In recent years, most of the literature on equilibrium problem is under the convexity condition or the function satisfies the triangle inequality property. In this paper, a kind of equilibrium problem based on the Ekeland variational principle is established. This problem weakens the convexity requirement of the function and its domain, and reduces the condition of triangle inequality. The function only requires the condition of cyclic antimonotonicity, but the equilibrium problem has good properties. On the one hand, we use nonlinear analysis method to study the uniqueness of solutions for nonconvex compact and nonconvex noncompact equilibrium problems. In the sense of Baire classification, we obtain the general uniqueness of the solutions to the equilibrium problems based on the Ekeland variational principle. On the other hand, the stability of the solutions of nonconvex compact and nonconvex noncompact equilibrium problems are studied by using bounded rationality model, and the structural stability and robustness to ε-equilibrium are obtained. Finally, as an application, we give the stability of Nash equilibrium point for bounded rationality model of nonconvex compact n-person noncooperative game.

Two-level factorial designs are widely used in various scientific experiments. To satisfy the requirement of these designs for practical applications, the general minimum lower order confounding (GMC) criterion was proposed to select factorial designs, called GMC designs. The theoretical construction results of GMC 2^n-m designs with N/4+2≤n ≤5N/16 are proposed, where 2^n-m is used to denote the two-level fractional factorial designs with n factors and N(=2^n-m) runs. This paper aims at studying the statistical properties of GMC 2^n-m designs with N/4+2≤n ≤5N/16. At first, we study the properties of the aliasing relations of these GMC designs. Then, we provide some results to illustrate the estimability of them, propose a new theoretical method to obtain their aliasing information and list the table containing the considered designs with the information of their statistical properties. Finally, we make the conclusion of this paper and discuss some future work.

We introduce the Bregman divergence as a general loss function for the generalized linear sparse model with group structures so that the parameter estimation and variable selection methods are not limited to a specific model or a specific loss function. We compare the characteristics of eight kinds of penalty functions, such as Ridge, SACD, Lasso, Adaptive Lasso, Group Lasso, Hierarchical Lasso, Adaptive Hierarchical Lasso and Sparse Group Lasso, and the methods of parameter estimation and variable selection with these penalties. The Coordinate Descent algorithm for Hierarchical Lasso and the Accelerated Full Gradient Update algorithm for Sparse Group Lasso are also detailed. The simulation study shows that the Group Lasso, Hierarchical Lasso, Adaptive Hierarchical Lasso, and Sparse Group Lasso can better utilize the group structure information of the data, Adaptive Hierarchical Lasso and Sparse Group Lasso in terms of variable selection accuracy and parameter estimation accuracy. Compared with other methods, the Sparse Group Lasso is optimal in model prediction accuracy. As an empirical example, we apply a logistic model with Sparse Group Lasso penalty to the analysis of gene expression levels in peripheral blood mononuclear cells of patients with osteoarthritis and selected 136 genes in 9 gene sets which affect osteoarthritis, in order to have a certain guiding value for the follow-up biomedical research.

In this paper, a kind of generalized polynomial complementarity problem is studied. Under certain conditions, we prove that the generalized polynomial complementarity problem has a unique solution. We show that the generalized polynomial complementarity problem can be converted to a smoothing unconstrained optimization problem by minimax technic, and we also propose a new smoothing conjugate gradient method for solving the proposed generalized polynomial complementarity problem. The proposed method can be proved to be globally convergent under standard assumptions. Finally, numerical results are given to illustrate that the proposed method can efficiently solve the generalized polynomial complementarity problem.

