a new Hardy-Hilbert-type integral inequality with a derivative function, gave the best constant factor and the equivalent properties of multiple parameters related to the inequality, and gave the similar case of the nonhomogeneous kernel and some special cases.

We discussed the Lipschitz continuity for solution mappings of parametric non-convex weak generalized Ky Fan inequality in real normed vector space. Firstly, we gave the concept of a class of nonlinear functional and its basic properties. Secondly, under appropriate assumptions, we established the non-convex separation theorem. Finally, we gave the sufficient theorem of Lipschitz continuity for the solution mappings of parametric non-convex weak generalized Ky Fan inequality under the conditions for the solution mapping without any convexity, monotonicity and univariate, and gave an example to prove the theorem.

Firstly, aiming at the modelling problem of complex integer-valued time series data, we proposed a first-order mixed integer-valued binomial autoregressive model. Secondly, we proved the strict stationary ergodicity of the model, gave the probabilistic statistics properties of the model, such as transition probability, expectation and variance, and estimated the model parameters by maximum likelihood method. Finally, the proposed model was applied to the fitting of harmonised index of consumer price (HICP)  data. The example analysis results show that the fitting effect of the proposed model is better than that of the existing model.

Firstly, by applying the corresponding infinitesimal generators of Heisenberg group on the factorable surfaces, we obtained four kinds of one-parameter surface families. Secondly, by using the analytical method, we established the mean curvature flow (MCF) equations of three nontrivial surface families, and obtained the conclusion that the existence time of the MCF and the soliton solution of the minimal MCF of two kinds of the surface families were saddle surfaces.

Based on the Black-Scholes-Merton option pricing model, we first gave a simplified algorithm of European option pricing equation under Vasicek model by using the method of conversion of valuation units, and then based on the simplified equation, we gave the iterative scheme for the numerical solution of the European option price by using the explicit difference method and the Crank-Nicolson difference method, and verified the stability of the iterative scheme.

Based on Lyapunov  method and stochastic differential equation theory,  we proved the global existence and uniqueness of positive solution for a class of  stochastic SIQR epidemic model with nonmonotonic incidence rate driven by Lévy noise,  investigated the  asymptotic behavior of the solutions for  the  model near  the disease-free equilibrium point and the endemic equilibrium point of corresponding deterministic model, repectively, and analyzed  the influence of random noise on the dynamic behavior of the model.

We considered a class of discrete quasi-periodic Schrodinger operators with some Gevrey potential energy, in which the potential energy could be written as a large valued analytical function having a Gevrey small perturbation on the one-dimensional torus. By using large deviation theorem and semi-algebraic theory, we proved that the operator satisfied the non-perturbative Anderson localization for any fixed phase and almost all frequencies under large coefficients.

According to  Salagean operator and subordination relationship, by  introducing a subclass L_m,n of starlike functions, which was subordinate to a leaf-like domain, we studied  the majorization properties of the function class.

Aiming at the problem of high ranking of commodity views and lagging sales in online shopping, based on purchasing intention of users, we proposed a collaborative filtering recommendation algorithm merged with enhanced rating matrix. Firstly, the penalty factor was used as the evaluation weight of the enhanced matrix, and the commodity portrait representing the user’s purchasing intention was weighted to obtain the prediction score of the enhanced matrix. Secondly, the basic rating matrix was obtained by combining the item-based collaborative filtering recommendation to establish the item similarity matrix between the potentially interested commodities. Finally, TOP-N result was used to recommend the top ranked commodities to target users with strong purchasing intention. The experimental results show that, compared with the traditional item-based collaborative filtering recommendation algorithm, the recommendation accuracy of the enhanced rating matrix collaborative filtering recommendation algorithm is improved by 2.48%, the recall rate is improved by 4.31%, and the comprehensive value F₁ is improved by 3.19%, which effectively solves the problem that the commodities of interest to users are ranked low and are not purchased or purchased less times, so as to achieve recommendation purpose of more accurate target users with strong purchasing intention, and then improve the recommendation accuracy.

