We considered the approximate controllability of the initial boundary value problems of a class of double coupled linear degenerate parabolic equations. We overcame the difficulty of the degeneracy of the equations,  constructed the control function by means of their conjugate problems, and proved the  approximate controllability of the initial boundary value problem  in L². In other words, for any initial value function and desired function  in L², we could  find a control function in L², which made the solution of the initial boundary value problem of the equation approximately reach the desired function in L² at the terminal time.

We considered the long-time dynamic behavior of solution for the Berger equation with nonlinear damping, and proved the existence of time-dependent global attractor by using the method of asymptotic priori estimation and contractive function.

We considered a class of quasilinear equations defined on a three\|dimensional semi infinite cylinder, in which the solutions of the equations were assumed to satisfy the nonhomogeneous condition at the finite end of the cylinder and the zero boundary condition at the side of the cylinder. By limiting the nonlinear terms and  using  the differential inequality technique, we gave the alternative theorem of the solutions of the equations on three different cylinders, and gave  the upper bound of total energy in the case of decay.

By using Krasnoselskii’s fixed point theorem and Gronwall inequality, we discussed the existence and uniqueness of solutions for fractional impulsive integro-differential equations, and obtained the exp-type Ulam-Hyers stability of these solutions. The applicability of the obtained conclusions was illustrated by an example.

The author defined the Rota-Baxter operator whose weight was λ for q-3-Lie algebras, and gave the necessary and sufficient condition for P to be Rota-Raxter operator with weight λ of q-3-Lie algebras. They could be derived from Rota-Baxter Lie-algebras, Rota-Baxter q-Lie algebras, Rota-Baxter pre-Lie algebras and Rota-Baxter group algebras.

We studied the structure of non-Abelian n-Lie algebras with β(L)=m-n+1, and classified n-Lie algebras that satisfied dim L¹<4. We  prove that there are only 2,6 and 11 classes in the cases dimL¹=1,2,3, respectively. Furthermore, if β(L)=m-n+1 and  Z(L)L¹, then (m-n+1)/2≤dimL¹≤m-n+1.

By using the finite group theory and elementary number theory, we ascertained the characteristics of elements of a class of non-Abelian finite group with order 10pⁿ, and built all homomorphic mappings from quaternion group to the class of non-Abelian finite group with order 10pⁿ. We verified that those two groups could suffice the conjecture of Asai and Yoshida by calculating the number of homomorphic mapping.

The graph formed by removing an edge from a complete graph k₄ was called chordal 4-cycle. By discharging method, the author discussed the weight of 4-cycle with chords in IC-planar graphs with degree restrictions. The author proved that each IC-planar graph with minimum vertex degree at least 5 and minimum edge degree at least 11 contained a light chordal 4-cycle v₁v₂v₃v₄v₁, and proved that the upper bounds of weight of this kind of light chordal 4-cycle was at most 37.

We used the cubic B-spline finite element method to solve a class of fourth order parabolic equations with variable coefficient, and proved the boundness and convergence of the solution of the semi-discrete scheme. For the discretization of time variable, the backward Euler scheme was
 constructed, and the convergence order of the solution of the fully discrete scheme was O(Δt+h⁴). A numerical experiment verified the effectiveness of theoretical analysis results and the B-spline finite element method.

Aiming at the problem of low computational efficiency of sequential quadratic programming (SQP) algorithms when dealing with semi-infinite minimax discretization problems with complex structures and large nonlinearities, we proposed a non-monotonic sequential quadratic constrained quadratic programming (SQCQP) algorithm, and proved the convergence of the algorithm under appropiate conditions. The results of numerical experiments show that the non-monotonic SQCQP algorithm is better than the SQP algorithm in reducing the number of iterations and calculation time when the discrete level is 100.

We proposed an efficient algorithm for the pricing problem of American multi-asset option. Firstly, by using the penalty method and the perfectly matched layer technique, we transformed the linear complementary model satisfied by multi-asset option into a nonlinear parabolic problem on a bounded domain. Secondly, a semi-implicit finite difference method was used to solve the transformed nonlinear problem, and we gave the error results of the method and the nonnegative proof of the numerical solution. Finally, numerical experiments were used to verify the practicability and effectiveness of the proposed algorithm.

