This paper develops an extended newsboy model and presents a formulation for this model. This new model has solved the budget contained multi-product newsboy problem with the reactive production. This model can be used to describe the status of entrepreneurial network construction. We use the Lagrange multiplier procedure to deal with our problem, but it is too complicated to get the exact solution. So we introduce the homotopy method to deal with it. We give the flow chart to describe how to get the solution via the homotopy method. We also illustrate our model in both the classical procedure and the homotopy method. Comparing the two methods, we can see that the homotopy method is more exact and efficient.

In the presence of external forces depending only on the time and space variables, the Boltzmann-Enskog equation formally conserves only the mass of the system, and its entropy functional is also nonincreasing. Corresponding to this type of equation, we first give some hypotheses of its bicharacteristic equations and then get some results about the stability of its global solution with the help of two new Lyapunov functionals: one is to describe interactions between particles with different velocities and the other is to measure the $L^1$ distance between two mild solutions. The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the $L^1$ stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.

It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in $C[a, b]$ or $L^2 [a, b]$. In this paper, the representation of the solution for Fredholm integral equation of the first kind is given if it has a unique solution. The stability of the solution is proved in the reproducing kernel space, namely, the measurement errors of the experimental data cannot result in unbounded errors of the true solution. The computation of approximate solution is also stable with respect to $‖·‖_C$ or $‖· ‖_{L^2}$. A numerical experiment shows that the method given in this paper is stable in the reproducing kernel space.

This paper deals with a Lotka-Volterra ecological competition system with cubic functional responses and diffusion. We consider the stability of semi-trivial solutions by using spectrum analysis. Taking the growth rate as a bifurcation parameter and using the bifurcation theory, we discuss the existence and stability of the bifurcating solutions which emanate from the semi-trivial solutions.

A continuous map from a closed interval into itself is called a $p$-order Feigenbaum's map if it is a solution of the Feigenbaum's equation $f^p (λx) = λf(x)$. In this paper, we estimate Hausdorff dimensions of likely limit sets of some $p$-order Feigenbaum's maps. As an application, it is proved that for any $0 < t < 1$, there always exists a $p$-order Feigenbaum's map which has a likely limit set with Hausdorff dimension $t$. This generalizes some known results in the special case of $p = 2$.

Given an irreducible plane algebraic curve of degree $d ≥ 3$, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.

Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional (with natural boundary conditions) is minimal. By the spline function methods in Hilbert space and variational theory of splines, the characters of the interpolation solution and how to construct it are studied. One can easily find that the interpolation solution is a trivariate polynomial natural spline. Its expression is simple and the coefficients can be decided by a linear system. Some numerical examples are presented to demonstrate our methods.

Under the assumption of sixth power large sieve mean-value of Dirichlet $L$-function, we improve Bombieri's theorem in short intervals by virtue of the large sieve method and Heath-Brown's identity.

The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem (NEP), in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players. This problem has been used to model various problems in applications. However, the convergent solution algorithms are extremely scare in the literature. In this paper, we present an incremental penalty method for the GNEP, and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs. We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.