In this paper, we study the existence of one-sign solutions for the following problem:#br#(-div(φ_p(▽u))=α(x) φ_p(u⁺)+β(x)φ_p(u^-) +ra(x)f(u), x ∈ Ω,
u(x)=0, x ∈ ∂Ω,)#br#where Ω is a bounded domain in R^N with a smooth boundary ∂Ω and N≥2, 1 < p < +∞, φ_p(s)=|s|^p-2s, a(x)∈ C(Ω) is positive, u⁺ = max{u, 0}, u^-= -min{u, 0}, α(x), β(x)∈ C(Ω); f∈ C (R, R), sf(s)>0 for s≠0. We give the intervals for the parameter r≠0 which ensure the existence of one-sign solutions for the above high-dimensional p-Laplacian half-quasilinear problems if f₀∉ (0, ∞) or f_∞∉ (0, ∞), where f₀= lim_|s|→0 f(s)/φ_p(s), f_∞=lim_|s|→+∞ f(s)/φ_p(s). We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

A general class of semiparametric hazards regression models for recurrent gap times is proposed to model the risk function of multiple type recurrent gap times. We further develop an estimating equation method to estimate the model parameters. The asymptotic properties of the proposed estimators are established, and the finite sample properties are also investigated by simulation studies.

Parallel projection algorithm is one of the most popular tools for solving convex set image reconstruction problem, which includes upper-relaxed and under-relaxed versions and expresses iteration-complexity O (1/k²) convergence. In this paper, inspired by the acceleration technique introduced by Nesterov, we present an accelerated under-relaxed parallel projection algorithm for the convex set image reconstruction problem and show its convergence with an iteration-complexity O (1/k²) under some suitable conditions. Furthermore, we propose its self-adaptive formalization by involving Arimijo-like technique. Finally, numerical simulation results for convex set image reconstruction problem are reported to show that our algorithm converges more quickly than the under-relaxed parallel projection algorithm.

The topology of mathematics can naturally represent the coding relationship structure, also known as topological coding, which is applied in many fields of science and real situations. Topological coding can be used to build up topological graph passwords (Topcodes), and possess provable security, computational security, computationally unbreakable and irreversible. For more new techniques of topological coding we combine graph proper total colorings with graph edge-magic labellings together for obtaining a new type of proper total colorings having the restriction of graph edge-magic labellings, these new proper total colorings are called edge-magic tcn-pure total colorings and equitably edge-magic tcn-pure total colorings in this paper. Our new proper total colorings induce extremum chromatic numbers (edge-magic tcn-pure total coloring numbers), which are related with the famous Total Coloring Conjecture in graph theory. We show: "Each tree admits an edge-magic tcn-pure coloring, or an equitably edge-magic tcn-pure coloring" and determine edge-magic tcn-pure total coloring numbers for some particular trees. Moreover, we get "Any connected graph G admitting an edge-magic tcn-pure coloring or an equitably edge-magic tcn-pure coloring is the result of doing vertex-coinciding operation to some tree T admitting an edge-magic tcn-pure coloring or an equitably edge-magic tcn-pure coloring, that is, T is graph homomorphism to G." It is noticeable, our results can be generalized to graphs with cycles, and make random Topcodes and random rooted Topcodes.

In this article, we propose a general class of semiparametric rates models for recurrent event data, which includes the proportional rates model and a semiparametric additive rates model as special cases. For the inference on the model parameters, estimating equation approaches are developed. The consistency and asymptotic normality properties of the proposed estimators are established. Jeep problem is an important optimization model related to logistics and transportation. At present, the focus of jeep problem is mainly on the problem of the longest distance, while the time efficiency problem, which is equally important in practical problems, has not been deeply studied. In this paper, the problem of how to achieve the shortest time for caravan to reach the longest distance is considered. By introducing the concept of driving task, the representation and solution of optimal time for the caravan are given. In the absence of depot quantity constraint, the shortest time needed to reach the maximum distance is obtained. Under the constraint of the minimum number of depots, an estimate of the optimal time for the caravan of two jeeps was obtained, and a guess about the optimal time for the caravan of two jeeps was put forward. The conclusions and methods of this paper are useful for many practical problems that require time efficiency and time optimization of other jeep problems.

In this work, the class of self-adjoint lubrication (SAL) equation is investigated, which is an important model of various nonlinear real situations describing the dynamics of thin liquid films. By applying a set of non-singular local multipliers and the classical Lie method, we systematically present the complete set of local conservation laws, nonlocally related PDE systems, Lie symmetries and some interesting analytical solutions of the equation for an arbitrary constant a. Here it is the first time to investigate the nonlocally related PDE systems for this model, which can be used to expand the solution space of the given PDE system. Furthermore, by using its local conservation laws and variational principle, we obtain four kinds of Lagrangians.