We used two-point G-J method and three-point G-J method to discretize Bermuda option based on the jump diffusion stochastic volatility model, gave the pricing of American barrier option and American option, and gave the numerical calculation and the result analysis.

We gave an algorithm for calculating the maximum eigenvalues and corresponding eigenvectors of irreducible nonnegative matrices with parameter variables by using Collatz-Wielandt function,  the parameters could  be selected appropriately at every step of iteration of the algorithm to optimize  the algorithm.

When the filtration equation was solved numerically by the standard finite volume method, numerical solutions suffered from the “numerical hot barrier” phenomenon, in which numerical interface could not propagate forward correctly, we proposed a modified finite volume method. The diffusion coefficient of the method was the algebraic mean of the values of density variable at two neighbour elements. The results of numerical experiments show that the new scheme can effectively avoid “numerical hot barrier” phenomenon.

We discussed the optimal conditions for lower semicontionuity of efficient solutions to parametric set-valued vector equilibrium problem in real Hausdorff topological vector space. Firstly, we gave the concepts of weak efficient solution,Henig efficient solution,Global efficient solution, super efficient solution and f-efficient solution for the parametric set-valued vector equilibrium problem. Secondly, on the basis of nearly cone-subconvexlike, we gave the characterization results of weak efficient solution, Henig efficient solution, Global efficient solution and super efficient solution by using the form of f-efficient solution and the separation theorem of convex sets. Finally, we established the optimality theorems of lower semicontinuity for the efficient solutions to parametric setvalued vector equilibrium problem under the weak-f-property of set-valued mapping.

Let {X_n, n≥1} be an identically distributed  pairwise negatively quadrant dependent (NQD) sequence  with mean zero. Under suitable conditions, using the cental limit theorem and moment inequality of pairwise NQD sequences, we gave general function formula for the precise asymptotics of the complete moment convergence of the partial sums for pairwise NQD sequences in the logarithm law.

We studied a class of singular perturbation initial-boundary value problem of the convection-diffusion equation by using the perturbed renormalization group method. Firstly, the delay differential equation was decomposed into left and right boundary value problems without delay. Secondly,  the asymptotic solutions of left and right problems were constructed by using renormalization group method. Finally, the left and right solutions were connected by smooth joint condition, and the asymptotic solutions of the original problem were obtained.

Aiming at the large error of optimization results in engineering problems, we proposed a multi-objective particle swarm optimization algorithm based on Kriging models. Firstly, the response information of the Kriging model was used to predict errors,  the prediction errors were introduced into the comparison of Pareto dominance relationship, the selection of global and local leaders, and the process of mutation mechanism. Then, combined with the infilling strategy in the text, the optimization process could quickly and accurately approach the Pareto frontier solution set on the premise of small number of samples . The performance test results show that the proposed algorithm can improve the optimization efficiency and accuracy of complex system models.

Aiming at the problem that the traditional blockchain consensus algorithms had low efficiency and waste of a lot of resources, we proposed an algorithm that used the directed acyclic graph data structure based on ID classification to replace the traditional blockchain’s chain data structure, which solved the problem that the traditional blockchain project could confirm a large number of transactions in a short time. The proposed algorithm used the data structure of directed acyclic graph based on ID classification could reach a consensus more simply, and this data structure was more suitable for multi-user transaction confirmation at the same time. The proposed consensus algorithm could confirm transactions in a shorter time than traditional blockchain projects, which saved a lot of hardware resources and improved the processing efficiency of blockchain transactions.

Aiming at the problem of natural noise in the recommended system, we proposed a collaborative filtering correction of natural noise method of sparse data based on concept lattice. Firstly, users and items were divided into three classes: strong, average and weak to detect natural noise. Secondly, the collaborative filtering of sparse data based on concept lattice was used to correct these natural noises. Finally, the unrated items were predicted from the obtained dataset without natural noise. The experimental results on the dataset containing natural noise show that the proposed method has high recommendation accuracy and good performance in the case of sparse data.

