The Darwin model is a very good approximation model for the Maxwell’s equa- tions when no high frequency or no rapid current change occurs in 3-D bounded simply connected domains. In this paper, we consider the adaptive finite element method for the Darwin model. The method based on a posteriori error analysis. In this paper, we provide the upper bound estimation based on a posteriori error estimates.

In this paper, by using the fixed point theorems in cones the existence of positive solutions for the second-order semipositone eigenvalue problems Lu+λf(t, u)=0, αu(0)-βu'(0)=0, γu(1)+δu'(1)=0 is obtained under the condition that f is super-linear(sub-linear). The results obtained in this paper essentially improve, generalize many well-known results.

The existence of positive solution for a class of p-Laplace equations with super- critical exponents is proved. Some properties of the solution are also given.

The method to obtain the integrating factors of the 3-rd order autonomous system based on two one-parameter Lie groups is given and shown.

The efficient solution of symmetric vector quasi-equilibrium problem for set- valued mappings is introduced. By using the scalarization method and fixed point theo- rem, existence theorems are obtained. As applications, existence theorems for new types of generalized vector saddle points, vector variational inequality, and vector complementarity problem are established.

In this paper, adopted a k-part partition of matrices index set, obtain some new criteria for nonsingular H-matrices, which include and generalize some relative results. Their effectiveness is illustrated by numerical examples.

This work concerns the development of stabilized finite element methods for the nonstationary stokes problem considering equal order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions, a Petrov-Galerkin approach is employed. Basing on multiscale functions that relate or don’t relate with the time, backward Euler method is employed for the time part. And we studied the stability and convergence of the two forms.

In this paper I discuss the existence of moment for the nonlinear autoregres- sive model: x_t=φ(x_t-1,···,x_t-p)+ε_t , when the following geometric ergodicity condi- tions are met: (i) |φk(y₀,y_-1,···,y_-p+1)-φk(y₀',y_-1,···,y_-p+1)|≤c_k|y₀-y₀'|;(ii) |φk(y₀,y_-1},···,y_-p+1)|≤M_p^k(|y₀|+···+|y_-p+1|)+c. I find that if for a real number r ≥ 1, E|ε_t|^r < ∞, then E|x_t|^r< ∞. This result complements the existing vacancy in the literature, that for (i)(ii) no corresponding existence of moment result was given.

In this paper, I first characterize the nonemptiness and boundedness of the weakly efficient solution set of convex vector optimization problem, especially a cone-constrained convex vector optimization problem in infinite-dimensional real reflexive Banach spaces. More specifically, I will consider the covex vector optimization problem when the objective space R_m is ordered by a nontrivial, polyhedral cone with nonempty interior instead of the nonnegative orthant R₊m. Then, I apply the characterizations to the convergence analysis of a class of penalty methods.

In this paper, we talk about the strong convergence law of partial sums and product sums for pairwise NQD random variables with different distributions under a set of extensive condition. Under weaker condition, we extend and improve some latest conclu- sions. The results in our paper have more superiority, they are the genelization of relevant classic conclusions in independent random variable case.

Using the energy estimate method, Sobolev embedding theorems and the bootstrap arguments, the global existence of classical solutions for a predator-prey-mutualist model with cross-diffusion is proved when the space dimension is less than 10.

Feng Dejun et al. introduced homogenous Cantor sets and partial homogeneous Cantor sets in their related papers. In this paper, we construct a class of homogeneousMoran sets between them and obtain an exact formula of Hausdorff dimension. In the meantime, we discuss the discontinuity of dimensions on parameters.

Original unknown functions are expressed by introducing auxiliary unknown functions and integral relations of auxiliary unknown functions. The dual integral equations are decoupled and reduced to Abel integral equations by using Sonine first finite integral formula and the it is further reduced to regularized Fredholm singular integral equations with logarithmic kernels of first kind by Abel anti-transformation. Thus general solutions of singular integral equations are given. And then analytic solutions of dual integral equa- tions are obtained. Simultaneously the equivalence between dual integral equations and corresponding Fredholm singular integral equations with logarithmic kernel of first kind, the existence and uniqueness of solutions are proved exactly.

The Cauchy problem of Helmholtz equation is severely ill-posed problem.In this paper,we consider the Cauchy problem for the Helmholtz equation where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x < 1. A semi-discrete difference schemes together with a choice of regularization parameter is presented and error estimate is obtained.

By means of probability measure space, the concept of probability which sat- isfied Kolmogorov axioms is introduced in n-valued Lukasiewicz logical system. The fun- damental theorem of probability logic is proved and generalized in n-valued Lukasiewicz logical system, so the estimate of uncertainty of conclusions in inference is improved. The basic methods of probability logic are introduced into quantitative logic and a more general logic metric space has been obtained; The MP ??HS rules of probability are proved by the fundamental theorem of probability logic, and it is the generalization of the MP ??HS rules of truth.

For the numerical difficulties on discontinuous diffusion coefficient of heat con- duction equations, the so-called “twin-fitting” adaptive method has been proposed. In this paper, the intrinsic properties of heat conduction equations are extended to high order, and a kind of “twin-fitting” algorithm which is more accurate and more economic is constructed, at last numerical experiments validate the algorithm’s advantages.

It is difficult to analyze nonlinear dynamic system because of its property. Generalized cell mapping is a powerful tool for analyzing nonlinear dynamic system by the way of discretizing continuous space and mapping. However, inferior accuracy, high complexity and low efficiency of the existing analysis algorithm restrict its application in large-scale and high quality requirement. In the present paper, the concept of cell degree is presented and cells are classified into different categories because of their various cell degree properties. Then from the respect of category of cells, an efficient and precise algorithm of analysis for generalized cell mapping is proposed. And at last, a series of experiments are given to show the validity and advantage of the algorithm.

