We considered the multiplicity of weak solutions for a class of p-Laplacian elliptic equations with signchanging weight function. By means of Nehari’s manifold and logarithmic Sobolev inequality, we proved  that there were  at least two nontrivial weak solutions to the problem.

By using Banach’s contraction mapping principle and Krasnoselskill’s fixed point theorem, we considered  a class of boundary value problems of Caputo type sequential fractional differential equations  with RiemannLiouville fractional integral  conditions, obtained  sufficient conditions for the existence of  unique solution and at least one solution to the problem, and gave an example  to illustrate the application of the result.

Based on the structure classification of three dimensional Lie bialgebras over C, we calculated the Atiyah classes of all three dimensional Lie bialgebras over C by using a sufficient and necessary condition for the Atiyah class of a Lie bialgebra to be zero.

By using the equivalence class of group action,  the set of cocycles and the group homomorphism were connected. By characterizing the number of homomorphisms between two finite groups through the set of cocycles,  we proved that for any finite groups A,G, if the number of elements in the set of cocycles of  A and G could be  divided by the greatest common divisor of |A| and |G|, the number of homomorphisms between A and G could be divided by the greatest common divisor of |A| and |G|.

Let R be a coherent ring. By using the divisible modules and hdivisible modules, we characterized the coherent domains with fPD at most one, and gave a set of equivalent conditions. Moreover, we gave homological characterization of the dimension fPD of R by using n-FP-injective modules.

Aiming at the bicyclic graphs, we designed an algorithm to determine the gracefulness of graphs verified the gracefulness of all bicyclic graphs with at most 17 vertices, and obtained all graceful and ungraceful graphs in this range. The results show that except ∞shape bicyclic graphs C_(m,n), all bicyclic graphs with at most 17 vertices are graceful, where (m+n)(mod 4)={1,2}. Finally, we gave a proof for the ungracefulness of this kind of graph, and further speculation that the conclusion still held true when the number of vertices was greater than 17.

Using the method of  graph transformation, we discussed  the HyperWiener indices of cacti. By comparison, we gave the secondminimum and thirdminimum HyperWiener indices of  cacti, and characterized the corresponding extremal graphs as well.

We considered  the Ghostburster system model of a class of weak electric fish neuron cells. Firstly, the equilibrium point of the neuron system was given by numerical calculation method. By analyzing the eigenvalues of the Jacobian matrix near the equilibrium point, the stability and its type near the equilibrium point were analyzed. Secondly, using the Hopf bifurcation existence theory and its analysis method, the direction of the model Hopf bifurcation and the approximate solution and the approximate period of the bifurcation period were given. The  results show that when the system parameters are controlled within a certain range, and the system model produces subcritical Hopf bifurcation, the periodic solution orbit is gradually increased and unstable.  Finally, the numerical simulation results of theoretical analysis were given by using MATLAB and other mathematical software. The parameters of the maximum conductance of potassium ion current of dendritic membrane and the injection current of the cell membrane were selected as the bifurcation parameters, and the dynamic behavior of the system under single parameter changes was investigated.

We gave  a new method to test the equality of  twosample highdimensional  covariance matrices based on random matrix theory in a highdimensional framework. The results show that using  the central limit theorem for the linear spectral statistics of highdimensional random F-matrices, the limit distribution of the test statistics is given, which is not only suitable for  highdimensional data but also valid for nonnormal cases.

We assumed that the stock price obeyed the CEV jump-diffusion model. Firstly, we gave the probability density function of the stock price by using It formula and Feller lemma of jump process. Secondly, we established the risk neutral measure by using the measure conversion of the compound Poisson process. Finally, under the condition of  the risk neutral measure, we gave the pricing formula of the European call option by using the riskfree discount of expected return.

Aiming at the problem that the step size of traditional multithreshold color image segmentation method was less than the fixed value of distance parameter, sometimes the step size was too large to exceed the optimal result, we proposed global segmentation method for multithreshold color images under adaptive step size. Firstly, the color image was preprocessed to solve the problem of reducing the total number of colors without reducing the quality of the color image in order to improve 
the segmentation efficiency. Secondly, according to the objective function of multithreshold color image segmentation, the chaotic optimization theory was combined with particle swarm optimization algorithm, the global segmentation objective function of multithreshold color image was solved by chaotic particle swarm optimization algorithm. Finally, combined with the  global segmentation method for multithreshold color image under adaptive step size, the optimal color image thr
eshold segmentation results were obtained. The experimental results show that the segmentation effect, accuracy, stability and convergence of this method are good.