Based on the judgment poststratified sample with censoring, we discuss the Kaplan-Meier estimator for survival function and its asymptotic property. With the rank information of judgment poststratified sample, we also propose several isotonized Kaplan-Meier estimators and their asymptotic properties. Moreover, we conduct the simulation studies to compare the performance of all estimators based on judgment poststratified sample with the Kaplan-Meier estimator based on simple random sample and show that the estimations based on judgment poststratified sample is more efficient than the estimator based on simple random sample, and the isotonic regression method can improve the efficiency of the estimation. Meanwhile, based on the several Kaplan-Meier estimators, we construct the corresponding mean estimators and compare their performance under various settings, the results present the similar patterns of estimators of survival functions. Moreover, we study the impact of imperfect ranking on these estimators by setting different relative coefficients between variable of interests and covariate. The estimators under imperfect ranking show similar performance with that under perfect ranking, and the increase of relative coefficient can improve the relative efficiency of estimators. A real data analysis is also conducted to show the efficiency of the proposed estimators.

Considering the transaction cost and jumping environment, the European option pricing model is established using mixed sub-fractional Brownian motion. Firstly, using the Delta hedging strategy, the stochastic partial differential equations satisfied by European call option are obtained. Secondly, the pricing formula of European call, put option and the parity formula of call and put are obtained by using the self-financing strategy. Finally, using the daily price data of the closing prices of "Shanghai Stock Index", "Shibei B-shares" and "Yaopi B-shares", the research shows that:the jump environment model is closer to the true value than the classic B-S model.

A graph Γ is said to be a bi-Cayley graph over a group G if Γ has an automorphism group isomorphic to G and acts semiregularly on V (Γ) with exactly two orbits. Furthermore, if G is normal in the full automorphism group of Γ, then Γ is said to be a normal bi-Cayley graph over G. In this paper, we prove that for all most non-abelain simple groups G, cubic vertex-transitive bi-Cayley graphs over G are normal bi-Cayley graphs over G.

Column augmented design is an important way to add experimental factors and arrange follow-up strategies to follow. In this paper, the four-level and five-level column augmented designs are constructed, respectively. The uniformity of this kind of design is studied based on the average mixture discrepancy criterion, and a new lower bound of the average mixture discrepancy of the four-level and five-level column augmented design is obtained, respectively. The numerical examples show that the constructed column augmented designs in this paper are highly efficient.

Based on the existence theorem of cooperative equilibria for n-person noncooperative games by Scarf and Kajii, more and more studies show that its necessary to study the cooperative equilibrium of non-cooperative games. We integrate the population games model by Sandhlom and the existence theorem of cooperative equilibrium for multileader-multi-follower games proved by Yang and Ju, aiming at studying the cooperative equilibrium of multi-leader-multi-follower games in detail. First, we introduce the concept of cooperative equilibrium in multi-leader-multi-follower population game and prove its existence theorem by Proposition 2 in Kajii. Then, we degenerate the conclusion of this paper into the existence theorem of cooperative equilibrium in population games and the existence theorem of cooperative equilibrium in single-leader-multi-follower population games. Finally, we give the corresponding examples analysis of population games and multi-leadermulti-follower population games.

In this paper, the stability of a bidirectional ring neural network system with two delays is analyzed. Firstly, the system is linearized around the trivial solution, and the characteristic equation of the linearized system is deduced. Then the characteristic polynomial of the system is decomposed into four first-order factors through the factorization method, hence, the characteristic values are equal to zeros of the four first-order factors. Secondly, by constructing auxiliary functions and discussing the different range of system parameters, we obtain the conditions that the zero of each factor has negative real part. Moreover, the delay-independent and delay-dependent stability results are established, and the allowable interval for delays are also given. Finally, we also study a linear neural network, which origins from the adjacent two neurons of the ring neural network is cut off. The stability and instability of the linear neural network are systematically discussed in the whole parameter space, and the range for system parameters are obtained. The numerical simulation results show that the linear neural network has a more wider range of parameters, that is, it is more easier to be stabilized than the ring architecture.

