Let φe(m) be generalized Euler function, where e is a positive integer. Aiming at the problem of the solvability of equation φ2(φ6(m))=2ω(m), based on the calculation formulas of generalized Euler function φ2(m) and generalized Euler function φ6(m), and combined with the properties of Euler function φ(m), the author gives all the 92 integer solutions of the equation.

Firstly, the correlation estimates of temperature and velocity were obtained by using differential inequality technique, especially the fourth-order norm estimates of temperature and the gradient estimates of velocity. Secondly, with the help of a priori bound, the energy expression was constructed and the differential inequality satisfied by the expression was derived. Finally, the continuous dependence of solutions of Brinkman-Darcy fluid equations on the boundary coefficients α was established.

By using Fourier analysis and Leray-Schauder fixed point theorem, we discuss the existence and uniqueness of solutions of the fourth-order periodic boundary value problem. The existence and uniqueness of the solution of the problem are obtained under the condition that the nonlinear term f(x,u,v) satisfies an appropriate inequality.

By using the methods of upper and lower solutions  and topological degree theory, the author discusses the existence and multiplicity of positive solutions for the nonlinear simply supported static beam equations  with a parameter. where λ>0 is a parameter, k₁<k₂<0, k₁ and k₂ are real constants, f: ［0,1］×［0,∞)→(0,∞) is a continuous function, and f(x,y)  monotonically increases with respect to y for a fixed x∈［0,1］.

By analyzing the position of the root of characteristic equation on the complex plane, we gave the local asymptotic stability and global stability of unique positive equilibrium point of the model and the existence of Hopf bifurcation, and numerically verified the obtained theoretical results by using the MATLAB software package and numerical methods for solving initial value problems of ordinary differential equations.

By using the theory of operator semigroup, estimation technique of noncompact measure and Darbo’s fixed point theorem,
we studied the existence of mild solution for conformable fractional non-instantaneous impulsive differential equation with nonlocal problem. Under the condition that the nonlinear term satisfied the  appropriate growth condition and noncompactness measure condition, and  the nonlocal term and non-instantaneous impulsive function satisfied the  Lipschitz condition, we obtained the existence result of solution of the problem, and gave  an example  to illustrate the effectiveness of the result.

By using the method of weight functions and the technique of real analysis, we obtained a new half-discrete Hilbert-type inequality involving higher-order derivative function. As applications, we also discussed the equivalent conditions of optimal constant factor related to a few parameters in inequalities and some special inequalities.

We considered several hybrid conjugate gradient methods, determined their step length by weak Wolfe line search, and generated the descent direction satisfying the sufficient descent condition by orthogonalization strategy. The effectiveness of these algorithms was verified by CUTEst test problems, and the numerical performance of these algorithms was analyzed.

Firstly, by using the relationship between generalized quadratic matrix A and idempotent matrix P, which was expressed as (A-dP)(A-eP)=0, we discussed the basic properties of generalized polynomial f_P(A) of A, and proved the rank invariance of operations for generalized polynomials. The results show that ranks of generalized polynomials are not only independent of the choice of combination coefficients, but also independent of the choice of polynomials in most cases. Secondly, as applications, the relevant results of idempotent matrix, involutory matrix, quadratic matrix and generalized quadratic matrix were summarized and generalized.

By using the properties of covariant derivative operator, gradient operator, Ricci curvature and strong maximum principle of second-order elliptic operator on multiply warped product manifolds, the author discussed gradient Ricci almost solitons on multiply warped product manifolds, gave the necessary and sufficient conditions that multiply warped product manifolds were gradient Ricci almost solitons and a rigid result of gradient Ricci almost solitons on multiply warped product.

We studied the w-module theoretic analogue of P-flat modules by using w-module theory over commutative rings. Firstly, we introduced the concept of w-P-flat modules over commutative rings and discussed several characterizations and properties of w-P-flat modules. Secondly, as applications of w-P-lat modules, we give some new characterizations of von Neumann regular rings and p.p rings (i.e., rings in which every principle ideal is a projective ideal).