Aiming at the problem that the traditional relation extraction model relied on machine learning methods such as feature engineering, which had low accuracy and complicated rules, we proposed a BERT+BiLSTM+CRF method. Firstly,  bidirectional encoder representations from transformers (BERT) was used to pre-train the corpus. Secondly, by using BERT to dynamically generate word vectors according to context features, the generated word vectors were encoded through bidirectional long and short-term memory network (BiLSTM). Finally, we input it into the conditional random field (CRF) layer to complete the extraction of causality. The experimental result shows that the accuracy of the model on the SemEval-CE dataset is 0.054 1 higher than that of the BiLSTM+CRF+self-ATT model, which improves the performance of the deep learning method in the task of causality extraction.

In order to realize more accurate collaborative clustering by using both attribute information and structure information, we proposed a semi supervised collaborative clustering framework based on attribute heterogeneous information network (SCCAIN) at the same time. Firstly, a learnable global association measure was designed, which integrated structural association and attribute association through meta path and attribute projection. Secondly, the three factor decomposition of constraint negative matrix was introduced into the constrained collaborative clustering nodes, the correlation measurement and collaborative clustering were combined, the collaborative clustering results were taken as the sharing factor, and a unified semi supervised learning framework was proposed to jointly optimize the given constraints of collaborative clustering and correlation measurement. Finally, simulation experiments were carried out on different data sets, the experimental results show that the clustering effect of the proposed
 method is good, which verifies that the attribute information and structure information can improve the effect of collaborative clustering.

Aiming at the problem that the traditional Yinyang algorithm did not use the data structure, resulting in low computational efficiency, we proposed an efficient Yinyang k-means clustering algorithm. The algorithm decomposed the original data layer by layer according to the data similarity, and established a full m-tree structure to store the data of each layer. The weighted data was established based on the data information stored in each leaf node of the tree structure, and the weighted Yinyang k-means algorithm was run to obtain the convergence center. In the original data, the convergence centers of the weighted data were taken as the initial condition to run the traditional Yinyang k-means algorithm to further optimize the objective function value. Comparative experiments with k-means, traditional Yinyang k-means and other two acceleration algorithms on five UCI data sets show that the proposed algorithm has a high acceleration ratio, and the solution accuracy is basically equivalent to Yinyang k-means clustering.

In order to effectively meet the requirements of h-adaptive mesh remeshing, we proposed a local tetrahedral mesh remeshing by advancing front method and Delaunay algorithm. Firstly, the order from line to surface and from surface to body was adopted to ensure the coordination of the overall network in the process of remeshing. Secondly, the local size function was used to ensure the smooth transition of mesh size. Finally, the projection method was used to make the mesh be shape-preserving. The simulation results show that the algorithm is suitable for the complex computer aided design (CAD) model with multiple parts, the mesh is closer to the real geometry in the process of h-adaptive encryption, and the mesh quality can be guaranteed after remeshing.

Firstly, we proposed a rational fractal interpolation functions with function scaling factors for curve modeling, which could accurately characterize irregular data with weak self-similarity. Secondly, we discussed the stability, the convergence and box-dimension of fractal curves. The experimental results show that the proposed method performs better than cubic spline interpolation, polynomial fractal interpolation with function scaling factors and rational fractal interpolation with constant vertical scaling factors, which is more suitable for reconstructing real data and irregular data.

Aiming at the problem of adaptive threshold setting of color image edge, by using the dynamic properties of the reaction-diffusion equation and optimizing the diffusion coefficient of the reaction-diffusion equation, the author proposed a cellular neural network color image edge detection method based on the reaction-diffusion equation to make the threshold more adaptive. The results show that the improved cellular neural network based on FitzHugh-Nagumo reaction-diffusion equation can better detect the edge of color image. Compared with several common methods, the experimental results also show that this method can not only ensure the accuracy of edge detection, but also maintain the color information of image edge, it is superior to other common methods in quantitative and qualitative analysis.