Firstly, the product inference engine formula of the consequent output fuzzy set of fuzzy rules was given by using the triangular fuzzifier. Secondly, the analytical expression of the Mamdani fuzzy system was introduced according to the triangular fuzzifier, product inference engine and central average solution fuzzifier, and when the antecedent fuzzy set value was triangular fuzzy number, the output algorithm of Mamdani fuzzy system was designed by using subdivision domain and a triangular fuzzifier method. Finally, the effectiveness of the output algorithm was verified by using the statistical t-hypothesis test in an example.

Aiming at the problem of low accuracy and poor effect of traditional target contour recognition algorithm for image target contour recognition, the author proposed a binary image target contour recognition algorithm based on deep learning. Firstly, the deep convolution network algorithm in the
deep learning algorithm was selected to recognize the target contour of binary image, and the binary image was divided into non overlapping sub block images with the same size and input into the first layer of the deep convolution network. Secondly, the filter (convolution kernel) in the convolution network used the cost function optimized by the traditional neural network algorithm to implement convolution filtering on the input sub block image, and sent the down sampling image after convolution filtering to the second layer. After the same processing, the second layer input the result into the third layer, and the output image of the third layer was the target contour recognition result of the sub block. Finally, after all the sub blocks were recognized, all the sub blocks were clustered in the output layer by the full connection method, and the final binary image target contour recognition results were output. The experimental results show that the average recognition accuracy of 15 binary images is 98.75%, the average signal-to-noise ratio is 2.42, and the recognition effect is better.

In order to improve the accuracy of heart sound recognition of artificial intelligence assisted diagnosis, according to the periodicity of heart sound signal, we proposed a fast principal component analysis algorithm to reduce the dimension of heart sound signal and extract features. At the same time, based on the simplex evolution algorithm, the output of BP neural network learning algorithm and the expected error function were optimized to improve the learning performance of BP neural network and realize the classification and recognition of heart sound signal with higher accuracy. Aiming at the normal heart sound and eight kinds of abnormal heart sound signals, the performance was analyzed and tested. The experimental results show that the average recognition rate of all kinds of heart sounds is 95.96%. Compared with other algorithms, the improved algorithm improves the recognition rate by 4.9%, 3.9% and 1.9% respectively. It shows that the proposed algorithm can effectively classify and recognize heart sound signals and improve the recognition rate of artificial assisted diagnosis.

Aiming at the problem that the detection method of rotational magnetic resonance was easy disturbed by the peak noise, we proposed  a collaborative filtering method to eliminate the peak noise in nuclear magnetic resonance (NMR) data. Firstly, the method of 3σ was used to determine whether the
re was peak noise in the measured data under a pulse moment. The measured data were divided into two groups: peak noise and non-peak noise. Secondly, the discrete cosine transform and the Hadamard transform were performed respectively to obtain two groups of transform domain coefficients. Wiener filtering coefficient was calculated by using the transform coefficient of non-peak noise data, and the coefficient was used to filter the data containing peak noise. Finally, the filtered coefficient containing the peak noise data was inversely converted by Hadamard and discrete cosine to eliminate the peak noise. The simulation results show that the method has high accuracy in eliminating the peak noise in the ground NMR data and improves the extraction precision of signal characteristic parameters.

Aiming at the problem that the frequency interval of emission current controlled by traditional method was fixed and the frequency domain was not completely controllable, in order to improve the efficiency of frequency domain electromagnetic sounding, we proposed a selective harmonic eliminated pulse width modulation (SHEPWM) control method based on improved particle swarm optimization algorithm. Firstly, the expected spectrum was cal
culated according to the exploration target. Secondly, the half-cycle symmetric SHEPWM nonlinear equations were established according to the time-frequency domain characteristics of the emission current. Finally, the improved particle swarm optimization algorithm was used to solve the nonlinear equations, and the corresponding switching time sequence was obtained. By controlling the inverter of the emission system by the switching time sequence, the emission current required for detection could be output. When the depth target was explored, the launch system could output the optimal launch current, and then improve the vertical resolution.