Firstly, by introducing the concepts of a weakly idempotent quantale and a weakly idempotent nucleus on a quantale, we gave an equivalent characterization for the largest weakly idempotent quantale quotient. Secondly, we proved that the category of weakly idempotent quantales was a reflective subcategory of the category of quantales, the category of idempotent quantales was a reflective subcategory of the category of weakly idempotent quantales. Finally, we obtained that the category of idempotent quantales was a reflective subcategory of the category of quantales.

We considered the problem of the reliability analysis of the dependent series system with masked data. Firstly, when the component life followed Burr Ⅻ distribution, the dependent structure of component life variables and the reliability model of dependent series system with masked data were established by introducing Copula function, and some probability results for the series system were derived. Secondly, Based on the progressively Type-Ⅱ censored system failure  data, we obtained the maximum likelihood estimation of model parameters, and the asymptotic confidence intervals and bias-corrected percentile bootstrap confidence intervals of model parameters were constructed based on asymptotic normality theory and bootstrap sampling algorithm. Finally, simulation and real data analysis were carried out. The results show that the proposed model  methods are feasible and effective for the reliability analysis of dependent masked data systems.

Under sublinear expectations, using the truncation method, we gave strong law of large numbers for a sequence of widely orthant de
pendent (WOD) random variables.

We considered the linearized compressible Navier-Stokes equations in a one-dimensional bounded interval. By using Hormander theorem, we first proved the unique continuity of the dual problem, and then obtained the approximate controllability of the Navier-Stokes equations through the unique continuity of the dual problem.

Firstly, we studied  homotopy category consisting of Ding-projective modules, and constructed some recollements in these homotopy categories.
Secondly,  we obtained  some related to stable t-structures by using relationship between t-structure and recollement.

The author considered the Eisenhart problem of symmetric parallel (0,2)-type tensor fields with respect to a quarter-symmetric metric connection  by using tensor analysis method and Kenmotsu manifold’s structural equations. Under suitable conditions, the author  proved that the  tensor field must be a constant multiple of the metric tensor, and gave a necessary and sufficient condition for a Ricci symmetric Kenmotsu manifold to be an Einstein manifold.

Aiming at the problem that a large number of software packages had no tags or imperfect tags on node package manager (NPM) platform, we proposed a deep learning method based on the seq2seq model to recommend tags for software packages. Firstly, we used ECMAScript tools to analyze the source code of software package, constructed function call graphs of the package, and traversed the function call graph, so as to convert the software package into a set of function call sequences with package semantic information. Secondly, we trained the seq2seq model and applied the trained model to tag recommendation of software package. The trained model could map the function call sequence of package to a group of predicted tag sequence, so as to complete the tag recommendation of software packge. The experimental results show that the method can recommend a reasonable set of tags for the software package, and the accuracy is 82.6%.

Aiming at the problem that most of the existing task assignment in spatial crowdsourcing only considered unilateral benefits, short-term benefits and single scenario, we proposed a spatial crowdsourcing task assignment algorithm based on multi-agent deep reinforcement learning. Firstly, a new spatial crowdsourcing scenario was defined, in which workers could freely choose whether to cooperate with others. Secondly, a multi-agent deep reinforcement learning model based on the attention mechanism and A2C (advantage actor-critic) method was designed for task assignment in the new scenario. Finally, simulation experiments were carried out, and the performance of the algorithm was compared with other latest task assignment algorithms. The experimental results show that the proposed algorithm can achieve higher task completion rate and worker profitability rate simultaneously, which proves the effectiveness and robustness of the algorithm.