A class of generalized vector variational inequalities over product of sets and system of generalized vector variational inequalities are considered. The relationships of solution sets among those variational inequality problems are presented. With the hemi- continuity and several kinds of generalized monotonicities with respect to a vector, several existence results for the solutions of problems mentioned above are established by fixed point theorems of setvalued mappings.

The value composition of the Warrant Bonds is discussed in a quantitative analysis. Under stochastic interest we get the pricing formula of Warrant Bonds by means of Martingle approach (risk-neutral valuation).

This paper discuss a type of predator-prey system with Holling III type func- tional response. The existence and stability of co-exist periodic solutions are investigated by using the bifurcation theory,the implicit function theorem and the method of asymptotic expansion.

In this article, inferences for the ratio of scale parameters of two Weibull distri- bution are studied. We propose the generalized confidence intervals and hypothesis testings for the ratio of scale parameters by generalized pivotal quantities and generalized test vari- ables. When shape parameters are equal, the studies show that the covered probability of 100(1-α)% generalized confidence interval for the ratio of scale parameters is 1-α, and the Type-I error rates is nominal significance level α. When shape parameters are unequal, we also study frequency properties of inference, and the numerical studies show that the proposed generalized confidence and generalized p-values inferential proceduers are satisfac- tory. At last, hypothesis testings for the ratio of shape parameters are studied, we prove that the Type-I error rates are nominal significance level α.

An equivalent condition about the nonsingularity of convex semidefinite pro- gramming, which can be viewed as a generalization of the equivalent condtion about the nonsingularity of linear semidefinite programming, will be presented in this paper.

It is well known that the classical parallel projection algorithm for convex feasi- bility problem in Hilbert space is weak convergent. In this paper, a modification of parallel projection algorithm is presented by introducing a parameter sequence for solving the con- vex feasibility problem. To prove the strong convergence in a simple way, we introduce a product space. Then, we transmit the modified parallel algorithm in the original space to a aternating one in the product space. Thus, the strong convergence of the modified parallel projection algorithm is derived from the alternating one under some parametric controlling conditions.

This paper first presents a kind of new definition of fractional difference, frac- tional summation, and fractional difference equations, gives methods for explicitly solving fractional difference equations of order (k, q) by use of Z-transform theory.

This paper investigates the stability of adapted solutions of stochastic Volterra integro-equations. Applying Lyapunov second method and generalized Itô formula, we ob- tain sufficient criterion of almost sure exponential stability and moment exponential stability of the adapted solutions to stochastic Volterra integro-equations.

In this paper, utilizing Perron value of bottleneck matrix for a branch based at a vertex and quadratic form of algebraic connectivity, change rules of algebraic connectivity on undirected weighted trees are studied roundly under changing or shafting branches (edges, vertices) and changing weights of edges.Algebraic connectivity could be considered as a measure of central tendency of edges and weights about a weighted tree. We introduce generalized tree and generalized characteristic vertex, and transform a Type II tree into a Type I tree with equal algebraic connectivity. Therefore, it only needs to investigate Type I trees when discuss algebraic connectivity of trees.

The paper presents a new class of supermemory gradient methods for uncon- strained optimization problems, and analyzes its global convergence and linear convergence rate. The algorithm generates new iterative points by means of a multi-step curve search rule. The search direction and the stepsize are determined simultaneously at each iteration of the new algorithm. It avoids the computation and storage of some matrices and is suitable to solve large scale optimization problems. Numerical results show that the new method is efficient in practical computation.

In this paper, we mainly consider Stokes approximation system of compressible flow in two dimensions. We prove the existence of global weak solutions to Stokes approxi- mation system on condition that γ = 1 and γ > 1 by using the similarly method of Eduard Feireisl.

Original unknown functions are expressed by introducing auxiliary unknown functions and integral relations of auxiliary unknown functions. The dual integral equations are decoupled and reduced to Abel integral equations by using Sonine first finite integral formula and the it is further reduced to regularized Fredholm singular integral equations with logarithmic kernels of first kind by Abel anti-transformation. Thus general solutions of singular integral equations are given. And then analytic solutions of dual integral equa- tions are obtained. Simultaneously the equivalence between dual integral equations and corresponding Fredholm singular integral equations with logarithmic kernel of first kind, the existence and uniqueness of solutions are proved exactly.

The Cauchy problem of Helmholtz equation is severely ill-posed problem.In this paper,we consider the Cauchy problem for the Helmholtz equation where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x < 1. A semi-discrete difference schemes together with a choice of regularization parameter is presented and error estimate is obtained.

By means of probability measure space, the concept of probability which sat- isfied Kolmogorov axioms is introduced in n-valued Lukasiewicz logical system. The fun- damental theorem of probability logic is proved and generalized in n-valued Lukasiewicz logical system, so the estimate of uncertainty of conclusions in inference is improved. The basic methods of probability logic are introduced into quantitative logic and a more general logic metric space has been obtained; The MP ??HS rules of probability are proved by the fundamental theorem of probability logic, and it is the generalization of the MP ??HS rules of truth.

For the numerical difficulties on discontinuous diffusion coefficient of heat con- duction equations, the so-called “twin-fitting” adaptive method has been proposed. In this paper, the intrinsic properties of heat conduction equations are extended to high order, and a kind of “twin-fitting” algorithm which is more accurate and more economic is constructed, at last numerical experiments validate the algorithm’s advantages.