内积模

The notion of the adjoint operator of amodule homomorph ism defined on a densely submodule of a comp lete random inner p roduct module is introduced and discussed, in particular the closedness of adjoint operators is p roved.

摘 要 对定义在完备随机内积模的稠子模上的模同态引入了共轭算子的概念并讨论其基本性质，尤其证明了共轭算子的闭性。

Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions: sM is minimal injective and MR is a M-minimal injective kasch-module; MR is a M-minimal injective kasch-module and for any two ideals LI and L2 ofSS rM(L1 n L2)-rw(L1)+rM(I2); sM is minimal injective and for any two submodules A and B of MR,Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.

我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：_sM是单内射的并且M_R是一个M-单内射kasch-模；M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；_sM是单内射的且对M_R的任意两个子模，有l_s=l_s+l_s。2 在第2.3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。

It is proved that a semiring is complemented if and only if it is a direct product of some Boolean algebra and Boolean ring by studying the relations of congruences on the complemented semiring, so the complemented semiring is the commutative semiring.

s-内射模。通过研究可补半环S的性质及S上的同余关系，证明出可补半环与布尔环和布尔代数的直积是等价的，因而可补半环是交换半环。

Material and MethodsRA Rabbit Model group: 15 early RA rabbits of the same weight and variety Control group:10 normal rabbits of the same weight and varietyMethods of making RA rabbit model:elect 15 normal rabbits of the same weight and variety, dissolve Ovalbumin in 0.9% sodium chloride solution, to make the solution of 20mg/ml concengtration, blend the same quantity of complete Freund′s Adjuvant equably ,inject the mixed solution into endermic tissue of the rabbit′s scapular section, making the rabbits allergic, inject 1 ml of the mixed solution to one rabbit every time, inject 1 ml of the mixed solution in 5 different places of the rabbit′s scapular section, inject the rabbit one time every week,inject 3 weeks continuously,it turned out to be that the rabbits will be allergic, inject Ovalbumin blent with the 0.9% sodium chloride solution into the knee joint cavity of the rabbits in the fourth week, 5 mg Ovalbumin every knee joint cavity,two knees of every rabbit of the 15 rabbits are both injected,the arthrosis diameter and the exterior temperature of the knees will be added obviously in 24 hours,and they will drop gradually,at the time of the 14th or 21th day, the arthrosis diameter and the exterior temperature of the knees will get to the balance time,the incidence rate of RA is 100%.after the RA model succeeds,it is the early time of RA from the first week to the fourth week,after the fourth week,it is the late time of RA, the cartilage of the femoral condyle and the subchondral bone cortices will be changed unrecoverily.

资料与方法RA模型组：早期RA家兔模型15只，品种及体重接近。对照组：正常家兔10只，品种及体重接近。RA家兔模型制作方法：选取15只成年家兔，体重、品种接近，将卵蛋白溶解于生理盐水，配成浓度为20mg/ml的溶液，与等量完全弗氏佐剂混匀，注入家兔肩胛区皮下致敏，每只家兔每次注射1ml，于肩胛区5个不同区域注射，每周一次，连续注射3周而致敏，第4周向膝关节腔注射卵蛋白生理盐水溶液，每只关节腔注射5mg卵蛋白，15只家兔膝关节全部注射，24h内此关节直径和表面温度大幅度上升，以后缓慢下降，至14～21d达到平台期，发病率达100%。造模成功后第1～4周为早期改变，第4周以后出现不可逆的关节软骨及骨破坏。采用高频超声对RA模型组与对照组的膝关节髌上囊液体厚度、滑膜及股骨髁软骨厚度及软骨下骨皮质的回声情况进行对比观察。结果RA组模型组膝关节髌上囊积液及滑膜的厚度明显厚于对照组，其股骨髁软骨的厚度与对照组相比无明显差异，其软骨下骨皮质与对照组相比无明显改变。

Thesolutions compose self-contained inner product space (weight function is constant 1),i.e., they are basic series of Hilbert space; hence any internal waves could beexpanded into generalized Fourier series. With difference method, Sturm-Liouvilleequation is transformed to matrix eigenvalue problem. Each eigenvalue is related to ahorizontal wave number, and relevant solution is a certain mode of internal wave.

其解系构成完备的加权(权函数为1)内积空间，即Hilbert空间，从而内波现象可以展成该解系的广义Fourier级数；采用差分方法，将 Sturm-Liouville 方程转化为矩阵特征值问题，每一个特征值对应一个内波波数，该特征值对应的解则是内波的某一模态。

The notion of a complete random inner product module is a random generalization of that of a Hilbert space.

完备随机内积模是Hilbert 空间的随机推广。

First ,it is proved that a positive Hermite form corresponds uniquely to a positive self2adjoint operator in a complete random inner product module ,then this result leads directly to the proof of the main result of...

值得指出的是：完备随机内积模上Friedrichs 定理的证明中所涉及的一系列基本概念与方法都是以随机共轭空间理论为出发点的，与经典情形完全不同。

In this paper, we describe the history of Mizar system and the method to use theMizar system. Then we give the definition of quaternion numbers, module, algebrastructre etc, and then we prove its properties.

本文首先介绍了Miza(来源：ABC论文61网www.abclunwen.com)r系统的计算机操作原理和使用方法，给出了在Mizar语言系统下四元数、四元数的运算、共轭、内积、模等代数结构理论的证明，所做结论都通过了Mizar系统的验证，发表在波兰的《Formalized Mathematics》杂志上，并被收录到MML数据库中。

The sufficient and necessary conditions which Fuzzy subalgebra of direct product Boolean algebra can be decomposed as direct product of two Fuzzy subalgebras are obtained.

进而分别定义了在对偶模意义下直觉模糊群的外直积与内直积。

Recently , the classical Riesz representation theorem on Hilbert spaces has been generalized onto complete random inner product modules ,in this paper the classical Friedrichs theorem on Hilbert spaces is generalized onto complete random inner product modules.

最近，经典的Riesz 表示定理已经被推广到完备随机内积模上，在此基础上本文将Hilbert 空间上经典的Friedrichs 定理推广到完备随机内积模上。首先，证明完备随机内积模上任一正Her2 mite 型惟一地对应一个正自共轭算子。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。