Aiming at the problem that the classification accuracy and generalization ability of model were low in missing attribute value dat
a classification algorithm, we proposed a classification algorithm of missing attribute value data based on fuzzy rule, namely “cyclereceive” model. The algorithm did not need an interpolation computation to the missing attribute value data and could directly classify the data set. The simulation experiment of UCI open data sets was carried out. The experiment results show that, compared with other algorithms, the “cyclereceive” model has higher classification accuracy and generalization ability.

Aiming at the problem that traditional image restoration algorithms were difficult to derive convergent splitting results when antiHermitian components dominated Hermitian components, which led to poor image restoration effect， the author proposed an image restoration algorithm based on displacement Hermitian splitting. Firstly, displacement parameter was introduced to define quasiHermitian splitting when matrix was split. Secondly, the splitting result was substituted by conjugate gradient normal residual (CGNR) algorithm for inner iteration to approximate each outer iteration, and each outer iteration was derived from the convergent splitting of coefficient matrix. Thirdly, the derived convergence splitting 
results were applied to the image restoration model. Finally, the algorithm was compared with the generalized minimum error method and the generalized preconditioned symmetric splitting method. The experimental results show that the iterative approximation results obtained by the proposed algorithm are better, the number of iterations and CPU time required are significantly reduced, the CPU occupancy time is only 0.25 s, and the image restoration effect is better.

Based on the matrix decomposition and clustering, we proposed a collaborative filtering recommendation algorithm. The algorithm first used alternating least squares (ALS) algorithm to decompose matrix, and then used improved k-means clustering algorithm to compensate for large amount of calculation caused by single ALS algorithm in the later stage of collaborative filtering. The problem that the recommendation accuracy was low due to the reduction of the high dimension and high sparsity of original matrix was solved, and the calculation speed and recommendation accuracy were greatly improved. Experimental results show that the improved algorithm significantly improves the recommendation accuracy.

Aiming at the problem that the density peaks clustering (DPC) algorithm had poor clustering performance in dealing with data with complex structure, high dimensionality and multiple density peaks in the same class, we proposed a density peaks clustering algorithm based on Knearest neighbors and classesmerging (KMDPC). Firstly, the sample distribution was described by the defined density calculation method, and the clustering center was obtained by using new evalution index. Secondly, an iterative assignment strategy based on the idea of Knearest neighbors was designed to classify the remaining data points accurately. Finally, a local merging method was presented to prevent the splitting of classes with mu
ltiple density peaks. Simulation results show that the performance of this algorithm is obviously better than that of DPC algorithm on 22 different datasets.

Firstly, using fractional Fourier transform algorithm, the color image was transformed in fractional frequency domain, the frequency domain information of the image was analyzed, and the coefficients suitable for embedding watermark were selected as the embedding position of watermark. Secondly, the concealment characteristics of the frequency domain coefficients were defined, and the concealment characteristics were classified, so as to achieve the optimal watermark embedding coefficient. Finally, the lifting wavelet transform and compressive sensing measurement of binary watermark images were made, and the measurement results were embedded into the carrier images as watermark information. It solved the problem of high redundancy of embedded information in most current watermark algorithm and insufficient antiattack ability of the watermark image. Experimental results show that the proposed algorithm has good performance in resisting geometric attack and noise interference, it significantly reduces the redundancy of watermark information, improves the efficiency of embedding and security of watermark information, and increases the reliability of watermark algorithm.

We proposed an image multiscale geometrical analysis method based on multistage directional guided filter, and applied it to the multisensor images fusion. Firstly, using the guided filter, an approximate subband and some detail subbands at the different scales were obtained by multistage edgepreserving decomposition of the image. Secondly, directional analysis of 
detail subbands was carried out by using wedgeshaped small size directional shear filter, multiple directionalsubbands were generated at the different scales. According to the different physical meanings of approximate subbands and the directional detail subbands, the different fusion criteria were used to merge the decomposed coefficients respectively. Finally, the reconstructed image was obtained by simple superposition calculation of the merged coefficients. This method effectively solved the problems that filters lacked selfadaptability and image analysis did not have translation invariance in the other multiscale decomposition. The experimental results of multigroup image fusion show that the performance of this method is better than other typical multiscale decomposition methods in the process of image fusion.