For a special class of multiobjective optimization problems, where each objective function is the sum of a second-order continuous differentiable function and a proper convex but not necessarily differentiable function, a proximal Newton method is proposed.We introduce the proximal Newton method with line search and the proximal Newton method without line search. We prove that each accumulation point of the sequence generated by these two algorithms is a Pareto stationary point of the multiobjective optimization problem. In addition, we give their applications in constrained multiobjective optimization and robust multiobjective optimization. In particular, for robust multiobjective optimization, we show that the subproblems of proximal Newton algorithms can be regarded as quadratic programming problems. Numerical experiments are carried out to verify the effectiveness of the proposed method.

In this paper, we consider an anisotropic curvature flow of convex hypersurface in the Euclidean n-space. This flow involves Gaussian curvature and functions of support function and its gradient. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a corollary, we give the existence of smooth solutions to the dual Orlicz-Minkowski problem.

Given progressively Type-II censored sample, reliability R=P (Y < X) and un-observed data are analyzed by using Bayesian technique when X and Y are independent BS (Birnaum-Saunders) distributions with unknown parameters. Firstly, under different loss functions, Bayes estimation of the BS unknown parameters and reliability R are derived based on Markov Chain Monte Carlo (MCMC) technique because they cannot be derived in closed form. Secondly, the un-observed data and corresponding credible intervals are predicted. Finally, two real data sets are presented for illustrative purposes.

The development factors play important roles in liability reserve valuation, since they reflect the laws of development from claim incurred to settled due to report delay or settlement delay. Most of researches on the chain ladder method assumed that development factors are a series of non-random parameters and estimated them according to chain ladder model. However, due to heterogeneity over policies in the concerned portfolio, the development factors are generally taken as random variables, and therefore, the estimation and statistical inferences on development factors fall into the Bayesian framework. In this paper, the idea from credibility theory is used. With the multiple claims triangle data, we build the Bayesian models for development factors and derive the inhomogeneous and homogeneous credibility estimates of development factors. Furthermore, the structure parameters in credibility estimate are given, as well as the statistical properties of these estimators are proved. We thus get the empirical Bayesian estimates for development factors. Finally, numerical comparisons and case analysis are given to show that the estimates we got are convenient to use and the results are good enough in practices.

In this paper, we prove a Talagrand's T₂-transportation inequality on the continuous paths space with respect to the weighted L₂-norm, for the law of a stochastic heat equation in the whole space R^d, d ≥ 1. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The Girsanov transformation for the space-colored time-white noise plays an important role. This extents the result in Boufoussi and Hajji (2018).

By using the concepts of Chapman Kolmogorov equation, fundamental solution matrix and state transition matrix, and combining with Floquet theory, a class of nonlinear neutral differential system with delays are considered. Firstly, a new expression of the system solution is obtained by an appropriate integral transformation. Further, by using Krasnoselskii s fixed point theorem, the existence of periodic solutions for the system is given. Some sufficient conditions for the uniqueness of periodic solutions and stability of zero solutions are obtained by constructing an appropriate contractive mapping under certain conditions, which improve the corresponding results in the literature.

In this paper, we study the $k$ -product facility location problem with linear penalties and submodular penalties when the cost of setting up any facitity is zero. In the $k$ -product facility location problem with linear penalties, each client has a linear penalty cost, the problem is to select a set of facility to be set up and determine an assignment for some of the clients to a set of $k$ facilities to provide it with $k$ distinct products and penalize the remaining clients. The aim is to minimize the sum of the penalty costs and the shipping costs from facilities to clients. For this problem, we design an approximation algorithm with an approximate ratio of $\frac{3k}{2}-\frac{3}{2}$ when $k\geq 3$. In the $k$ -product facility location problem with submodular penalties, each subset of the clients has a submodular penalty cost.By exploiting the special structure of the problem, when $k\geq 3$, we design a primal-dual approximation algorithm with approximation factor $\frac{3k}{2}-\frac{3}{2}$.

