In view of the practical problems of malaria vaccine in malaria transmission, a dynamical model of malaria with vaccination and vaccine failure is constructed, and the control reproduction number $\mathcal{R}_{c}$ is calculated. The existence conditions of malaria-free and malaria equilibria in terms of $\mathcal{R}_{c}$ are given. By using the Lyapunov function method and the generalized Lyapunov-Lasalle theorem, the sufficient and necessary conditions for the global asymptotic stability of malaria-free and malaria equilibria with respect to $\mathcal{R}_{c}$ are established.

Experiments are indispensable in many fields, for example, well-designed scientific experiments are needed in the research, development and testing of new products. When it is very difficult to change the level of factors in the experiment, how to arrange the run order reasonably is a very important problem. In this paper, some basic theories of run order with the minimum and maximum level changes are studied, and the optimal run order construction methods are discussed for full factorial designs, nonregular fractional factorial designs and uniform designs. As for some designs are widely used in practice, the run order with the minimum and maximum level changes and the corresponding of level changes are given by using the results of this paper.

This paper focus on the multiplicity problem of positive solutions of integral boundary value problems for an impulsive fractional differential equation with positive homomorphism operator. By using the classical Guo-Krasnosel’skii fixed point theorem, some sufficient conditions for the existence of at least two positive solutions of impulsive fractional differential equation are derived. Finally, one example is shown to illustrate the theoretical results.

An adjacent vertex distinguishing edge coloring of graph $ G $ is a proper edge coloring of $ G $ such that any pair of adjacent vertices have distinct sets of colors. The mininum number of colors required for an adjacent vertex distinguishing edge coloring of $ G $ is denoted by $ \chi_{a}{'}(G) $. This paper proves that $ G $ is a normal planar graph with girth at least $ 6 $, $ \chi_{a}{'}(G) \leq \max\{6, \Delta(G)+1\} $.

In this paper, the Lévy process is used to model the surplus level of an insurance company. It is assumed that the insurance company observes the surplus level at a fixed time and makes a decision after each observation. If the observed surplus level is less than a given critical level and is non-negative, the shortfall is injected once to bring the surplus level back to the critical level. If the observed surplus level is negative, ruin is declared immediately. Using the Fourier cosine series expansion method, we propose some numerical methods for computing the finite-time expected total discounted cost of capital injections before ruin and the finite-time expected discounted penalty function. The accuracy and efficiency of the method are demonstrated through error analysis and numerical examples, and the effect of each parameter on the results is studied.

As an important part of the financial market system, it is very important for insurance companies to choose the optimal investment and reinsurance strategies. This paper considers an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. Assume that insurers diversify their risk by purchasing proportional reinsurance, which the surplus process is a diffusion process similar to the classical Cramér-Lundberg model. In addition, insurers increase their income by investing in risk-free assets and risky assets, which the prices of risky assets follow the Volterra Heston model. Due to the non-Markovian and non-semimingale properties of the Volterra Heston model, the classical stochastic optimal control framework is no longer applicable. By constructing an auxiliary stochastic process, we obtain the optimal investment and reinsurance strategies and the efficient frontier, which depend on the solution to a Riccati-Volterra equation. Finally, we numerically analysis the relationship among optimal strategies, effective frontier, volatility roughness and reinsurance factors. It is found that the rougher the volatility of stocks, the greater the demand of insurance companies for stocks and reinsurance.

This study is concerned with numerical solutions of the two-dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation (Fisher-KPP) by a class of weighted structure-preserving finite difference methods (W-SP-FDMs) combined with Richardson extrapolation methods (REMs). By using the discrete energy analysis method, it is shown that as the parameters $\alpha$, $p$ and $\theta$, and the ratios of temporal meshsize to spatial meshsizes satisfy certain conditions, the current W-SP-FDMs possess many properties, such as, preserving positivity, preserving boundedness, preserving monotonicity, and has a convergence order of $O(\tau+h_x^2+h_y^2)$ in $L^{\infty}$- norm. Also, by using the discrete energy analysis method, it is shown that the REMs, which are developed by the asymptotic expansion formula of the numerical solutions, can make the final solutions convergent with an order of $O(\tau^{2}+h_{x}^{4}+h_{y}^{4})$ in $L^{\infty}$- norm, thus improving computational efficiency. Finally, numerical results confirm the correctness of theoretical findings and high performance of the current methods. It is worthwhile to mention that additional condition for the ratios of temporal meshsize to spatial meshsizes is not supplemented as REMs are used.

Under investigation in this work is the integrable coupled Sasa-Satsuma equation, which can be used to describe the propagation dynamics of two ultrashort pulses in the birefringent or two-mode fiber. Through the Darboux-dressing transformation, we obtain a family of semirational solutions. This family of solutions exhibits various scenes of superimposition between rogue waves and breathers. These results may contribute to enriching and explaining some related nonlinear phenomena in optical fibers and dispersive media.