Based on the principle of minimum energy consumption of each ring in the wireless sensor network, we proposed a clustering algorithm for ring wireless sensor networks based on optimal cluster head number. Firstly, we calculated the optimal number of cluster heads in each ring in the network. Secondly, on the basis of this optimal number of cluster heads, the network was divided into several clusters of different sizes. Finally, when selecting the cluster head, the ratio of the optimal number of cluster heads of each ring to the number of nodes in the corresponding ring, the remaining energy of the node, the shortest distance from the cluster member node to the cluster head node and the distance from the cluster head node to the base station were considered. This solved the problem of excessive communication energy consumption of nodes in the wireless sensor network cluster, and balanced the energy consumption of network nodes. Simulation results show that the algorithm improves network energy efficiency and scalability, balances network energy consumption, and extends the lifetime of the network.

Aiming at the shortcomings of the current wireless sensor network routing algorithm, such as low data transmission success rate, long network time, high packet loss rate, and in order to obtain better data transmission results, we proposed wireless sensor networks routing algorithm based on weighted fusion of evidence theory. Firstly, the clustering analysis algorithm was introduced to cluster the wireless sensor network, which made the distribution of cluster heads more uniform, and solved the problem that the cluster heads were too centralized and the distribution of cluster member nodes was unreasonable. Secondly, the weight of the remaining energy, communication distance between nodes, communication energy consumption were calculated by using evidence theory, the performance of each node was comprehensively evaluated according to the weight, and the most reasonable cluster head was selected according to the comprehensive evaluation results. Finally, the comparison test was carried out with other wireless sensor network routing algorithms. The test results show that, compared with the comparison algorithm, the data delay average and packet loss rate of the wireless sensor network algorithm are greatly reduced,  the data transmission success rate is improved, the energy consumption between nodes is more balanced, the lifetime of wireless sensor network is extended, and the routing reliability of the established wireless sensor network is higher.

We proposed a novel data completion algorithm based on the nonnegative matrix factorization model. Using the iterative traversal, the algorithm determined the best construction matrix and rank value, which solved the problem of missing value in single cell transcriptome sequencing data (RNA-seq), and avoided the influence of deficiency of the single cell sequencing depth on cell typing analysis. The experimental result on chronic myeloid leukemia data shows that after the missing values are recovered by the completion algorithm, the cell typing is more clear, which verifies the effectiveness of the proposed algorithm.

Using the B3LYP method of density functional theory (DFT), the author optimized the geometry of gas phase phenylalanine (Phe) molecule at the level of 6-311+G(2df) basis set, and analyzed the characteristics of intermediate and transition state of each elementary reaction in the process of chiral transition of Phe molecular system. Under PBE0/def2-TZVP, the time-dependent density functional theory (TDDFT) was used to obtain the electronic excitation characteristics of transition state in the process of chiral transition of Phe molecular system under gas phase conditions. The results show that according to the criteria of Δr index and the hole-electron index of the electronic excitation characteristics, the obvious excitation characteristics
 can be identified effectively, and the judgment results of the excitation characteristics of the transition state molecular system in the elementary reaction are given directly.

We studied self-adaptive sliding mode synchronization of fractional-order nonlinear chaotic system, and considered the outer disturbances and uncertainty. We gave the construction and  self-adaptive rules of sliding mode functions and controllers, and obtained the sufficient conditions for self-adaptive sliding mode synchronization of fractional-order nonlinear chaotic systems. The conclusions of the same order were extended to different cases, and the correctness of the conclusions was verified by using MATLAB numerical simulations.

We proposed an optical fiber vibration sensing system based on φ-optical time domain reflectometer (φ-OTDR). Eleven characteristic parameters, such as main wave energy, main lobe delay, main peak value, main wave Q value, signal spectrum, 3 dB time delay, peak value of main lobe, center frequency, second return loss, wave number and half wave length were extracted as reference criteria, and the target recognition method of multi-parameter fuzzy evaluation matrix was used to recognize and classify similar targets in five groups of 200 experiments. The results show that the comprehensive recognition rate is 90.87%. This method improves the accuracy of similarity recognition by using multi-angle parameters to measure and discriminate targets.