In order to give full play to the advantages of fuzzy theory in uncertainty prediction and retain the interpretability of the fuzzy time series (FTS) forecasting model, this paper improves the widely used fuzzy C-means clustering (FCM) algorithm and proposes a method based on Cuckoo Search-FCM (CS-FCM) algorithm. A fuzzy time series forecasting model based on CS-FCM algorithm is established, and the CS-FCM algorithm is used to partition the universe of discourse and fuzzify the data. The CS algorithm can find the global optimization of the clustering center, reduce the error caused by the traditional FCM algorithm that is easy to fall into the local optimum, and improve the prediction accuracy of the model. The empirical analysis results show that the fitness of the CS-FCM algorithm is better than the FCM algorithm, and the forecasting error of the proposed model in this paper is smaller than that of the classic fuzzy time series forecasting model, which verifies the effectiveness of the new forecasting model.

Under the background of big data, people pay more attention to mining the hidden information of large-scale high-dimensional data. In this paper, the main purpose of this paper is to adopt the distributed optimization method to solve the problem of parameter estimation and variable selection in high dimensional linear regression with SCAD and Adaptive LASSO penalty. The main method is to convert the optimization problem based on the global loss function into the optimization problem based on the surrogate loss function by constructing an communication-efficient regular surrogate loss function. The modified ADMM algorithm designed in this paper only requires the worker machine to calculate the gradient based on local data, and the master machine to perform parameter estimation and variable selection. In terms of communication complexity between worker and master machines, the estimation error obtained by the proposed method converges to the estimation error obtained by optimizing the global loss function. The feasibility and practicability of the distributed computing method proposed in this paper are further verified through simulation and empirical research.

Let $G$ be an undirected multigraph. The oriented diameter of $G$ is the minimum diameter of any strongly connected orientation of $G$. Dankelmann, Guo and Surmacs[J. Graph Theory. 88 (2018) 5——17] showed that every bridgeless graph $G$ of order $n$ has an oriented diameter at most $n-\Delta+3$, where $\Delta$ is the maximum degree of $G$. Let $H$ be a spanning subgraph of $G$. By defining $N_G(H)=\bigcup\limits_{v\in V(H)}N_G(v)\setminus V(H)$ and using the above conclusion, they also proved that the oriented diameter of a graph with a given edge $e$ is at most $n-|N_G(e)|+5$, and for every bridgeless subgraph $H$ of $G$, there is such an oriented diameter at most $n-|N_G(H)|+3$. Let $P_3=uvw$ be a path of length 2 in $G$. It is easy to see that $P_3$ contains two edges and both edges are its bridges. In this paper, we adopt the method of contracting the path to a vertex to prove that the upper bound of the oriented diameter of a graph $G$ with given $P_3$ is $n-|N_G(P_3)|+5$. In particular, if $P_3$ is in a $4$ circle or $P_3$ is not in a circle but both edges $uv$ and $vw$ are in some two triangles, the oriented diameter of $G$ is at most $n-|N_G(P_3)|+4$. Finally, examples show that the above bounds is tight.

Some singularly perturbed quasilinear boundary value problems are studied. Under certain conditions, solutions are shown to exhibit shock layer behavior at turning point t=0. The formal approximation of problems is constructed using the method of composite expansions, and then approximation solutions of left and right sides at t=0 are jointed by joint method which exhibits boundary layer behavior respectively. As a result a approximate solution which exhibits shock layer behavior at t=0 is formed. And the existence and asymptotic behavior of solutions are proved by the theory of differential inequalities.

This paper is concerned with the initial boundary value problem of a coupled system of viscoelastic wave equation with nonlinear damping term and source term. Firstly, under certain condition on the initial and boundary data of the system, we prove that the solution is global existence by using the potential well theory. Secondly, under suitable restriction on the relaxation function of the system, we show that the solution energy has a general decay by using the perturbed energy method and inequality techniques.

Option pricing is one of the most complex problems in all financial mathematics. With the establishment of axiomatization of uncertainty theory, the research of option pricing based on uncertainty theory is gradually expanded and the fractional derivative term of fractional differential equation can well describe the memory characteristics of the market. Uncertainty and randomness are two basic types of indeterminacy. Chance space is initialized for modelling complex systems with not only uncertainty but also randomness. In order to model the financial market with uncertainty, randomness and memory characteristics, this paper presents a new version of uncertain market model on a chance space by assuming that stock price satisfies an uncertain fractional differential equation for Caputo type and stochastic interest rate satisfies a stochastic differential equation. By using the Mittag-Leffler function and α-path of the differential equation, the pricing formulas of butterfly option and European spread option based on the proposed model are investigated as well as some numerical examples.

