In this paper, we introduce the concept of $k$-path cover about the directed graph of matrix. With the help of the $k$-path cover we obtain a new distribution theorem on eigenvalues of matrix, and improve the classical Brauer's theorem. Moreover, we obtain lower bounds for the rank of some matrices, and improve Shemesh's theorem.

Consider the neutral differential equation with positive and negative coefficients $$ [x(t)+C(t)x(t-\gamma))'+P(t)x(t-\tau)-Q(t)x(t-\sigma)=0 $$ where $C,P,Q \in C([t_1,\infty],R^+),\tau,\sigma\in[0,\infty), \ \gamma>0.$ This paper gives some estimates of the distance between adjacent zeros of the solutions, and the sufficient conditions for oscillations of the all solutions of above equation; and improves the known results.

A fractional-step implicit finite difference scheme is introduced for two-dimensional equations of heat conduction with three temperatures. The optimal rate of convergence and the stability in discrete $H^1$-norm for this scheme are derived.

In this paper, we prove the existence of saddle-node invariant loop in the model of mitosis in frog eggs and its bifurcation and present the spatial region and the parameter region where there exists saddle-node invariant loop in the model of mitosis in frog eggs proposed by Borisuk M T and Tenson T Tin [1]. The model is a cubic system on the plane. The results in this paper embrace the numerical results in [1].

In this paper, we discuss the parameter estimation of the constant stress accelerated life tests under the Weibull distribution. Some new estimates are given, and approximate confidence intervals are also derived with nonconstant shape parameter. The simulation results show that the new estimates and approximate confidence intervals are effective.

This paper investigates the algorithms for evaluating two new types of generalized Ball curves/surfaces and their applications. First, the conversion algorithms from evaluating B\'{e}zier curves/surfaces to evaluating these two types of curves/surfaces are presented, and thus the time complexity is greatly reduced. Secondly, a unified representation of the B\'{e}zier curves and these two types of generalized Ball curves is given. On the basis of this representation some recursive algorithms for conversions among one another of these curves are presented.

Parameter identification problem of a three-dimensional species ecological dynamical system with reaction-diffusion phenomenon is investigated in this paper. The mathematical model of this problem is constructed and the continuous dependence of the solution to the direct problem on the parameters identified is obtained on the basis of investigation of the direct problem. Meanwhile, a result on the existence of optimal solution of the parameter identification problem is given.

In this paper, we investigate the problems on fixed points and hyper-order of solutions of four types of second order differential equations with meromorphic coefficients. Because of the restriction of differential equations, we obtain the precise properties of the fixed points of solutions.

Let $\{X_{i}\}$ be a stationary normal sequence, $EX_{1}=0, \ EX_{1}^{2}=1, \ \rho_{n}=EX_{1}X_{n+1}$. For the levels $u_{n}=\frac{x}{a_{n}}+b_{n}$ and $v_{n}=-\frac{y}{a_{n}}-b_{n}$, write $$ N_{n}(B)=\sum\limits_{i/n\in B}\,I_{\{X_{i}>u_{n}\} \cup \{X_{i} \leq v_{n}\}}, \qquad S_{n}=\sum\limits_{i=1}^{n} X_{i}. $$ The asymptotic joint distribution of $N_{n}(B)$ and $S_{n}$ is obtained; the asymptotic joint distribution of extreme value and $S_{n}$ is also given under the condition $\lim\limits_{n\rightarrow\infty}\rho_{n}\log n=\gamma \ (0<\gamma<\infty)$.

By using the abstract continuity theorem, we study the problem of periodic solution of the nonlinear neutral differential equation $x'(t)=f(t,x(t),x(t-\tau),x'(t-\tau))+p(t)$ and obtain sufficient conditions for the existence of periodic solutions related with delay $\tau$.

n this article, we establish the lower bound estimate of moderate deviation for NA random variables with stationary distribution,and then we establish the moderate deviation principle for NA random variables with stationary distribution.

In this paper, the long-time behavior of non-autonomous Navier-Stokes equations with non-homogeneous boundary condition in Lipschitz domains is studied. The existence of uniform attractor is proven when external force is quasi-periodic in time. Moreover, the upper bounds of the uniform attractor' dimensions are given.

By using the Krasnosel'skii fixed point theorem of cone expansion-compression type, the existence and multiplicity of solutions and positive solutions are considered for a class of fourth-order two-point boundary value problems with bounded-below nonlinearity. This class of problems usually describes the deformations of elastic beams with both fixed points.

