充分条件

Firstly, The sufficient con-ditions of the existence and uniquity of the positive equlibria by applying functional derivative isobtained; Secondly, the global attractability of the positive equlibria is investigated by charateristicroots theory and oscillation theory; Thirdly, regarding the delay as a parameter, conditions of theexistence of Hopf bifurcation and the peridic solution, furthermore, the form of the approximateperidic solution are obtained; Finally, Some specific examples are given and the solution diagrameappears by Matlab.

首先利用导数性质，得到了该模型正平衡态存在惟一性的充分条件；其次，利用特征值和振动性理论得到了该模型正平衡态全局渐近稳定性充分条件；然后，应用Hopf分支理论证明了该模型Hopf分支及近似分支周期解的存在性，并给出了周期解的近似表达式；最后，借助于MATLAB数学软件，举例并绘出了模型数值解的拟合图象，验证了文中定理条件的可行性。

It is proved that the choice function must satisfy the axiom of consistency if the choice function meets the axiom of substitutability together with quotafilling condition; then, it is driven that the choice function is a path independence choice when it satisfies the substitutability and consistency condition.

首先证明了在替代性原理的前提下，选择函数定额充分是满足一致性原理的充分条件；得出了当选择函数满足替代性和定额充分条件时，必定满足路径无关性选择的结论。

In the first chapter, we study a sufficent condition for a operator is compact on Bergman space of the unit ball.

本文在第一章中就复超球上Bergman空间上的算子为紧的充分条件进行了研究，在附加了一定的条件，给出了算子为紧的充分条件。

The method used to check for completeness is based on a set of sufficient conditions proposed by Guttag and Horning We weakened the definition of the measure function and the conditions under which an extension function can be converted to constructors and generators.

完备性的验证方法是，以Guttag-Horning的一组完备性充分条件为基础，放宽其中测度函数的定义，以及扩充函数可以转为生成函数与构造函数的判别条件，拓广了充分条件的适用范围。

And it makes a detailed and brief exposition, and offers some extremely targeted examples of the application, in order to understand the integrability conditions and enhance the understanding and application capability.

文章最后专门讨论了复合函数的黎曼可积性和可积函数列的逐项积分，得出了如何根据特定条件来判断一个复合函数可积性的定理和判定一个函数列可逐项积分的一个充分条件，并将其推广，得到一个更弱的充分条件。

Furthermore, when the system is a periodic system, sufficient conditions are obtained that guarantee the existence, uniqueness and global attractivity of the positive periodic solution by applying the Brouwer fixed theorem and the method of constructing Liapunov function.

利用比较原理给出了系统一致持久性的充分条件；当系统是周期系统时，应用Brouwer不动点原理和构造Liapunov函数的方法给出其存在惟一且全局吸引的ω-周期正解的充分条件，并通过实例验证了文中定理条件的可实现性。

By the theory of difference equation and the construction of proper vector Liapunov function, we for the first time get the sufficient conditions for the permanence, extinction and balancing survival for this ndimensional system, which are all independent on the delays, hence under peroper conditions the time delays of this system are both harmless and profitless.

通过运用差分方程理论以及构造适当向量Liapunov泛函的方法，首次得到了此类系统多维自治的情形下系统持续生存、灭绝和残存的充分条件，这些充分条件均不依赖于时滞，因而在一定条件下，此系统中的滞后因素既是无害时滞也是无益时滞。

If ris also irreducible in Z_nand f~n~κ≡fmodn,r(xfor all f∈Z_nthen we call n a Carmichael number of orderκmodulo r.Denote the set of all such numbers by C_κ,r(x.Define C_κ=∪_r(xC_κ,r(x,where rpasses through all monic irreducible polynomials over Z of degreeκ.We call elements of the set C_κCarmichael numbers of orderκ.ZHU and SUN[Journal of Sichuan University,42;1(2005), 47-51]first gave a necessary condition for Carmichael numbers of order 3.Then they gave a sufficient condition for Carmichael numbers of order 3,but did not find any such numbers less than 10~8 satisfying the sufficient condition.At last,they asked three questions;1.is the necessary condition also sufficient?

f∈Z_n，就称n是模r的κ阶Carmichael数，用C_κ，r(x表示所有这种数的集合，定义C_κ=∪_r(xC_κ，r(x，这里r跑遍Z中所有κ次首一不可约多项式，C_κ里面的元素就称为κ阶Carmichael数，C_κ称为κ阶Carmichael数集，朱文余和孙琦[四川大学学报，42；1(2005)，47-51]首先给出了3阶Carmichael数的一个必要条件；然后又给出了这种数的一个充分条件，并发现10~8内没有满足此充分条件的这种数；最后他们提出三个问题；一、问必要条件是否也是充分的？

A hypothesis to ensure the tenability of a second-order-sufficient-condition of a linear support vector classifier is presented in the paper, which is a weak one for the solution of a linearly separable support vector classifier must meet it.

针对线性支持向量分类机优化问题，研究了其解的二阶充分条件，给出了二阶充分条件成立的假设条件。

These conditions do not relate to the unknown parameters but the dimension of model, so it is easy to apply in practice. Next, we consider the relative efficiency of LSE against the beast linear unbiased estimator.

我们给出了Pitman准则下，最小二乘估计优于Within估计的一个充分条件，最后又得到了广义均方误差意义下两步估计一致的优于Between估计和Within估计的充分条件，并且这些条件只是和模型维数有关而和未知参数无关的一些不等式关系，因此便于实际应用。

In the United States, chronic alcoholism and hepatitis C are the most common ones.

在美国，慢性酒精中毒，肝炎是最常见的。

If you have any questions, you can contact me anytime.

如果有任何问题，你可以随时联系我。