Aiming at the problem that the current convolutional neural network could not make full use of the shallow feature information, 
and it was difficult to capture the dependency between each feature channel, resulting in the loss of high-frequency information, we proposed a new generative adversarial network for image super-resolution reconstruction. Firstly, the WDSR-B residual block was introduced into the generator to fully extract the shallow feature information. Secondly, the GCNet module and pixel attention mechanism were combined into the generator and discriminator to learn the importance and high-frequency information of each feature channel. Finally, using spectral normalization instead of batch normalization that was not conducive to image super-resolution, and reduced computational overhead and stabilize training. Experimental results show that compared with other classical algorithms, the proposed algorithm can effectively improve the utilization of shallow feature information, better reconstruct the detailed information and geometric features of the image, and improve the quality of super-resolution images.

Aiming at the problem that extraction accuracy and speed were limited when using principal component analysis (PCA) algorithm only in face image high-dimensional data dimensionality reduction, we proposed a hybrid feature extraction algorithm based on wavelet transform and improved PCA. Firstly, the face image was decomposed by wavelet, and low-frequency component was selected for feature extraction. Secondly, the improved PCA algorithm was used for principal component extraction to obtain the feature vectors representing face features. Finally, the algorithm was applied to the image classification of Olivetti Faces  dataset. The experimental results show that the recognition accuracy is improved by 10% and the recognition speed is improved by about 37% when the image feature data processed by the hybrid algorithm are classified and recognized by convolutional neural network (CNN) algorithm.

Network based on protein-protein interaction (PPI) had a obvious modular characteristics, which played an important role in predicting protein functions and explaining specific biological processes. Network motif was an important topology for the evolution of complex networks. It represented important functional units in complex systems and had the characteristics of evolutionary conservation. We proposed a new protein complex recognition algorithm based on network motif as the core node group.  According to the topological characteristics of the PPI network, the algorithm took  motifs as central structures of the protein complexes, and expanded the two-layer nodes based on central structure, which could accurately and effectively identify protein complexes. The 2D network of the protein complexes was transformed into 3D network, so as to show  the structural characteristics of the protein complexes more intuitively and clearly. The experimental results verify the effectiveness and feasibility of the algorithm.

By optimizing and reconstructing the traditional SEIR (susceptible-exposed-infectious-removed) algorithm, we proposed an improved SEIR algorithm. The algorithm added attenuation function to the infection rate β, and the data were fitted to get the infection rate β that changed with time. Simulation results show that the average MSE (mean square error) of the algorithm on the test set is only 0.11, which significantly reduced the error value compared with other existing algorithm.

Aiming at the problem that normal and abnormal sounds might have great similarity and sometimes could not distinguish normal and abnormal sounds by the size of reconstruction error of autoencoder, we proposed a generative adversarial single classification network method for abnormal sound detection. Through multiple training, the method learned the distribution characteristics of normal samples. In the test process, the normal sample could be reconstructed with minimal error, while the abnormal sample had poor reconstruction effect, and distortion occurred in some frequency bands, so as to give the discriminant classification results. In the experiment, UrbanSound8K public data set and measured motor sound data set were used for testing, and the accuracy of this method is 86.3% and 98.1%, respectively, which is 5.0% and 3.0% higher than main deep learning methods such as convolutional autoencoder.

Aiming at the problem of node energy limitation in wireless sensor networks, we proposed a clustering routing protocol (EAPGA) which combined the affinity propagation (AP) algorithm and genetic algorithm. Firstly, the optimal cluster head was determined according to the residual energy, the distance between nodes, the distance from node to base station and the node centrality. Secondly, in channel selection, fitness function was constructed by energy consumption deviation between cluster heads, and each channel was optimized by genetic algorithm. The experimental results show that the protocol ensures energy balance and load balance, and maximizes the network lifetime.