Aiming at the problem that it was difficult for the four-way shuttle storage system to process multi-order tasks at the same time, we proposed an adaptive genetic annealing algorithm to solve the operation time model of the system, and designed a chromosome individual containing the delivery order of multi-order tasks and the selection information of cargo elevator, which solved the difficult problem of processing multi-order tasks at the same time in the four-way shuttle system, and realized the scheduling optimization of multi-order tasks. The experimental results show that the algorithm optimizes the outbound order of outbound goods and the selection information of cargo elevator, and effectively solves the problem that the four-way shuttle storage system processes multi-order tasks at the same time.

In view of the problems that traditional network representation learning method could not learn the correlation of network structure of nodes, we proposed a learning model for network structure representation based on neighborhood information. Firstly, the model defined a calculation method of structural similarity between nodes based on neighborhood information, and modeled the structural similarity between nodes in different neighborhood ranges. Secondly, the deep autoencoder was constructed, the node structure similarity was used as supervised information to optimize the network representation, and the node structure information was learned
 in the process of network embedding. Compared with three related algorithms such as node2vec,SDNE,struc2vec, the experimental results show that the proposed method has better ability of identifying the network structure, can learn the structural correlations between nodes, and the obtained network representation can be suitable for role identification related tasks. In addition, the experimental results of cross-network classification also demonstrate the potential of this method in transfer learning.

Aiming at the problem that the simple linear iterative clustering (SLIC) algorithm in the superpixel segmentation of multi-spectral remote sensing images underutilized the image feature information and the fixed size and number of superpixels leaded to low segmentation accuracy, we proposed to introduce manifold SLIC (MSLIC) algorithm into the task of superpixel segmentation of remote sensing images and improve it. Firstly, we proposed an improved texture feature extraction method for multi-spectral remote sensing images based on color local binary pattern (CoLBP). Secondly, we expanded the spectral space of the MSLIC algorithm so that the algorithm could adapt to high-dimensional image data. Finally, we improved the clustering distance measurement of the MSLIC algorithm, fused the multi-spectral features, spatial features and texture features of the image to perform iterative clustering of pixels to achieve content-sensitive superpixel segmentation. The experimental results show that compared with the existing methods, the proposed algorithm has more accurate superpixel segmentation results of multispectral remote sensing images, and improves the edge recall rate, under segmentation error and subdivision accuracy. It can improve the problems of low accuracy in the preprocessing method of multispectral remote sensing image segmentation.

Aiming at the problem that the traditional recommendation algorithm in commodity session sequence recommendation relied too much on adjacent clicks and lost the overall commodity access trend to a certain extent, we proposed a new commodity session sequence recommendation algorithm based on global directed graph. Firstly, the global directed graph of commodity conversation sequence was constructed. The nodes in the graph were commodities, the arcs between nodes represented the click order, and the directed graph was stored in the graph database. Secondly, the global preference propagation strategy on the directed graph was given, and the important influence of click time on recommendation was considered. Finally, the score of the commodity to be recommended was obtained. On Diginetica and Yoochoose standard data sets, the recommended accuracy of the algorithm was improved by 6.12% and 30.25% respectively compared with the traditional Item-KNN method according to the P@20 standard, according to the MRR@20 standard, the recommended accuracy was improved by 15.04% and 33.88% respectively. The experimental results show that the proposed global directed graph search and scoring strategy is effective.

Aiming at  the problem that the traditional iris location algorithm was difficult to achieve accurate location, resulting in unstable recognition effect, we proposed an iris location algorithm based on improved YoloV4 network. Firstly, YoloV4 combined with MobileNetV3 was used for rough positioning of the inner and outer circles of the iris, and then the gray difference between the pupil, iris and sclera was used for fine positioning of the inner and outer circles of the iris. At the same time, K-means++ clustering algorithm was used to generate a priori box. The fast soft-DIoU-NMS algorithm was used to remove the redundant boxes in the prediction process and improve the detection rate of the algorithm. The Focal Loss was used as a category loss function. The comparative experimental results show that the algorithm runs faster, the location accuracy is higher, and the effect of recognition algorithm is more obvious.