In this paper, we establish the optimality conditions for the nonsmooth multiobjective optimization problem by using Fréchet subdifferential and the extremal principle. First, the necessary conditions of semi-infinite multiobjective optimization problems are studied. And then, we establish the necessary condition of Henig proper efficient solution for the nonsmooth multiobjective optimization problem.

Viral hepatitis C (simply referred to as hepatitis C) is a form of viral hepatitis caused by infection with the hepatitis C Virus (HCV). HCV will cause chronic inflammation, necrosis, and fibrosis of the liver, some patients may develop cirrhosis and hepatocellular carcinoma (HCC). In this paper we take advantage of the hepatitis data set to construct penalized trinomial logit models to diagnose the disease stages of patients. Firstly, we select 12 physiological indicators of patients as a predictor vector, and choose 3 disease stages of hepatitis C as the response variable. Secondly, we apply the 70% data as the training set to learn LASSO/Ridge/ENet penalized trinomial logit model, and take advantage of the coordinate descent algorithm to complete variable selection and obtain parameter estimations. Thirdly, we apply the remaining 30% data as the testing set, and combine three-class confusion matrix, the ROC (receiver operating characteristic) surface, HUM (hypervolume under the ROC manifold), PDI(polytomous discrimination index) to assess the prediction accuracy to disease stages. Finally, we introduce some machine learning methods such as artificial neural network (ANN), support vector machine (SVM) and random forest (RF)to compare with the proposed penalized trinomial logit models, and found that penalized trinomial logit models possess the best three-class prediction performance. They can not only improve the diagnostic accuracy to disease stages, but also reduce the cost of hepatitis C detection.

As an important problem in the field of information science such as machine learning and image processing, low rank matrix completion has been widely studied. The first-order primal-dual algorithm is one of the classical algorithms for solving this problem. However, the data processed in practical applications is often large-scale. Therefore, based on the framework of the primal-dual algorithm, this paper proposes a modified primal-dual algorithm for large-scale matrix completion problems by exploring correction strategy with the variable step size. In each iteration of the new algorithm, the primal and dual variables are firstly updated by the primal-dual algorithm, and then the correction strategy with the variable step size is used to further correct the two variables. Under certain assumptions, the global convergence of the new algorithm is proved. Finally, the new algorithm is verified to be efficient by solving some random low rank matrix completion problems and examples of image restoration.

The uniform designs are accepted widely because of its robust and easy to use, flexible characteristics. In order to distribute the points evenly in the experimental domain, many criteria ($L_2$-discrepancy) have been forwarded to measure the uniformity of the design array. At present, centered $L_2$-discrepancy, wrap-around $L_2$-discrepancy, mixed discrepancy and so on are widely used. Symmetric $L_2$-discrepancy has better geometric sense, but the poor performance at projection uniformity limits the use of SD. To refine the projection properties of SD, a projection weighted SD is proposed. The SD was exponentially weighted. The projection weighted SD can retain the excellent properties of the original discrepancy, and overcome the original defects effectively, and has better performance. The foldover is a useful technique in construction of factorial designs. In this paper, the projection weighted symmetric $L_2$-discrepancy is used as the optimality criterion to evaluate the quality of the foldover scheme. Lower bounds for projection weighted symmetric $L_2$-discrepancy on combined two-level U-type designs under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans.

In this paper, the partial functional linear multiplicative model is considered. This model, which becomes partial functional linear regression after taking logarithmic transformation, is useful in analyzing data with positive responses. Based on B-splines, two estimation methods are proposed by minimizing the least absolute relative error (LARE) and the least product relative error (LPRE), respectively. The dimension of the B-spline bases is selected using the Schwarz information criteria. Consistency and asymptotic normality of the two methods are investigated. For the slope function, we prove that its convergence rate achieves the optimal rate of nonparametric function. Monte Carlo simulations are conducted to evaluate and compare the finite sample performance of the proposed estimators with the least squares (LS) estimator and the least absolute deviation (LAD) estimator under the different random error settings. Simulation results show that the proposed methods are comparable to other methods. Finally, an example of real data analysis is given to illustrate the application of the model.

Blow up phenomena of solutions to a dissipative generalized Tricomi equation with variable coefficients and nonlinearity of derivative type in the subcritical case are considered. By constructing some time-dependent functionals associated with test function methods and Bessel equations, an iterative frame and the first lower bound of the time-dependent functional are obtained. Then, blow-up of solutions and upper bound estimate for the lifespan to the Cauchy problem are proved via iteration arguments.

