当且仅当

Then R is an Archimedean domain if and only if a is contained in some minimal prime ideal of R for every nonzero nonunit a.

则有R是阿基米德整环当且仅当a含在R的某个极小素理想中。

The problem then becomes constructing such an ordering, and it turns out that every set can be well-ordered if and only if the axiom of choice is true.

问题就变成构造这样一个排序，而它使得所有集合可以是良序的，当且仅当选择公理为真。

Tarski's definition of truth of the calculus of classes is: x is true if and only if x is a sentence of the calculus of classes and every infinite sequence of classes satisfies x .

塔斯基得到的真概念定义是：x是类演算语言的任一语句，x为真当且仅当每一个类的无穷序列都满足x。

In order to study the popular curve shortening flow in the plane, Gage[3] has shown in 1983 the following inequality, if k is the signed curvature of a closed convex plane curve γ with length L and enclosing area A, then one gets Gage calls it an isoperimetric inequality in [3], but he does not show that the equality holds if and only if the curve γ is a circle, while as an isoperimetric-type inequality, one should prove this kind of result.

Gage称之为等周不等式，但他没有证明其中等号成立当且仅当，γ为圆周，而作为等周型的不等式是应该证明这种结果的，本文我们利用单位速率外法向量流，通过努力证明了这种结果，从而把Gage不等式加强成更加等周的形式。

Some concepts in quantale, such as c-nuclei, r-nuclei, closed map closed filter were introduced and Some properties of these new nuclei are discussed.

幂等左侧Quantale是单纯Quantale当且仅当它是离散一致简单Quantale。二。研究Quantale中的核映射和闭滤子。

Some concepts in quantale,such as c-nuclei,r-nuclei,closed map closed filter were introduced and Some properties of these new nuclei are discussed, we abtain the characterizations of c-nuclei in different quantales and the equivalance between c-nuclei and r-nuclei.

幂等左侧Quantale是单纯Quantale当且仅当它是离散一致简单Quantale。二。研究Quantale中的核映射和闭滤子。引入了c-核映射，r-核映射，闭滤子和闭映射等概念，讨论了它们的一系列性质。

A linear mapping between two Banach spaces and is continuous if and only if its graph is a closed subset of.

巴拿赫空间之间的线性映射是连续的，当且仅当这个映射的图像是闭集合。

Some basic properties of σ- LFSR over F4 are studied, such as nonlinearity, cycle structure distribution of state graph, the largest period and counting problem related. The conclusions are as follows:The coefficient ring of σ-LFSR is isomorphic to the matrix ring over F,. The cycle structure of σ- LFSR is consistent with that of the determinant of the corresponding polynomial matrix if and only if the feedback polynomial of - LFSR does not contain nontrivial factor over F2,. The counting formula of the number of σ- LFSR with inconsistent cycle structure is also showed in that part. The period of σ-LFSR with degree n is maximum if and only if the determinant of the corresponding polynomial matrix is the primitive polynomial with order 2n over F2,.

本文研究了有限域F_4上的σ-LFSR的一些基本性质，如非奇异性、状态图的圈结构的分布、最大圈的充要条件及相关的计数问题等，得到以下结论：σ-LFSR的系数环同构于F_2上的矩阵环；σ-LFSR的状态图的圈结构与对应的多项式矩阵的行列式的圈结构一致的充要条件为σ-LFSR的反馈多项式不含有非平凡的F_2上的因式，给出了圈结构不一致的σ-LFSR的计数公式； n次σ-LFSR周期达到最大，当且仅当对应多项式矩阵的行列式为F_2上的2n次本原多项式。

Xu Jinzhong proved that a simple module over commutative ring is injective if and only if it is flat.

徐金中证明了交换环上单模是内射的当且仅当它为平坦的。

We will prove that the only compact dual Toeplitz operator is the zero operator, and that a densely defined dual Toeplitz operator with square integrable symbol is bounded if and only if its symbol is essentially bounded.

首先，我们证明了只有零算子是紧的对偶Toeplitz算子，同时给出了以平方可积函数为符号稠定义的对偶Toeplitz算子有界当且仅当它的符号函数是本性有界的。

Since historical times,England ,where the early inhabitants were Celts, has been conquered three times .

从有历史以来，英国，在此地早期居住的是凯尔特人，已经被征服了三次。

Bluetooth OBEX File Transfer Enables the sending and receiving of files on your phone via Bluetooth.

蓝牙OBEX文件移动允许经过蓝牙传送和接受文件。。。。