Aiming at the shortcomings of current wireless network channel allocation methods, such as interference and low network throughput, etc, we designed a wireless network channel allocation method based on combinatorial intelligent algorithm. Firstly, the principle of wireless network channel allocation was analyzed, and the wireless network channel allocation model was constructed. Secondly, the genetic algorithm was used to generate the initial allocation scheme of wireless network channel,  the particle swarm optimization algorithm was introduced to search the initial allocation scheme of wireless 
network channel, and a reasonable allocation scheme of wireless network channel was obtained. Finally, the throughput, network delay and packet loss rate of data transmission of wireless network were simulated and tested on the platform of MATLAB 2016. The simulation results show that this method greatly improves the throughput of wireless network. The network delay and packet loss rate of data transmission are much less than that of a single genetic algorithm or particle swarm optimization algorithm, which improves the communication performance of wireless network.

Using the B3LYP method of density functional theory, we optimized the geometric configuration of the gasphase Sphenylalanine (S-Phe) chiral enantiomers at the 3-21G basis set, calculated the Phe molecular orbital wave function 
and charge decomposition analysis (CDA) of fragment orbit, and gave the orbital interaction diagrams of fragments COOH,C₆H₅ and NH₂ during the formation of fragments. The results show that the maximum positive values of r_i of fragments COOH,C₆H₅ and NH₂ based on CDA are r₁₉,r₁₇,r₁₅, indicating that the formation of molecular orbits leads to the transfer of electrons from the fragment orbit to overlapping regions. The maximum negative values of ri of fragments COOH and NH₂ based on CDA are r₄₄ and r₃₅, indicating that the formation of molecular orbits results in the removal of electrons occupying fragment orbits from overlapping regions.

Using the CAMB3LYP method of density functional theory, we optimized the geometric configuration of R-Cysteine (R-Cys) molecule in gas phase at 6-31G (d) basis set, theoretically studied the charge transfer characteristics between fragments of R-Cys system during the electron excitation, and calculated the charge transfer percentage between fragments based on the excited state density of relaxation and nonrelaxation. The results show that for SH fragment, the charge transfer percentage from S0 to S3 is 57.96%. For COOH fragment, the charge transfer percentages of the excited states from S0 to S1—S5 are negative, and the qualitative results of charge transfer are consistent. For NH₂ fragments, the charge transfer percentages from S0 to S1 and S4 are 698% and 3145%, respectively.

The FeN thin films were prepared on Si(100) and SrTiO3(100) single crystal substrates by DC magnetron sputtering, keeping the flow rate of argon unchanged, and controlling the volume fractions of nitrogen at 10%,12.5% and 15%, respectively. The structure and magnetic properties of FeN thin films grown on the different substrates were characterized by using Xray diffraction (XRD) and vibrating sample magnetometer (VSM) respectively. The results show that the singlephase γ′-Fe₄N thin films are obtained on the SrTiO₃(100) single crystal substrate. Oriented growth of the γ′-Fe₄N films  induced on SrTiO₃(100) substrate is easier than Si(100) substrate. When the volume fraction of nitrogen is about 12.5%, the crystallinity is better and the saturation magnetization is higher of the singlephase γ′-Fe₄N thin films, the coercivity is lower than that of the Fe_N thin films obtained on Si(100) substrate, and the soft magnetic properties are better.

We designed a coplanar waveguidefed 2.45 GHz flexible microstrip antenna. The threelayer human tissue model (skinfatmuscle) was constructed in the high frequency structure simulator (HFSS) software, the influence of human tissue on the key parameters of the antenna was analyzed, the structural parameters of the flexible antenna were optimized, and the sensing performance of the flexible microstrip antenna under bending condition was studied. The test results show that the return loss of the antenna is -26 dB at 2.40 GHz and the frequency band is 2 030—2 770 MHz. When the antenna is bent, the centre frequency is offset, which can be used to meature different radius of curvature in the bending state.