Currently, the coronavirus disease (COVID-19) is becoming a pandemic around the world, threatening the human social and economic activities and life safety. It is scientifically interested to investigate the evolution process of COVID-19 with confirmed positive cases in specific region and time period. We propose a time-varying coefficient Hawkes process to study the trend of COVID-19 transmission over time and evaluate the effect of government intervention measures as well as some emergent events. Adopting the spline approximation technique, we develop a semiparametric estimating approach, whose finitesample performance is assessed through simulation study. As an illustration, we apply the time-varying coefficient Hawkes process to examine the transmission dynamics of the pandemic in Japan, South Korea, Beijing and Wuhan, showing that the government intervention policy and emergent public health event can affect the evolution mechanism of COVID-19.

In this paper, we first construct a bounded rationality model by defining a rational function, and study the stability of the NTU core for population games under bounded rationality. We next introduce the concept of another cooperative equilibrium for population games, that is strong equilibrium. Moreover, by using the similar method, we study the stability of the strong equilibria under the bounded rationality. Our result shows that the NTU core and strong equilibria of most of population games in the sense of Baire category are stable under our framework of the bounded rationality models.

Based on Lotka-Volterra competition model, a dynamic model including cancer cells and healthy host cells is proposed to study the effect of continuous chemotherapy on the steady-state solution of the system. Three possible situations of the system are discussed:cancer cells coexist with healthy cells, cancer cell eradicate and the cancer win the fight with healthy cells. Through further theoretical analysis of the solution of the system, sufficient conditions for the existence and global asymptotic stability of the solution are obtained. The theoretical results are further verified by numerical simulation. It is also found that the steady-state solution of the system is related to the infusion rate of chemotherapeutic drugs and the killing rate of chemotherapeutic drugs on cancer cells. Studies have show that in our model, high concentrations of chemotherapeutic drugs may completely eradicate cancer cells, but the threshold depends on the killing coefficient of drugs on cancer cells (regardless of cell mutation an acquisition of drug resistance). And for cancer treatment, enhancing the killing rate of cancer cells is more effective than increasing the drug dose.

As a measure for the centrality of a point in a set of multivariate data, statistical depth functions play important roles in multivariate analysis. Many famous depth functions have been developed in the last decades. Nevertheless, these depths mainly serve for descriptive/inferential purposes in the location setting, not from the regression point of view. In this paper, we consider the possibility of extending some of them into the regression setting. A general concept of regression depth function is provided to guide the possible extensions. Illustrative examples are also presented to show the proposed regression depth functions.

Based on a new search direction, a full-Newton step feasible interior-point algorithm is proposed to solve the linear weighted complementarity problem (LWCP), which is the more general optimization of the Fisher market equilibrium. By a continuous differentiable function in interior-point algorithms, we present the algebraic equivalent transformation of the smooth central path and obtain the new search direction for LWCP. By extending a full-Newton step interior-point algorithm for linear optimization, we propose a full-Newton step feasible interior-point algorithm for solving LWCP. At each iteration the algorithm uses only full-Newton steps and does not require any linear search, which saves computation and memory. We show the polynomial complexity of the algorithm for LWCP and the Fisher market equilibrium problem. The numerical results illustrate the efficiency of the algorithm.

Nonlinear evolution equations are widely used in the field of engineering technology. To get the exact solution by solving nonlinear differential equations is always the focus and difficulty. At present, many methods for solving equations have been proposed, such as scatter inversion method, Lie group method, Backlund transformation method and some direct expansion methods, including bilinear method, mixed exponential method, homogeneous equilibrium method, hyperbolic function expansion method, Jacobi elliptic function expansion method and so on. In this paper, we proposed a coupled trial equations method based on the trial equation method. The Hirota-Satsuma coupled KdV equations describing the interaction of two long waves with different dispersion relations are solved by it. Then the classifications of traveling wave solutions of the equations are given by the complete discrimination system for polynomial, including four groups of solitary wave solutions, two groups of discontinuous periodic solutions and seven groups of Jacobi elliptic function solutions are obtained. Compared with the solutions obtained in other literatures, our solutions not only include them,but also include new solutions,which have not been obtained by other methods.