This paper studies the problem of testing the mean of high-frequency functional data. For functional data with infinite number of principal components and spiked eigenvalues of covariance operators, the classical Chi-square or mixed Chi-square test constructed based on the dimension reduction method using functional principal components will become invalid due to insufficient sample size and strong conditions of covariance operators. Therefore, this paper proposes a randomized test to solve this problem, and proves the large sample properties. Further, the numerical simulation of limited samples is used to verify the effectiveness of the proposed test. Finally, this method is applied to the phoneme data.

In this work, aiming at the problem of lacking analytical methods and it is typically difficult to find explicit exact solutions to the population balance equations with fragmentation processes. We consider the admitted Lie groups, group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions of two classes of integro-partial differential equations (population balance equations with fragmentation processes) by use of the method of scaling transformation group analysis, the method of classical Lie group analysis and the method of developed Lie group analysis. Firstly, the admitted scaling transformation Lie groups of the integro-partial differential equations are explored by using the method of scaling transformation group analysis. Secondly, the integro-partial differential equations are transformed into pure partial differential equations, the admitted Lie groups of the pure partial differential equations are calculated by use of the methods of classical Lie group analysis. Thirdly, the admitted Lie groups of the original integro-partial differential equations are determined by use of the methods of developed Lie group analysis combining with the related results obtained by method of scaling transformation group and the method of classical Lie group analysis. Finally, the admitted Lie groups of the original integro-partial differential equations are successfully found. All group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions are given. The related analysis for dynamic behavior characteristics of a solution with evolution of the size distribution are also presented.

In this paper, we study the 1-dimensional linear Navier-Stokes-Fourier equations and obtain the pointwise estimates of the decay properties of the solution under the appropriate initial value conditions, and describe the decay direction of the solution, and verify that the generalized Huygens’ principle holds. To this end, we divide the Fourier transform of the Green function of the equations into low-frequency, mid-frequency and high-frequency parts by means of Fourier transform, and prove the decay properties of the Green function in the corresponding frequency parts, and then obtain the decay estimates of the solutions by means of the Fourier inverse transform and the properties of the fundamental solutions.

The research on customer strategy behavior in queueing system combined with game theory is a hot topic in the current queueing theory. This paper studies the strategic behavior of risk-sensitive customers in discrete-time queueing systems. Different from the classical economics of queues, the utility function in this paper is an expectation-variance quadratic utility function. Based on the Nash equilibrium and Markov process theory, we study the game behavior of Geo/Geo/1 queueing system with risk-sentitive customers under fully observable case and fully unobservable case, respectively. The individual optimal joining strategy, the joining strategy for the social net welfare and the server’s profit optimization are obtained. It is found that the smaller the risk sensitivity coefficient is, the more customers like to take risks and the stronger the willingness to join the system. Some numerical experiments are provided to illustrate the effect of the risk sensitivity coefficient on the customer strategic behavior.

In this paper, we introduce the notion of higher-order generalized weak Studniarski epiderivatives for a set-valued map without lower-order approximating directions and obtain chain and sum operation rules of the epiderivative. Then by applying the higher-order epiderivative, we establish the optimality conditions of the weakly efficient solution for unconstrained composite set-valued optimization problems. Some illustrative examples are provided as well.

Considering the uncertainty of financial market data volatility, a new logmean reversion jump diffusion 4/2 random volatility (LMRJ-4/2-SV) model was proposed in this paper. Firstly, the LMRJ-4/2-SV model was constructed, and the European option pricing formula based on LMRJ-4/2-SV model was obtained by using FFT and other methods. Secondly, descriptive statistical analysis of the actual market data was carried out to discuss the price change characteristics of the underlying asset and the applicability of the LMRJ-4/2-SV model, and the model parameters were estimated by particle swarm optimization algorithm. Finally, European options were priced based on the option pricing formula and parameter estimates under the LMRJ-4/2-SV model, and the pricing results were compared with the 4/2, 3/2, Heston model estimates and market prices. The results show that the pricing error of European option based on LMRJ-4/2-SV model is minimal, and the pricing results have obvious advantages over other stochastic volatility models.

In this paper, it is proved that if a dynamical system has the periodic $\mathscr{M}_{\alpha}$-shadowing property or periodic $\mathscr{M}^{\alpha}$-shadowing property, then the dynamical system restricted on its measure center has the same shadowing property. Conversely, if a dynamical system restricted on its measure center has the periodic $\mathscr{M}_{\alpha}$-shadowing property (resp., periodic $\mathscr{M}^{\alpha}$-shadowing property), then the dynamical system has the periodic $\mathscr{M}_{\beta}$-shadowing property (resp., periodic $\mathscr{M}^{\beta}$-shadowing property) for any $\beta\in [0, \alpha)$. Moreover, it is obtained that for an equicontinuous system, many shadowing properties are equivalent to the condition that it has a trivial measure center.