We gave the probability density, probability current density and topological phase of photon in one-dimensional photonic crystal
s (AB)^N by using the quantum theory of photon. When the refractive index of media B was n_b=1.12, we gave the band Zak phases by using the quantum method. The results show that when the incident angle θ and periodic number N change, the probability density and probability current density of photonic crystals change approximately periodically, and its amplitude increases with the increase of incident angles θ and periodic number N. When the frequency of incident photon corresponds to transmissivity T=100%, the amplitude of the probability density is the largest. When the frequency of incident photon corresponds to transmissivity T=0, the probability density is not zero, but the amplitude of the probability density attenuates to zero rapidly, so there is a quantum tunnel effect of photon in photonic crystals. 

CoFe₂O₄ nanoparticles were prepared by organic synthesis of acetylacetone salt, CoFe₂O₄ nanoparticles were compounded with MXene, C₃N₄ and AC to construct magnetic composites. The degradation rates of rhodamine (RhB) at the absorption peak (λ=555 nm) under ultraviolet (UV) and visible light irradiation were measured by UV-visible spectrophotometer, respectively. The experimental results show that the matrix/load composites are obtained, which can separate the photocatalyst from the liquid environment under strong magnetic action. Under UV light, the order of the photocatalytic activity of the four samples is AC@CFO>MX@CFO>CFO>C₃N₄@CFO. Under 400—780 nm visible light, the order of photocatalytic activity of the four samples is AC@CFO>C₃N₄@CFO>CFO>MX@CFO.

We studied the adaptive sliding mode synchronization of three-dimensional fractional-order chaotic systems, gave the design of sliding mode functions and construction of controllers, and obtained the sufficient conditions for adaptive sliding mode synchronization of three-dimensional fractional-order uncertain chaotic systems. The relevant conclusions of fractional-order were extended to the case of integer-order, and the correctness of the conclusions was verified by MATLAB.

Based on the principle of laser drilling, we designed and developed a precise micro laser drilling device for diamond anvil cell metal sealing material. The experimental results show that this device system can meet the requirements of laser drilling of diamond anvil cell metal sealing material in micron scale.

The precursor was ultrasonically pretreated by the layer-by-layer (LbL) assembly method, and a new type of visible light photochromic inorganic organic multilayer film composed of phosphotungstic acid (PWA) and polyvinylpyrrolidone (PVP) was synthesized. Under visible light irradiation, the colorless film turned blue with a reversible photochromic effect. The bleaching process occurred when the film was in contact with oxygen in the dark or heated in the air. In the X-ray photoelectron spectroscopy (XPS), the W⁶⁺ in the W of the ultrasound-assisted LbL film was further reduced to W⁵⁺, indicating that a photoreduction reaction occurred between the PWA particles and the PVP matrix. Ultrasound assisted dispersion of PWA particles could enhance two photochromic properties, reduce the size of PWA to the size of quantum dots (10 nm to 3.4 nm), induce conversion effects and dielectric confinement effects. Within the same irradiation time, the intensity of the absorption peak increased (0.23 to 0.45), the excitation energy decreased (2.51 eV→2.23 eV), which expanded the range of absorption wavelength and absorption intensity of characteristic peak, and a higher coloring speed was obtained.

A wearable and self-powered perovskite photodetector was obtained by drop-coating method on polyester-cotton cloth substrate. The samples were characterized by scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS) and X-ray diffraction (XRD). The photo response performance of the device under various conditions was measured. The results show that the perovskite crystals are preferentially distributed between the gaps of the fibers. At a bias of 1 V, the responsivity of the device is 82 mA/W, the response time is 32 ms and the recovery time is 33 ms. The device can also work without bias, exhibiting a high on/off ratio of 3.57×10³. The device shows good stability under bending conditions. The device fabricated directly on the garment shows good performance.  The device possesses long-term stability, maintaining its original high performance after at least 60 d.