Aiming at the problem of chromatic aberration and poor definition between fog image and dust image, we proposed an enhancement 
algorithm of fog image and dust image. Firstly, the grayscale world algorithm was used to solve the chromatic aberration problem of dust image in Lab color space, the dust image and fog image after adjusting the chromatic aberration were converted to HSV color space to improve the saturation and contrast of the image respectively. Secondly, the Laplacian operator was used to detect the blur degree of the image. For the image with high blur degree, the weighted least squares filtering algorithm was used to remove the fuzzy features and further improve the visibility of the image. Finally, a comparative experiment was carried out with the existing fog removal algorithm and dust enhancement algorithm. The experimental results show that the algorithm can better restore the color of the image and effectively improve the image definition.

Aiming at the shadow detection problem of single image in complex environment, we proposed a fast automatic shadow detection algorithm based on multi-scale super-pixel fusion. Firstly, the depth image was used to calculate the normal vector and spatial coordinates of every point, and the simple linear iterative clustering algorithm was used to complete multi-scale super-pixels segment for the color image. Secondly, the shadow confidence algorithm was used to estimate the shadow confidence of the super-pixels in each scale combined with the chromaticity, the normal and the spatial position information of the image. Finally, the trained Adaboost classifier was used to fuse the super-pixels shadow confidence in each scale, and the final judgment result was obtained. The experimental results show that the accuracy of the proposed algorithm is significantly higher than the original shadow confidence algorithm, and the running time is about 10% of the original shadow confidence algorithm. It performs more prominently in the detection of small shadows, large shadows, and soft shadows with unclear edges, which is suitable for pre-processing of the images in complex light environment.

Aiming at the problem that it was difficult to select the parameters of gradient boosting decision tree (GBDT), we proposed a GBDT regression prediction algorithm based on improved whale optimization algorithm (IWOA). Firstly, an improved whale optimization algorithm was proposed, which initialized the population by using chaotic mapping to improve the diversity of the population, and the inertial weight and the mutation crossover strategy of differential evolution algorithm were introduced to solve the problem that it was easy to fall into the local optimization in the later stage of iteration. Secondly, IWOA was used to optimize the key parameters of the GBDT to avoid the blindness of parameter selection and improve the generalization ability of the regression prediction model. Finally,
 the IWOA-GBDT regression prediction model was established and verified by the UCI dataset. The experimental results show that compared with decision tree, support vector machine, Adaboost and GBDT algorithms, the proposed model algorithm has better fitting effect and certain practical value.

Aiming at the problems of large number of parameters, unstable gradient and low segmentation accuracy of the original SegNet network model, we proposed an improved model by constructing the SegNet combined with bottleneck block with residual, depthwise separable convolution and skip connection structure. The experimental results on aerial and satellite remote sensing image data sets show that the improved network model obtains better results in performance evaluation indexes such as accuracy, recall and F₁ value, which shows that the improved network model has good practical value in building segmentation task of remote sensing images.

To address the problem that features unrelated to language identification in broadcast audio have an impact on the language 
identification results, an improved language identification method based on gamma frequency cepstrum coefficients with improved feature parameters is proposed. By extracting the energy spectral envelope of each frame, the speaker-related features are removed, filtered by a Gammatone filter banks, and then by the discrete cosine transform and cepstrum lifting to obtain the improved gamma frequency cepstrum feature parameters. The feature parameters extracted from broadcast audio signal were input into hidden Markov model for training and testing， and the language identification results were obtained. The results show that the proposed method can effectively improve the language identification accuracy for broadcast audio, which is better than the currently used gamma frequency cepstrum coefficient features and their derivatives.

Firstly, aiming at the problem of facial expression recognition, we proposed a new multi-scale feature selection network recognition method. The network fully combined the advantages of multi-scale network structure and feature selection structure, which could extract the spatial information in the facial static images more effectively. Secondly, in order to verify the recognition performance and generalization ability of the proposed multi-scale feature selection network, we carried out comparison and cross validation experiments with some common methods on two classical facial expression recognition databases. The experiment results show that the proposed network achieves better recognition effect and has good generalization ability, it can be flexibly embedded into the facial expression recognition and analysis system.