The dressed states of the single atom-cavity system were described and Hilbert space was truncated t-photon transition process. By solving the master equation of the system, we calculated the energy level shift caused by two-photon resonance transition, and calculated the second-order correlation function of photon, which confirmed the existence of photon anti-bunching effect under given parameters.

ZnO nanowires were prepared by microemulsion synthesis method. The morphology and structure of ZnO nanowires were characterized by transmission electron microscopy (TEM) and Raman spectrum, and the third-order nonlinear optical properties of ZnO nanowires were studied by ultra-short pulse Z-scan experiment. The results show that the prepared materials have the same size, good crystallinity and good excitonic photoluminescence characteristics in ultraviolet region. The third-order nonlinear optical refractive index of ZnO nanowires is positive, and the third-order nonlinear optical susceptibility is almost three orders larger than that of bulk ZnO materials. Therefore, the ZnO nanowires have good optical and nonlinear optical properties.

The stannic anhydride (SnO₂) nanoparticles were synthesized by simple hydrothermal method and used as sensing materials after calcination at high temperature. In the process of synthesis, glucose and palladium were used as structure directing agent and material modification agent. The test results show that in the atmosphere of 100 μg/L n-butanol, the sensitive material of Pd doped SnO₂ with mole fraction of 0.04 has the highest sensitivity, the response value is 44.2, the response time is 37.8 s, and it has good selectivity and stability.

Using FeCl₃·6H₂O，FeCl₂·4H₂O and NaOH as raw materials, the magnetic fluid Fe₃O₄ was prepared firstly, and then sodium dodecyl benzene sulfonate (SDBS) modified sodium alginate (SA) magnetic spheres were prepared.  The effects of pH value, initial mass concentration, adsorption time and temperature on the adsorption of Cu²⁺ in the water were investigated by single factor experiment, the coordination characteristics of SA anion and Ca²⁺ and Cu²⁺ were simulated at the level of B3LYP/6-31G^**. The results show that at 25 ℃, the adsorption time is 80 min, the mass concentration of Cu²⁺ is 100 mg/L and pH=4, the maximum adsorption amount is 116.6 mg/g, and the bonding ability of Cu²⁺ is stronger than that of Ca²⁺.

The complex of trimesic acid-nickel was synthesized at room temperatuer with 1,3,5-benzoic acid as ligand, Ni²⁺ as center ion and water as solvent. X-ray diffraction was used to characterize the crystal structure. We studied the frequency characteristics, response recovery characteristics and humidity sensing characteristics, tested and analyzed the humidity sensing mechanism. The results show that the crystal unit structure is ［Ni₃(BTC)₂·14H₂O］·2H₂O(CCDC: 1990438), and the complex has zero dimensional structure. The linear relationship of humidity sensitive characteristic curve is good in the medium and high humidity area（54%—97% RH）, the recovery time is 2 s, and the humidity sensing performance is excellent, which is suitable for the detection of humidity sensing materials in medium and high humidity environment.

In order to study the atmospheric PM_2.5 pollution characteristics and sources in winter and spring in Changchun City, environmental samples were successively collected from 2018-01-06—2018-05-14, we analyzed inorganic elements and water-soluble anions in PM_2.5. The results show that the average mass concentration of PM_2.5 in Changchun City during the sampling period is (46.4±24.4) μg/m³, and the average mass concentration in winter and spring is (51.0±25.8) μg/m³ and (32.6±11.5) μg/m³, respectively. The rate of exceeding the standard is 11%, all of them occur in winter. The mass concentration of PM_2.5 is low and remains stable during the Spring Festival（2018-02-15—2018-02-21）. The mass concentration of all water-soluble anions and some inorganic elements (Al,As,Pb,Se and Ti) are higher in winter than that of in spring. The inorganic elements mainly come from coal combustion, traffic pollution and fugitive dust. The NO^-₃ and SO^2-₄ in PM_2.5 are the result of the combined action of vehicle exhaust and coal combustion,  and the latter plays a more important role. The main sources of PM_2.5 in winter and spring are secondary aerosol (28.2%),  soil dust (12.6%),  traffic emissions (10.7%), coal combustion and industrial dust (28.6%),  industrial dust and others (19.8%).