In this paper, a Legendre spectral method for solving the second Fredholm integral equation is presented. Firstly, the integral transformation is performed, and then the Legendre-Gauss quadrative formula combined with series expansion method are used to approximate the integral term and the unknown function, and then the approximate solution of the transformed integral equation is obtained, and the error analysis is carried out. Finally, the feasibility and effectiveness of the method are verified by numerical examples.

In this paper, the concept and properties of Schwarz derivative are reviewed, and the Taylor ex-pansion with Lagrangian coterms, Lagrangian interpolation and coterms are given. Finally, a new definition of higher order Schwarz derivative is proposed to establish a relationship with ordinary derivatives.

HÖlder inequality plays an extremely important role in analysis and elementary in- equality theory. As an important aspect of related research, the study of its equiva-lence relationship with other inequalities has also received increasing attention. This paper proves the H¨older inequality and the AM-GM inequality in measure space, and its equivalence relationship with the power mean inequality. These results generalize the discrete inequality equivalence relationship recently established by Li etal.

Due to the high cost for the health care resources for the patients in serious condition which require intensive health care either in waiting or treatment state, this paper aims for optimal design and control for this intensive health care system with a given patients’ arrival rate. Firstly, this stochas-tic two-stage intensive health care system has been characterized by a two-node queueing model. Secondly, based on the balance equations of the discrete-time Markov chain, the form of the sta-tionary distribution of this two-stage stochastic system has been analyzed. Deploying the perturbed random walk which has a product-form solution and the Markov reward process based approxima-tion scheme, this paper finds the mean number of patients waiting in each stage, which can be used to determine the optimal design for the beds in the intensive health care system. Thirdly, when the medical resources are determined, using the queueing game theory, the equilibrium arrive rate for the intensive health care system has been determined; finally, based on the numerical example, it has been shown that our approach can be used to design and control for the intensive health care system effectively. Our results can also be applied to the optimal design and control for other two-stage stochastic system with high cost in waiting.

Urban traffic congestion has become a major problem affecting people’s livelihood. Thus, traffic en-gineers have carried out many useful explorations and achieved good results. Urban traffic signal control system, which automatically detects the traffic flow on the road and gives the optimal con-trol scheme of traffic lights at intersections, has also been a research hotspot. The practice of traffic “green wave belt” in some cities gives citizens a good sense of participation. At present, however, the “one-way traffic green wave belt” is still difficult to meet the two-way coordinated control of trunk traffic, and too much emphasis on the green wave control in specific directions will also bring re-verse traffic obstruction or “red wave belt” phenomenon, sometimes seriously affecting the traffic efficiency of the intersection correlation phase. Whether we can use the spatio-temporal graph to optimize the algorithm of green wave control signal period and green signal ratio to further im-prove the traffic efficiency and realize the “bidirectional green wave band” is the focus of this paper. Through the study of Minhang District Xinzhen Road intersection, the traffic signal control strategy is optimized.

Toeplitz matrix is one of special forms of structural matrix, and its study plays an important role in the theory of matrix and computational mathematics. This paper mainly discusses the solution of the determinant of a special Toeplitz matrix, three recursive relations are obtained by using the properties of the determinant, and the solution of the determinant of the special Toeplitz matrix is transformed into the solution of the determinant of the tridiagonal Toeplitz matrix, the arithmetic difference and quasi-arithmetic sequence are constructed by combining the recursive relations to find the general term expression, and the exact solution of the determinant of the special Toeplitz matrix is given. As application, the positive qualitative determination problem of this special Toeplitz matrix is solved.

Employee performance appraisal will always affect the operation and development of enterprises; therefore, we should ensure the perfection of the appraisal system and the accuracy and objective of the appraisal results. By analyzing the background of the new energy automobile industry and the problems existing in the performance appraisal of the employees of such enterprises, this paper further analyzes the optimization strategy of the performance appraisal system of the employees of new energy automobile enterprises based on statistical methods.

This paper introduces the characteristics, functions, basic layout, characteristics and functions of container terminal yard and other basic information. This paper focuses on the storage process of container terminals, the “turning over” problem of container yards and its causes, the comprehen-sive construction of the mathematical model of spatial resource optimization of container terminals (including the hypothesis description of related problems, the establishment of mathematical models, and the application of branch and bound method) for reference.

This paper selects 399 policy documents of cruise industry at national and local levels, and adopts the methods of policy text quantification and social network analysis. This paper analyzes the evo-lution law and development trend of cruise industry policy in China from two aspects: policy struc-ture (including policy quantity, policy form, policy department and Policy Tool) and policy hot con-tent. The results show that: in the evolution of the policy structure, the cruise industry policy of our country has a certain scale in the quantity, form, promulgation departments and the use of policy tools, the number of policies issued by the state and the local government is approximately the same, the form of the local policy is slightly higher than that at the state level, the promulgation departments of the Industrial Policy are diversified, and there is a certain degree of coordination; however, most of them are issued by a single department, and the number of joint publications is limited. The frequency of using supply-oriented policies is higher than that of environmental and demand-oriented policies; at each stage of the policy hot content has a certain degree of change; that is, with the development of the cruise industry, the corresponding policy is also constantly making adjustments.

Accurate prediction of concentration can reduce the occurrence of many health problems and im-prove people’s quality of life. Most of the existing studies have established machine learning models or single deep learning models to predict concentration, and often only focus on the impact of single factor and ignore the comprehensive impact of multiple factors. Based on this, this paper adopted the method of combining one-dimensional convolution and long-term and short-term network (CNN-LSTM), using the pollutant concentration and meteorological data to predict the future short-term single-day concentration, and the results were compared with a single LSTM model. The results showed that the combined CNN-LSTM model has better prediction effect than the single LSTM model.