We propose a distributionally robust index tracking model with the conditional value-at-risk (CVaR) penalty. The model combines the idea of distributionally robust optimization for data uncertainty and the CVaR penalty to avoid large tracking errors. The distribution ambiguity is described through a confidence region based on the first-order and second-order moments of the random vector involved. We reformulate the model in the form of a min-max-min optimization into an equivalent nonsmooth minimization problem. We further give an app-roximate discretization scheme for the possible continuous random vector of the nonsmooth minimization problem, whose objective function involves the maximum of numerous but finite nonsmooth functions. The convergence of the discretization scheme to the equivalent nonsmooth reformulation is shown under mild conditions. A smoothing projected gradient (SPG) method is employed to solve the discretization scheme. Any accumulation point is shown to be a global minimizer of the discretization scheme. Numerical results on the NASDAQ index dataset from January 2008 to July 2023 demonstrate the effectiveness of our proposed model and the efficiency of the SPG method, compared with several state-of-the-art models and corresponding methods for solving them.

Focusing on 10 typical rural tourism scenic spots in different cities in Shanxi province, we construct an index system of the rural tourism public service. Constituted through the data of two questionnaires is a third-order tensor of ``surveyed tourists $\times$ survey indicators $\times$ scenic spots". Using the high precision low rank tensor (matrix) algorithm to fill missing data, we then use the Fmincon function to calculate the weights for the five most important indexes in scenic spots and satisfaction and overall satisfaction scores. The analyses show that tourists are generally dissatisfied with the self-guided tour service, and the satisfaction is significantly lower than its importance. Tourists' satisfaction and attention to theatrical performances are low. Tourists are significantly more satisfied with the information about safety risks than they take seriously. We should improve the shortcomings of tourism public services from a small point and continue to increase investment in rural tourism public services.

Aiming at the shortcomings of slow convergence and poor solution accuracy in the optimization process of the usual cuckoo search algorithm, novel cuckoo search algorithm is proposed and applied to image registration. Firstly, this paper analyzes the deficiency of L\'evy flight in usual cuckoo search algorithm; and using information sharing, local enhancement operators and new methods to build nests to improve population diversity, local exploration capability and convergence speed. Through 15 test functions and 2 medical image registration experiments, we show that improved cuckoo search algorithm has better convergence speed, calculation accuracy and robustness.

In spatial econometric models, the specification of the spatial weight matrix is of notable influence on the model estimation result. Therefore, how to specify a reasonable spatial weight matrix has long been one of the important issues in spatial data modeling. In this article, we construct a distance decaying spatial weight matrix with a controlling parameter. This spatial weight matrix has better adaptability and interpretability. Based on the constructed spatial weight matrix, a new spatial lag model is proposed and its maximum likelihood estimation procedure is formulated. Furthermore, a simulation study and real-life data analysis are conducted. The results demonstrate that the maximum likelihood estimation of the new model performs well in estimating the model parameters and the constructed spatial weight matrix can efficiently fit the spatial autocorrelation among the observations of dependent variables.

Time series with change-points always are important topics in economics, engineering and statistics, which have been widely applied in financial, meteorological, industrial and other fields. In this paper, we study the statistical inference of the AR(1) models with single change-point. For the AR(1) models with single change-point, we provide the estimators and the consistency condition of the autocorrelation coefficient estimators based on maximum likelihood (quasi-likelihood) methods. Under the provided sufficient conditions, we establish that the asymptotic distribution of the estimators, the hypothesis test on whether there is a change-point in the models and the increment of autoregressive coefficients are discussed. Finally, some simulation results and the analysis of the daily trading data of Shanghai Composite Index show the effectiveness of the proposed theories and methods.

The study of numerical methods for stiff functional differential equations is mostly carried out in the inner product space based on the assumption that the one-sided Lipschitz constant has a moderate size, whereas for some stiff problems, the one-sided Lipschitz constant inevitably takes very large positive values. Therefore, it is necessary to break through the limitations of inner product space and one-sided Lipschitz constant and study the related numerical methods directly in the Banach space. For the nonlinear composite stiff Volterra functional differential equation in Banach space, the non-stiff part is solved by the explicit Euler method and the stiff part is solved by the implicit Euler method, obtained from which is the implicit-explicit Euler method for solving the problem. The stability and asymptotic stability about the proposed method are established, and the numerical results verify the correctness of the obtained theory.

