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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。

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。

In particular, we compute Hochschild homology and cohomology groups of infinitedimensional path algebras and some of their quotient algebras, and we prove that for a general monomial algebra (not necessary finite-dimensional), all Hochschild cohomology groups of positive degrees vanish if and only if its Gabriel quiver is a finite tree.

特别地，我们计算了无限维路代数以及某些商代数的Hochschild同调群和上同调群，而且给出了一般单项代数的各正次Hochschild上同调群为零的充分必要条件，即它的Gabriel箭图是有限树。

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。

It is proved that a complete lattice is completely distributive lattice if and only if it is both semi-Smooth lattice and semi-continuous lattice.

证明了完备格L是半Smooth格和半连续格当且仅当L是完全分配格。

For example, one of our conclusions is that a nonsingular QSIC has one affinely finite component if and only if the characteristic equation of the two quadrics has two distinct real roots and a pair of complex conjugate roots.

由于两二次曲面的特征方程实根个数及其重数可用精确的计算方法得到，本文分类结果可用于实现对任意两二次曲面的非平面交线拓扑形态的判断。

The suite of the if clause, expr_true_suite, will be executed only if the above conditional expression results in a Boolean true value.

74用于判断结果真假的条件表达式，以及当表达式为真或者非零时执行的代码块。

In the third section,we study the relationship between the APOTP for continuous map / on a compact metric space X and that for the lift map / of / on a covering space X and show that has the APOTP if and only if has the APOTP.

在第四节中，我们研究逆极限空间上移位映射的APOTP，讨论紧致度量空间上连续映射APOTP与逆极限空间上移位映射APOTP间的关系。

Then f has β-differentiability property if and only if each level set of f〓 has ω〓-β-exposed property.

则f具有β-可微性质的充分必要条件是f〓的每个水平集具有ω〓-β-暴露性质。

This paper investigates topological properties of the general gradient system, and demonstrates by results of differential topology that the boundedness of the general gradient system is equivalent to the boundedness of the potential energy boundary surface. Moreover, a general gradient system is bounded if, and only if the system has both sink point and source point.

该文研究了广义梯度系统的拓扑性质，应用微分拓扑学原理证明，广义梯度系统的完全稳定性与势能界面的有界性是等价的；广义梯度系统完全稳定当且仅当该系统既有渊点又有源点。

What are your goals and strategies for growth?

你的成长目标和策略是什么？

And unto the angel of the church in Sardis write; These things saith he that hath the seven irits of God, and the seven star I know thy works, that thou hast a name that thou livest, and art dead.

3：1 你要写信给撒狄教会的使者，说，那有神的七灵和七星的，说，我知道你的行为，按名你是活的，其实是死的。