Al³⁺ composite polymeric flocculant （PAM-AA） was synthesized by  aqueous solution polymerization with acrylamide (AM), acrylic acid (AA) and Al₂(SO₄)₃ as raw materials. The optimum ploymerization conditions were determined by orthogonal experiment  as  m(AM)∶m(AA）=5∶3, reaction temperature  65 ℃ and  stirring time  5 h. The yield could reach  96.3% when the amount of the initiator was 3% of the mass of monomer.  On this basis, Ca²⁺,Mg²⁺ and Fe³⁺ composite polymeric flocculant were synthesized. The structure and morphology of Al³⁺ composite polymeric flocculant were characterized by infrared spectrum (IR), X-ray diffraction (XRD), scanning electron microscope (SEM), respectively. Al³⁺ composite polymeric flocculant and N,N-dimethyl, dodecyl, benzyl ammonium chloride (1227) were applied in municipal sludge dewatering. Through the filter cake moisture content (FCMC) and the specific resistance in filtration (SRF), we compared the effects of  the commercial flocculant polyaluminium chloride (PAC),  the mixture between Al₂(SO₄)₃ and PAM-AA on the dewatering performance of municipal sludge. The results show that the combination of Al³⁺ composite polymer flocculant and 1227 can reduce the FCMC of sludge filter cake to 63.8%, and the effect is good.

The  commonly available and stable catalysts and phosphorus ligands were used  as the catalytic system, we optimized the reaction system of  palladium-catalyzed copper-free Sonogashira coupling  to prepare β-(2-phenylethynyl)cyclobutanone compounds.  We obtained a series of alkynyl cyclobutanone compounds with different functional groups in high yields. Furthermore, the ring-opening-metathesis of these compounds catalyzed by copper was realized.

Aiming at the problems of the solid-liquid separation difficulties and the poor adsorption performance when the Al₂O₃ and expanded graphite (EG) were used separately as adsorbants, the nano Al₂O₃/EG was prepared with EG as aggregate,   Al(NO₃)₃9H₂O as aluminum source,  NH₃H₂O as precipitant and anhydrous ethanol as dispersant.  The adsorbents were characterized by Fourier transform infrared   (FT-IR) spectrosopy,   X-ray diffraction  (XRD) and  scanning electron microscopy (SEM). The effects of ammonia water addition,  activation temperature,  activation time,  the mass ratio of Al₂O₃ to EG and impregnation time on the adsorption properties of Al₂O₃/EG  on Congo red were investigated. The results show that nano Al₂O₃ is successfully loaded on the EG,  and when ammonia water is added drop by drop,  the activation temperature is 170 ℃,  the activation time is 3 h,  the mass ratio of Al₂O₃ to EG is 0.06∶1,  the impregnation time is 20 min,  the adsorption effect is best,  and the decolorization rate is about 86%. Compared with EG and Al₂O₃,  respectively,  the decolorization rate of Al₂O₃/EG is obviously increased,  which is more than 2 times that of EG, and Al₂O₃/EG is easier to separate than Al₂O₃ in separation operation after adsorption.

Chloromethylated polysulfone with high degree of substitution was prepared from polysulfone as raw materials by chloromethylation reaction,  after reacting with trimethylamine solution,  the quaternary ammonium polysulfone  with bacteriostatic properties and water-soluble was obtained by  dialysis and freeze-drying. The  quaternary ammonium polysulfone  reacted with  polyvinyl alcohol aqueous solution,  the  quaternary ammonium polysulfone/polyvinyl alcohol hydrogels could be obtained by  cross-linking γ-ray radiation without addition of  cross-linking agent. The problem of residual organic solvents and other auxiliary reagents in the polysulfone hydrogels was solved,  at the same time, the  hydrogels had bacteriostatic properties and could be used for preparing medical wound dressings.

Using microcrystalline cellulose (MCC) as raw materials, nano cellulose carrier (NCC) was prepared by ultrasonic assisted enzymatic hydrolysis. The morphology, thermal stability and particle size of NCC were characterized by scanning electron microscopy (SEM), particle size analysis, Fourier transform infrared  (FT-IR) spectroscopy and thermogravimetric analysis (TGA). The results show that the diameter of NCC prepared by MCC enzymatic hydrolysis ranges from 10.10 to 18.17 nm, and the length ranges from 531.20 to 1 106.00 nm, its thermal performance is more stable.