PM algorithm, an unconventional numerical method for initial value problems of ordinary differential equations, is investigated. In order to extend the algorithm to the two-point boundary value problem, the following improvements are made in this paper: Firstly, the separation principle is proposed to transform the multi-objective optimization problem corresponding to ordinary differential equations into several relatively independent single objective optimization problems; Then in the framework of the first order ordinary differential equations, the equivalence of the computational schemes for the I/II type boundary value problems of the second order ordinary differential equation is established; By maintaining the derivative relation and assuming the initial value condition or introducing independent parameters to approximate the initial value conditions, the improved PM algorithms are derived and can solve the boundary value problems of I/II/III types and even mixed type, with the second-order convergence speed.

A graph is said to be a Bi-Cayley graph over a group $H$ if it admits $H$ as a semiregular automorphism group with two vertex-orbits. The study about the symmetry of  Bi-Cayley graph is an important topic in algebra graph. Using the structure of the cubic tetracirculant graph,  we study cubic connected edge-transitive bi-Cayley graphs over semidihedral groups  and give  a classification of this kind of graphs.

The $r$-hued coloring of graphs is a hot topic in graph theory, which can be applied to fields like the communication of multi-agent systems in the optimal reconfiguration of power networks. An $(k,r)$-coloring of $G$ is a proper coloring with $k$ colors such that for every vertex $v$ with degree $d(v)$ in $G$, the color number of the neighbors of $v$ is at least min$\{d(v),r\}$. The smallest integer $k$ such that $G$ has an $(k,r)$-coloring is called the $r$-hued chromatic number and denoted by $\chi_{r}(G)$. In this paper, we study the $r$-hued coloring of Cartesian product of cycle and path, and obtain its $r$-hued chromatic number.

The length of the busy period for a multi-server queueing system is a key performance measure. It has important and wide applications in optimization problems related to queueing systems. Limited by mathematical methods, the busy period probability distribution function of multi-server finite-buffer queueing systems has not been sufficiently investigated for a long time. Using an importantly quantitative relationship between the sub-full busy period distribution of a multi-server finite-buffer Markovian queue and the full-busy period distribution of its corresponding finite waiting space queue, an elegant recursive formula for finding the Laplace-Stieltjes transform of the full busy period distribution of a multi-server Markovian queue with finite capacity is presented. Meanwhile, to calculate the full-busy period distribution function for any given time, the above Laplace-Stieltjes transform in the $s$-domain can be inverted into the time-domain by using the partial fractional decomposition method of rational functions and the rules for calculating residues. A semi-analytic inversion formula for the full busy period distribution function based on the characteristic roots of transform is also presented. Programming with Mathematica and comparing results with the exiting literature in the numerical examples validate the feasibility and simplicity of the proposed approach. This work provides a more intuitive and easy-to-implement way to determine the full-busy distribution function of a multi-server queueing system with finite buffer space.

Length-biased data widely exist in life research. However, there are relatively few studies on the mean residual life under complete length-biased data. As one of the important indicators to evaluate individuals’survival, mean residual life attracts more and more attention from statistical researchers. In order to construct nonparametric estimations of the mean residual life function under complete length-biased data, two estimations are derived by utilizing the idea of moment estimation. It is proved that the proposed estimations converge in distribution to two zero-mean normal variables under moderate conditions, respectively. In order to evaluate the performance of the proposed estimators in finite samples, a series of numerical simulations are carried out, and the simulation results are compared with those of existing methods. The numerical results show the rationality of two proposed estimations.

Since there are few tail data when the generalized Pareto distribution is applied to the data beyond the threshold, the Bootstrap confidence interval for the parameters is derived by combining the maximum likelihood estimation of the parameters with the Bootstrap method. The Bootstrap method can not only make full use of the sample information by resampling, but also avoid the estimation of the variance of the estimation when there are fewer samples. Bootstrap confidence intervals for parameters are proved being asymptotically efficient. Numerical simulation results show that the interval length and coverage are reasonable under a certain confidence level. The established Pareto model is used to analyze the magnitude data of the Bayan Har seismic zone. The results show that under the same confidence level, the length of the confidence interval determined by the Bootstrap method is shorter than that calculated by the approximate covariance matrix, which supports the applicability of the generalized Pareto model to describe the earthquake magnitude.