等价的

Firstly,a involution on Universal Clifford algebra 〓 is introduced.then we equip a inner product and the corresponding two equivalent norms onUniversal Clifford algebra 〓,and then we prove that Universal Cliffordalgebra 〓 is a Banach algebra with one of the above norms,these resultsare very important for further study in this field.

第一，本文创造性地准确地给出了在少Clifford代数〓上的一个对合运算表示，由此，我们给出了在泛Clifford代数〓上的一个内积，然后给出了相应的在泛Clifford代数〓上的两个等价的范数，并证明了若在〓上赋予其中的一个范数，则〓是一个Banach代数，这为在泛Clifford分析中进一步开展工作奠定了初步的基础。

By getting Lebesgue characteristic of integrable function of Riemann from the definition of Gather zero measure, discussing the relation between almost continuous everywhere and existent of limit, it gets the theory which is from the function integrability to the consecution and from consecution to the limit existence .i.e. the almost limit existence is equal to the almost continuous everywhere in the integrable function of Riemann. It also gets a unified condition which has a wider range than regulated function and comes to the conclusion that the function of bounded variation is the integrable function of Riemann.

通过定义零测度集给出了可积函数的特征，讨论了其几乎处处连续与极限存在的关系，从而得到了从函数可积性到连续性，从连续性到极限存在性的函数特性理论，即可积函数中极限的几乎处处存在与几乎处处连续是等价的，得出比正规函数更加宽泛的统一条件，得出了有界变差函数是可积函数的结论。

Thereby, these facts indicate that the SDE not only have better statistic properties but also have uniformly smaller mean square errors than ANOVAE in an available group of linear mixed models. The other significative following work is to generalize the idea of SDE to multivariate linear mixed model and obtain the SDE of variance component matrices which uniformly dominate the ANOVAE either.

对于含有一个随机效应的混合模型，目前文献中都仅讨论了方差分量的谱分解估计和ANOVA估计等价的充要条件，本文首次给出了方差分量的谱分解估计在均方误差意义下严格的优于ANOVA估计的理论结果，并推广到广义谱分解估计，从而说明谱分解估计不仅较ANOVA估计有更好的统计性质，而且在一类常用的模型下有更小的风险损失。

Suppose C is a Graph with k loops andf: G 梸- G is a continuous map with at least a peri- odic point. We prove the following three conditions are equivalent:(1) f is equicontinuous;(2) One of the following End + 2k + 1 conditions holds: in FJm.~0(E~+2L!= flJ1,m = l,2,挃,End J=1 + 2k; 1 0 = flf is a ioop on Gandf I~:0?- 0 is conjugate j=1 to a rational rotation;(3) One of the following End + 2k + 1 conditions holds: k m Lfl.m/~oE~(G+2i!~ is uniformly convergent, m = 1,2,,End+2k; n There is a poistive integer number n,~~~ is uniformly convergent.

假设G是含有k个圈的图，f是G到自身上的连续映射，并且至少有一个周期点，我们证明了下列三条陈述是等价的：（1）f是等度连续的；（2）下列End+2k+1个条件之一成立： m）F l）O=是G的一个圈，且fl_o：O→O共轭于一个有理旋转；（3）下列End+2k+1个条件之一成立： m）是一致收敛的，m=1，2，…，End+2k； l）存在某个正整数n，是一致收敛的。

In this paper,four aspects of the above problems were studied:Firstly, the problem of solving continuous linear operator equation inseparable Hilbert space is studied .

通过把第一类连续不适定算子方程转换为等价的无穷维方程组，然后利用投影法，给出了解存在唯一的充要条件；给出了连续线性不适定算子方程解析解，解决了不适定情况下解的表示问题，由此给出了算子方程的数值求解公式；进一步证明了，在解不唯一情况下，此表达式给出的解为算子方程的最小范数解，同时表示出了连续线性不适定算子方程的解集。

The latter is essentially derived from the geometric realization of Happels triangulated equivalence between stable module category of repetitive algebra and bounded derived category of finite dimensional algebra. In terms of this realization, we deduce that the Lie algebra realized by derived category of a finite dimensional algebra is isomorphic to the Lie algebra realized by stable module category of the corresponding repetitive algebra.

后者本质上是Happel关于重复代数的稳定模范畴和导出范畴的三角等价的一个几何实现及其应用，使用这种几何实现，我们可以证明在重复代数的稳定模范畴上定义的李代数同构于相应的导出范畴上实现的李代数。

Chapter 3, in the above system, gives out the definitions of topological new transitivity, topological transitivity, topological strong transitivity and topological conjugate of a sequence of maps, studies basic properties of the above topological conjugate, and obtains some main results, proves topological new transitivity and topological transitivity are equivalent u-under a compact metric space, a sequence of full maps and interchangeable with each other, and that some conclusions associated with topological conjugate.

第三章，在上述系统中给出了一列映射的拓扑新传递、拓扑传递、拓扑强传递和拓扑共轭的定义；研究了一列映射的拓扑共轭的基本性质，得到了一些主要结果；证明了在底空间是紧致度量空间、一列映射是满映射并且两两可交换的条件下拓扑新传递和拓扑传递是等价的；还证明了几个与拓扑共轭相关的结论。

Based on a kind of orthogonal complete functions in L2[0,1], another one (called V-system) which has equivalence relation with U-system is created. It is a kind of new and effective mathematical tool on signal multi-resolution analysis. Besides that V-system keeps the favorable properties of U-system and can accurately reconstruct geometry information which expressed in piecewise polynomials by finite sum of functions in V-system, it has more special properties, such as simple construction, abundant levels, convenience of application, quick calculation, local support, and so on. V-system is the generalization of Haar orthogonal functions, and to be a class of wavelet base. It has a good future for some applications in the fields of wavelet analysis and signal processing.

从L2[0，1]空间的一类正交完备函数系U系统出发，构造了另一类与之等价的正交完备函数系，称之为V系统，它是信号多分辨分析方面的一种新型的、有效的数学工具。V系统不仅保持了U系统的优良特性，对多项式表达的几何信息能够做到有限项精确重构，并且较之U系统，它更有结构简单、层次分明、计算快捷、局部支集等特点，应用起来将更加灵活方便。V系统可以看作是Haar函数系的推广，是一类小波基，在某些数字信号处理及小波分析问题中有良好的应用前景。

For a finite group G, a subgroup H is called c-normal in G if there is a normal subgroup K such that G=HK and H∩K≤H, the largest normal subgroup of G contained in H. c-normality was replaced by c-π-quasinormality or c-sub-normality. The following were showed equivalent. First, there is a solvable maximal subgroup M such that M is c-π-quasi-normal in G.

推广了有限群中的c-正规性概念，引入了c-次正规性和c-π-拟正规性概念，并利用新概念给出了有限群可解的几个条件，证明了：设G是有限群，那么，下述条件是等价的：G有一个极大子群M在G中是c-π-拟正规的而且是可解的。

As a consequence of an important theorem of algebraic topology, the invariants constructed by those diversity ways can be considered as identified via their unique isomorphisms. This concludes that the procedure of invariant construction gains a virtue of the model presented in this paper as being unrelated to whether there exists a symbolic represantation of images.

根据代数拓扑的重要定理，这些不同的构造方法所得的不变量在同构意义下是等价的，因此我们推论这种不变量的构造过程与图象是否有符号表示无关。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失