We study the existence of travelling wave solutions in a biological model for chemotaxis in a bacteria-substrate mixture which was first proposed by Keller and Segel, where the coefficients are of the form of power functions. For the case $D=0$, we give a complete proof of the necessary and sufficient conditions for the existence of travelling wave solutions by the phase plane method. The existence of travelling wave solutions is also considered for the case $D>0$, and we extend the results of Nagai and Ikeda to a more general case.

In this paper, a class of predator-prey diffusive system of two species with time delay and ratio-dependence is studied.By using the coincidence degree thoery, we establish the existence result of positive periodic solution for the system.

This paper discusses a type of damped nonlinear Schr\"odinger equation with harmonic potential appearing in attractive Bose-Einstein condensates. With reference to the physical properties of Bose-Einstein condensates, we prove the collapse of its initial value problem in finite time.

In this paper, a new method with explicit direction is presented by combining the concept of conjugate projection gradient with the idea of SQP method, and its global convergence and superlinear convergence are obtained under certain conditions. The numerical results show that the method in this paper is effective.

New sufficient conditions for the non-existence of positive solution to a nonlinear difference equation with unbounded delay and to its dual equation are obtained, and some of the results in the literature are improved.

In this paper, we consider the global attractivity of the zero solution of the super logistic type functional differential equation $$ x' (t)+\big[1+x (t)\big] F \big(t, \big[x (\cdot)\big]^\al\big)=0, \qquad t\ge 0, \ \al\ge1.$$ The theorem obtained in this paper improves all the related results in the literature.

A complete analysis of topological structure of ecological system $$ \dot {x} = x(1 - k_1 x - k_2 x^2) - \frac{xy}{1 + ax}, \qquad \dot {y} = y( - \delta _0 - \delta _1 y) + \frac{\gamma xy}{1 + ax} $$ is given in this paper. The authors discussed some topological structure types with limit cycles for this system again. Then this paper improved and enriched the work of [1].

In this paper, the unified existence theorem of essential components is first given. From this theorem, it is easy to derive existence theorems of essential components of the set of fixed points and Nash equilibrium points. Moreover, the unified stability theorem of essential components is first given.

In this paper, a system of differential equations related to an ecological model of microbes in anaerobic digestion process is studied. In its nonautonomous perturbation, by using the continuation theorem of coincidence degree theory, some sufficient conditions for the global existence of periodic solutions of this system are obtained.

For the multivariate Stancu polynomials $M_n(f,x)$, which are generalization of the Bernstein polynomials defined on the simplex, the optimal approximation order and its characterization of these polynomials approximating the continuous functions will be given. That is, we will find the class of functions for which $\|M_nf-f\|=O(n^{-1})$ in term of the behavior of a certain $K$-functional. Moreover, we also give the direct and converse theorems of approximation.

The extended systems of symmetry breaking bifurcation point is constructed, Splitting Iteration Method is introduced for symmetry breaking bifurcation point, numerical approximation of 2-Box Brusselator system is given.

Sufficient criteria are established for the existence of positive periodic solutions of a class of mathematical models arising in bioeconomics. The approach is based on the coincidence degree and related continuation theorem.

There are no relaxation parameters in the algorithm of the alternating subdomain Schwarz method without intersection given by Matsokin and Nepomnyaschikh$^{[1,2]}$. We know that the method converges geometrically, and the convergence is independent of $h$. Because of containing no relaxation parameters, the method cannot accelerate the convergence under any particular cases. In this paper, we give an algorithm which can accelerate the convergence. We add two relaxation parameters to the known algorithm. It has been proved that we can accelerate the convergence besides the exceptions, and $\overline {\theta}_1$, $\overline {\theta}_2$ are the optimum parameters under the balance condition. The last example shows the efficiency of our new algorithm.

Multivariate wavelets are powerful tool for multi-dimension signal processing. But tensor product wavelets has a number of drawbacks. In this paper, we give an algorithm of parametric representation compactly supported bivariate orthogonal wavelet filter. This representation simplifies the study of bivariate orthogonal wavelets. Examples are also given to demonstrate the method.