The ranking set sampling method is suitable for the situation where the sample measurement is difficult but the ranking is easy, and the sample contains the order information. Exponential distribution plays a very important role in life test. In order to improve the estimation efficiency of paremeter for exponential distribution, this paper presents the best linear unbiased estimator of paremeter under ranked set sampling, and calculates the variance of the new estimator. The new estimator is shown to have asymptotic normality. The research results of relative efficiency and simulated efficiency show that the estimation efficiency of the new estimator is not only higher than the uniformly minimum variance unbiased estimator under simple random sampling, but also higher than the sample mean and the modified maximum likelihood estimator under ranked set sampling. Finally, the proposed method is applied to the clinical study of metastatic renal carcinoma to verify the effectiveness of the method.

Considering the tracking, isolation and screening measures implemented in China, a COVID-19 complex network model considering 31 provinces, autonomous regions and municipalities (except Taiwan) was established based on the data of the national and Jiangsu epidemic and the data of Baidu migration. The purpose of this study was to analyze the trend of epidemic, the effectiveness of the prevention and control measures and the risk of the spillover of the outbreak. Firstly, the model is fitted based on the data of confirmed cases, tracking and isolating close contacts and the number of spillover areas. The development trend and infection scale of the epidemic were estimated, and the effects of routine prevention and control and emergency response measures on the epidemic development trend were discussed. Secondly, we focus on the risk of spillover infection in other areas when the initial outbreak area is given, and gives the identification method of spillover high-risk areas and the ranking of high-risk areas. By comparing with the epidemic spillover in Wuhan, Beijing and Liaoning in 2020, the reliability of the method is verified, and the high-risk input early warning of each province is released. Our findings can provide early warning for the prevention and control of sudden epidemic situation and spillover high risk areas in the future.

In this paper, a class of inverse Semidefinite quadratic programming(SDQP) problem is solved. It can be described as that the parameters in both the objective function and the constraint set of a given SDQP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate the inverse problem into a problem with linear constraint and semidefinite cone complementary constraint. By using the duality theory, the above problem is transformed into a problem with only semidefinite cone complementary constraints which is a rather difficult problem. By introducing a non smooth penalty function to penalize the complementary constraint, the inverse problem is transformed into a DC programming problem. We use sequential convex programming method to solve it and give the convergence analysis of sequential convex programming method and penalty method. Finally, numerical experiments show that our method is very effective for solving the inverse problem proposed in this paper.

The (list) dynamic coloring, an important research direction in graph theory and theoretical computer science, can be used to solve some critical problems on the channel assignment problem. Kim and Park (2011) announced that the list dynamic chromatic number of every graph with maximum average degree less than 8/3 is at most 4. However, this result is incorrect since the cycle C₅ on five vertices has maximum average degree 2 and list dynamic chromatic number 5. In this paper, we correct this flaw by proving that the list dynamic chromatic number of every graph with maximum average degree less than 8/3 is at most 4 (being sharp) if it is a normal graph, which is a graph having no component isomorphic to C₅. Meanwhile, we prove that every series-parallel graph has list dynamic chromatic number at most 4 if it is a normal graph, and exactly 5 otherwise, which improves a result of Song et al. (2014) that states that the list dynamic chromatic number of every series-parallel graph is at most 6.

In this paper, we present an inexact gradient mirror descent algorithm for nonsmooth convex optimization problem. It is a generalization of gradient mirror descent algorithm for smooth convex optimization problem which is proposed by Allen-Zhu in 2016. At each iteration, it allows the errors of gradient calculations of smooth terms and proximity operator calculations of non-smooth terms in the objective function. Moreover, the convergence rate of the proposed algorithm is analyzed under mild conditions. Finally, the algorithm is used to solve the lasso and logistic problems, numerical results illustrate the efficiency of the proposed algorithm.