We proposed a fractional-order Lü chaotic system with nonlinear delay term according to the fractional-order Lü chaotic system. Firstly, the fractional-order Lü chaotic system was numerically solved by Adomian-decompositio-method (ADM). Secondly, the phase trajectory diagram of the system was drawn by MATLAB software. Finally, by using simulation technology and dynamic analysis tools such as bifurcation diagram, complexity and phase trajectory, the effect of system parameters on the system was analyzed. The numerical simulation results show that the system has rich dynamic characteristics.

The synchronization of uncertain atmospheric fractional-order systems was studied based on the sliding mode theory. The controllers and adaptive control laws were designed, and two sufficient conditions for adaptive sliding mode synchronization of atmospheric chaotic systems were obtained. The results show that the atmospheric chaotic systems can achieve adaptive sliding mode synchronization under certain assumptions.

 The VASP (Vienna ab-initio simulation package) software package based on density functional theory was used to calculate the effect 
of five metal and non-metal doping on the crystal structure and electronic properties of two-dimensional Ga₂O₂. The results show that the doping of B,C,Mg,Si and P can change the crystal structure and electronic properties of Ga₂O₂. Compared with the intrinsic Ga₂O₂ crystal, the band gaps of these doping systems are all decrease, which is due to the charge redistribution between the doped atom and the adjacent atom.

The hydrogel was prepared by free radical polymerization and freeze-thaw cycle using hydroxypropyl acrylate， ι-carrageenan，  polyvinyl alcohol and calcium chloride as raw materials. We tested and analyzed the mechanical property，  self-recovery property，  swelling property，  self-healing property，  anti-fatigue property and the conductivity property of the hydrogel. The results show that the hydrogel has high mechanical strength, the fracture stress is 625.77 kPa， the fracture elongation is 604.48%, the swelling rate of hydrogel is low, and has excellent self-recovery property，  anti-fatigue property  and  conductivity property.

Amyotrophic lateral sclerosis (ALS) was a neurodegenerative disease characterized by motor neuron apoptosis. There was no effective treatment and drug at present. The discovery of new related genes or target genes played an important role in the study of pathogenesis and clinical treatment of ALS,  and provided a new direction and target for clinical prevention,  diagnosis and treatment. We collected the data information of ALS related genes from the website database,   used a variety of pathogenic gene prediction tool software for bioinformatics analysis,  and predicted the pathogenic genes. After the integration of data information,   39 candidate genes were obtained. The results show that the candidate  genes  have a certain correlation with ALS.

Based on the internal ecological restoration mechanism of ammonia nitrogen polluted groundwater, we studied the purification characteristics of ammonia nitrogen polluted groundwater by using indigenous nitrogen bacteria in groundwater loaded by scoria,  an ecologically safe natural mineral material. The results show that the biomass of indigenous nitrogen bacteria loaded by scoria is about 2.12×10⁷ per gram. The loading material can effectively remove hydrochemical factors while removing ammonia nitrogen from groundwater. The ammonia nitrogen removal rate is 83.39%~98.84%. The removal capacity of hydrochemical factors from large to small is Fe²⁺,HCO^-₃,Ca²⁺,Mn²⁺,CO^2-₃,SO^2-₄,S^2-,Mg²⁺,  in which Fe²⁺,Mn²⁺,S^2-,SO^2-4 promote ammonia nitrogen purification to a certain extent,  and CO^2-₃,HCO^-₃,Ca²⁺,Mg²⁺ inhibit ammonia nitrogen purification. The microstructure of the loading material becomes smooth after purification,  and the small protrusions are covered. The research results provide an experimental  basis for the research and development of internal ecological regulation and restoration technology of nitrogen polluted underground.