For the inference of the sensitivity of explosive products, one of the fundamental works is to give the accurate inference for the standard deviation of the response distribution. Aimed at this purpose a method to conduct approximate confidence intervals about the scale parameter of Logistic response distribution is presented by used binary response data and the saddlepoint approximation. A simulation study about these confidence intervals is conducted. The proposed method is also applied to two real data sets of QD-8 electric detonator. Numerical results show that the proposed method provides more precise inference for the scale parameter with moderate or small sample sizes. It observably improves the asymptotic normality method that is being used.

Let $1\leq p<+\infty$, and $f(x)$ be a function whose $k$th derivative is $p$ power intergrable over $[-1,1]$, with usual $L_{p}$ norm. Denote by $\Pi_n$ the class of polynomials of degree $n$. The present paper is devoted to the research on a class of functions $f$, which, for a fixed inner point $a$ of the given interval, possesses the following property: $$\|f-p_n(f)\|_{L_p[a-\frac{r}{n},a+\frac{r}{n}]}\geq C E_n(f)_p,$$ where $C$, $r$ are constants independent of $n$, $E_n(f)_p=\inf\limits_{p_n\in \Pi_n} \|f-p_n\|_p=\|f-p_{n}(f)\|_p.$ It is a quite surprising phenomenon that the $L_{p}$ mean approximation, especially mean best approximation, of some functions, such as power functions mentioned in \S 3, the products of power functions and ``slowly increasing" functions, can ``concertrate" in a small interval with an inner point as the center and $r/n$ as the radius. This phenomenon is thus referred as the ``concertration phenomenon".

In present paper, The existence of explosive solutions to a class of semilinear elliptic equations is obtained in bounded domain by the change of varible and the pertrubed method combining sub-supersolutions method.

In this paper, the generalized derivative nonlinear Schr\"{o}dinger equation is considered. By using the qualitative theory and bifurcation theory of planar dynamical systems, all of the explicit exact solutions of solitary waves in the explicit form of $a(\xi)e^{i(\psi(\xi)- \omega t)}$, \ $\xi=x-vt$, are obtained, by qualitatively seeking the homoclinic and heteroclinic orbits for a class of Li\'{e}nard equations. Our results in this paper have improved all of relative results.

This paper is concerned with the global dichotomy behavior of the non-periodic second-order differential equation: ${x}'' = f(t,x,{x}')$. Our results improve and generalize the corresponding ones on periodic equations.

In this paper, we studay the Poincare bifurcation of quadratic system with two centers and two unbounded heteroclinic loops once again. There bounds are formed by the hyperbola and the equatorial arc. We construct some concrete examples, in which there are three limit cycles with (0,3) distribution or a triple limit cycle.

In this paper, we propose a symmetric loss function for scale parameter with form as $L(\s, d)=\frac{d}{\s}+\frac{\s}{d}-2.$ This loss function can be viewed as a generalization of entropy loss and squared loss for scale $\s$. For the normal distribution $N(0, \s^{2})$, we give the minimum risk equivariant estimator (MRE) and the Baysian estimation of $\s$. Furthermore, the admissibility and inadmissibility of the estimators with form $(cT+d)^{\frac{1}{2}}$ are studied.

This paper discuss the relationship between the existence of fractional factor and the graphic isolated toughness $I(G)$ or the parameter $I'(G)$. We give a serious of results on the relationship between $I(G)$ or $I'(G)$ and the vertex (edge) delectable, fractional $k$-factor extensible and the existence of fractional $[1,b]$-factor.

In this paper we show the existence and continuity of the self-intersection local time of a class of $S'(R^1)$-OU processes, and the dependence of the local time on the branching density and the migration interaction parameter.

The existence of nontrivial homoclinic orbits of periodic Hamiltonian systems: $$ \ddot {q} + V_q '(t,q) = 0 \tag HS $$ is proved, where $q = (q_1, q_2, \cdots, q_n ), \ n > 2$, \ $V(t,q): R^1\times R^n\backslash \{ e \} \to R^1$ is a potential with a singularity, i.e.\, $-V(t,q) \to \infty$, as $q \to e$. Our main assumptions are lack of Gordon-Strong Force condion and the uniqueness of a global maximu of $V(t,q)$.

A system of retarded functional differential equations is proposed as a predator-prey model in two-patches. The invariance of non-negativity, nature of boundary equilibria, permanence and global stability are analyzed. We show that positive equilibrium is locally asymptotically stable when time dalay $\tau$ is suitable samll, while a loss of stability by a Hopf bifurcation can occur as the delay increases. That is, a family of periodic solutions bifurcates from positive equilibrium as $\tau$ passes through the critical value $\tau_0$.