证明性的

Firstly, we prove the existence and the uniqueness of the hyperfocal subalgebra when the coefficient field is split for a defect pointed group; then, considering the structural pattern of source algebras over arbitrary fields, by means of G-acted groups, we reduce the existence and the uniqueness of the hyperfocal subalgebra over an arbitrary field to the case that the coefficient field is split for a defect pointed group and the inertia group stablizes a hyperfocal subalgebra.

首先我们证明了如果系数域对亏点群是分裂的，则源代数的超聚焦子代数存在且共轭唯一；然后通过考虑任意域上源代数的结构模式，以G-作用群为手段，将超聚焦子代数的存在性和唯一性归结为系数域对亏点群是分裂的情况下惯性群稳定的超聚焦子代数的存在性和唯一性。

This article compared the integrand of infinite integral and unbounded function integral by comparing the infinitesimal with the infinity, obtained the corresponding equivalence theorem of criterion of convergence of abnormal integral, and gave out the proof.

将无穷积分及无界函数积分的被积函数运用无穷小和无穷大比较的方法进行比较，得到了相应的反常积分敛散性极限审敛法的等价定理，并给予证明，从而可运用等价定理灵活的判断反常积分的敛散性。

By increasing the condition of the integral mean value theorem,we prove that the existence of intermediate point and the existence of interval are corresponding to each other.

给出了积分中值定理的一个注记，证明了中值点的存在性与覆盖中值点的区间的存在性是相互对应的。

For its two special cases: N = 0 and R=0, based on nongradient and gradient methods respectively, neural networks for the two cases are constructed. With the Lyapunov theorem and the LaSalle invariant set principle, the proposed networks are proved to be

推广了R_0性质的定义，并证明无约束优化问题目标函数的水平集的有界性与矩阵对的R_0性质是等价的，最后运用Lyapunov稳定性理论和LaSalle不变原理，严格分析了网络模型的稳定性和收敛性，这些模

Finally, after defining an evaluation module of the corresponding Loop superalgebra (Section 4), two major results of the paper -Theorem 4.land Theorem 4.2 are proved: Theorem 4.1 reduces the irreducibility of the tensor product of finitely many evaluation modules to the irreducibility of the tensor product of finitely many irreducible modules of a nilpotent Lie superalgebra; Theorem 4.2 gives a criterion for the tensor product of such modules to be irreducible.

第4节在定义了相应的Loop超代数的赋值模之后，证明了本文的两个主要结论：定理4.1和定理4.2。定理4.1将有限多个赋值模其张量积的不可约性归结为一幂零李超代数的限多个不可约模其张量积的不可约性；定理4.2利用不可约指标给出了一幂零李超代数的限多个不可约模其张量积仍不可约的判别准则。

In this paper, a class of dynamical system in biochemistry reaction is discussed by using the method of qualitative analysis. And, at the same time,the existence, nonexistence and uniqueness of a limit cycle are proved, so that the entire results for the system is obtained.

研究微分方程定性理论的应用，利用定性分析的方法，讨论了一类在生化反应中的动力系统，证明了其极限环的存在性、不存在性及极限环的存在唯一性，从而得到了完整的结果。

Based on ear recognition, an improved NMFSC(Nonnegative Matrix Factorization with Sparseness Constraints) method was proposed by imposing an additional constraint on the objective function of NMFSC, which could capture the semantic relations of coefficient matrix as orthogonal as possible. The interated rules to solve the objective function with the constraint were presented, and its convergence was proved.

针对人耳识别问题，提出了一种改进的稀疏性受限的非负矩阵因子方法，通过增加一个使系数矩阵尽可能正交的约束条件来定义原目标函数，给出求解该目标函数的迭代规则，并证明迭代规则的收敛性。

Based on ear recognition, an improved NMFSC (Non-negative Matrix Factorization with Sparseness Constraints) method was proposed by imposing an additional constraint on the objective function of NMFSC, which could capture the semantic relations of coefficient matrix as orthogonal as possible. The interated rules to solve the objective function with the constraint were presented, and its convergence was proved.

针对人耳识别问题，提出了一种改进的稀疏性受限的非负矩阵因子方法，通过增加一个使系数矩阵尽可能正交的约束条件来定义原目标函数，给出求解该目标函数的迭代规则，并证明迭代规则的收敛性。

The convergence of the mean shift procedure to the closest mode of the underlying distribution is proven, both for the Epanechnikov kernel and the general case of kernels with convex and monotonic decreasing profile. The smooth trajectory property of the mean shift is also demonstrated.

均值平移算法对于Epanechnikov 核函数的收敛性在本文中得到了证明，进而推出均值平移算法收敛的充要条件是核函数具有凸的、单调递减性质的轮廓；揭示了均值平移过程运动轨迹的平滑性。

At the same time,Archimedes gives an exact proof on the formula of area of circle,with the method of exhaustion method depending on the Cyclotomic.

刘徽之前，希腊的阿基米德用穷竭法也证明了圆的面积公式。其中，刘徽的证法强调计算的程序性和构造性，而阿基米德则倾向于演绎的严谨性。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新