CaAl₄O₇∶Tb³⁺ and CaAl₄O₇∶Sm³⁺ fluorescent materials were prepared by the selfpropagating combustion synthesis (
SPCS) technology. The structure, morphology and light absorption property of the two fluorescent materials were investigated by Xray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), scanning electron microscope (SEM), and ultravioletvisible diffusereflectance spectrum (UVVis DRS), etc. The results show that SPCS technology can significantly reduce the synthesis temperature, crystallinity of samples is good, and the doping of Tb³⁺ and Sm³⁺ does not change monoclinic system structure and space group characteristics of CaAl₄O₇, but it results in the slight lattice distortion and the redshift of absorption peak and absorption band edge. The CaAl₄O₇∶Tb³⁺ and CaAl₄O₇∶Sm³⁺ emit green and pink light, respectively, and their excitation spectra and emission spectra show fine structural characteristics.

T-2 toxin liposome (LPT2) was prepared by thin film ultrasonic method. Entrapment efficiency was determined by high performance liquid chromatography (HPLC) method, and the linearity, additive recovery, precision and repeatability of the method were investigated. Effects of organic solvent, phospholipid species, m(drug)∶m(lipid), hydration volume, m(phospholipids)∶m(cholesterol) on encapsulation efficiency were investigated by single factor method, and the prescription of T-2 toxin liposome was optimized by response surface method. The liposome was granulated by high pressure homogenization and particle size distribution was determined. The inhibitory effects of T-2 toxin and LPT2 on growth of 5 tumor cells and one normal cell were detected by thiazolyl blue (MTT) method. The results show that the best preparation process of LP-T2 are egg phosphatidylcholine (EPC) as membrance material, thin film ultrasonic method, m(drug)∶m(lipid)=3.5, m(phospholipid)∶m(cholesterol)=374, 4hydroxyethyl piperazine ethylsulfonic acid (Hepes， 20 mmol/L, pH=7.4) hydration, hydration volume 11.8 mL, ultrasound for 10 min. The average particle size after high pressure homogenization is 267 nm. LPT2 can significantly inhibit the proliferation of tumor cells, and its
 activity is similar to that of mitomycin. IC₅₀ value of normal cell (1 881.54 ng/mL) is higher than that of T2 toxin (1427.83 ng/mL). The entrapment efficiency and particle size of LPT2 conform to the standard of drug, and it has antitumor activity in vitro.

Macroporous resin separation, LH20 gel chromatography and high performance liquid chromatography were used to isolate three polyphenols from  the mature yellow fruiting bodies of Inonotus hispidus. The structures of  the compounds were identrfied by nuclear magnetic resonance (NMR), mass spectrometry combined with their physical and chemical properties, and the anti-inflammatory activity and cell toxicity of the compounds were investigated. The results show that the obtained compounds are phellibaumin A (compound [STHZ]1[ST]), phelligridin D (compound 2) and3′,4′-dihydroxy-5-［［11-hydroxyphenyl］-6,7-vinyl］-3,5dioxafluoren5one (compound 3), of which compound 3 is a new compound. 
The three compounds inhibit the NO generation by lipopolysaccharide-induced RAW264.7 cells in a dose-dependent manner. 
The inhibition ratio of each compound on RAW2647 cell proliferation shows a dose-dependent trend. The inhibition ratio is much lower than 50% when the effective dose of anti\|inflammation is 50 μmol/L.

The changes of relative water content (RWC), net photosynthetic rate (A), transpiration rate (E) and water use efficiency (WUE) in leaves under calcium magnesium acetate (CMA) stress, artificial simulated freezethaw stress （10,5,0,-3,0,5,10 ℃） and combined stress of freezethaw and CMA were studied with alfalfa Dongmu1. The results show that: RWC,A,E and WUE values in CMA group are lower than those  in control group (CK); RWC, A, and E values in freezethaw group and combined stress group decrease first and then increase; WUE value in freezethaw group decreases first, then increases and then decreases, and reaches the minimum at T2 （5 ℃） and the maximum  at T5 （0 ℃）; WUE value in combined stress group reaches the maximum at T5; RWC, A and E values of alfalfa seedlings show the change that CMA group<freezethaw group<combined stress group, it may indicate that combined stress has greater effect on seedlings. Under combined stress, the water consumption of leaves decreases by closing the stomata, so the RWC, A and E values reduce significantly to improve the stress resistance of alfalfa.