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if and only if
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  当且仅当
     The question that how to determine a module M over a commutative ring R with invariant factors was discussed,and the result indicated that M was such an R-module if and only if the fractional module S-1M over the fractional ring S-1R had invariant factors whenever the multiplicative subset S of R was invertible in R/AnnR(M) being the annihilator of M in R.
     探讨了交换环R上具有不变因子的模M之判别问题,证明了只要R的乘法子集S在R/AnnR(M)中可逆,则M为具有不变因子的R-模当且仅当分式模S-1M为分式环S-1R上的具有不变因子的模.
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     2. N is △( M )-injective if and only if N is △( Mn)-injective.
     ②N是△(M)-内射模当且仅当N是△(Mn)-内射模.
短句来源
     In this paper we prove that the equation xn+yn=zφ(n) has positive integer solutions(x,y,z) with gcd(x,y)=1 if and only if it is n≤3.
     证明了方程xn+yn=zφ(n)当且仅当n≤3时有正整数解(x,y,z)适合gcd(x,y)=1.
短句来源
     It is proved that the ring [[Rs,≤,λ]] is a reduced left PP-ring if and only if R is a reduced left PP-ring and every S-indexed subset C of B(R) has a least upper bound in B(R);
     证明了[[Rs,≤,λ]]是reduced左PP-环当且仅当R是reduced左PP-环,且B(R)的每个S可标子集C在B(R)中有最小上界;
短句来源
     R is a ring without nonzero zero-divisors,then the ring [[Rs,≤,λ]] is Dedeking finite if and only if R is Dedeking finite.
     若环R无非零零因子,则[[Rs,≤,λ]]是Dedekind有限环当且仅当R是Dedeking有限环.
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  “if and only if”译为未确定词的双语例句
     In this paper we proved that the equation gcd(a_1,a_2,…,a_n)+1cm(a_1,a_2,…,a_n)=a_1a_2…a_n,has positive integer solutions (a_1,a_2…a_n) if and only if n=2 and it has only the solution (a_1,a_2)=(2,2).
     证明了:方程 gcd(a_1,a_2,…,a_n)+1cm(a_1,a_2,…,a_n)=a_1a_2…a_n 仅当 n=2时有正整数解(a_1,a_2)=(2,2).
短句来源
     In the paper we prove that a feasible graph G=G_1∪G_2∪G_3 for (X_1, X_2, X_3) is minimum if and only if G_i ∩G_j∩G_k are all trees for X_i∩X_j∩X_k≠ φ(i, j, k=1, 2, 3).
     本文证明,关于(X_1,X_2,X_3)的可行图G=G_1∪G_2∪G_3是最小可行图的充分必要条件是:当X_i∩X_j∩X_k≠φ(i,j,k)=1,2,3)时,G_i∩G_j∩G_k是树。
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     This paper proved that population X with EX=DX<∞, and nonsingular distribution is Poisson distribution if, and only if, the statistic T 2-T 1 has constant regression on T 1, where T 1==1n∑ni=1X i is sample mean, and T 2=1n-1∑ni=1(X i-) 2 is sample variance.
     证明了满足EX =DX <∞的具有非退化分布的母体X服从Poisson分布的充要条件是T2 -T1关于T1有常回归 ,其中T1= X =1n∑ni =1Xi,T2 =1n- 1∑ni =1(Xi- X) 2 分别为子样均值和子样方差
短句来源
     It is proved in this parper that population X of nonsingula distribution with EX 2<∞ is of Poisson type distribution P(λ,α,1) if and only if the statistic T 2-T 1 has constant regression on T 1 ,where T 1==1n∑ni=1 X i is sample mean and T 2=1n-1∑ni=1(X i-) 2 is sample variance.
     证明了满足EX2 <∞的具有非退化分布的母体X服从Poisson型分布P(λ ,α ,1)的充要条件是T2 -T1关于T1有常回归 ,其中T1= X =1n ∑ni =1Xi,T2 =1n - 1∑ni=1(Xi- X) 2 分别为子样均值和子样方差
短句来源
     Any RC_4S(v,λ) can be embedded in an RC_4S(u,λ) if and only if u≡v≡0(mod 4),λ≡0(mod 2) and u≥2v.
     任一个RC4S(v,λ)都可以嵌入到一个RC4S(u,λ)之中的充分与必要条件为u≡v≡0(m od 4),λ≡0(m od 2)且u≥2v.
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  相似匹配句对
     (c) If s(M) (?)
     (c)若s(M)(?)
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     R if m
     R。
短句来源
     ItlocatedbytheCanobicRiveranditsmodernnameisKomGe'if.
     它位于尼罗河的支流卡诺比克河口,其遗址的现代名字是考姆盖伊夫(KomGe'if)。
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     If-Conditionals and Stereotype
     If条件句与常规关系
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     R_P if and only if I (?)
     R_P当且仅当I(?)
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  if and only if
This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
      
TheS-arithmetic group г is of typeFn, resp.FPn, if and only if for allp inS thep-adic completionGp of the corresponding algebraic groupG is of typeCn resp.CPn.
      
We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable.
      
Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions.
      
We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis.
      
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In this paper, we give some results about the complete continuity of the Polynomial operators together with the analytical operators. As its application we consider the Liapunof-Lichtenstein operators. The main results are: Th. 2 completely continuous in some spher if and only if for every n, unx~n completely continuous. Th. 6. unx~n completely continuous if and only if un(x_1,…,x_n)completely continuous in n Variables. Th. 8 If kn(s,t_1,…,t_n)measurable, symtritical respect t_1…,t_n and maps...

In this paper, we give some results about the complete continuity of the Polynomial operators together with the analytical operators. As its application we consider the Liapunof-Lichtenstein operators. The main results are: Th. 2 completely continuous in some spher if and only if for every n, unx~n completely continuous. Th. 6. unx~n completely continuous if and only if un(x_1,…,x_n)completely continuous in n Variables. Th. 8 If kn(s,t_1,…,t_n)measurable, symtritical respect t_1…,t_n and maps completely continuous in its definite area.

本文考虑B型空间x入z解析算子其中u_n是有界对称n线性算子,设P_u为后端一级数的一致收歙半径第一部分证明了以下两个主要结论: 定理:F(x)在内完全连续的必要且充分条件为对任何n=0.1,2,……,u_nx~n完全连续。定理:u_nx~n完全连续的必要且充分条件为u_n(x_1,…,x_n)按n变元完全连续。作为以上结论的应用,第二部分讨论求得F(x)映 P_1(G)入 P_2(G)完全连续的条件。

This short paper gives two remarks on the singularities of the scattering amplitudes in a perturbation theory. First, it is pointed out that if the condition of stability is violated at several corners of a Feynmann diagram so that analytic extensions with respect to several external masses must be made, different sequences of carrying out the analytic extensions may lead to different positions of the singularities with respect to the final position of the integration contour (in a dispersion integral) and hence...

This short paper gives two remarks on the singularities of the scattering amplitudes in a perturbation theory. First, it is pointed out that if the condition of stability is violated at several corners of a Feynmann diagram so that analytic extensions with respect to several external masses must be made, different sequences of carrying out the analytic extensions may lead to different positions of the singularities with respect to the final position of the integration contour (in a dispersion integral) and hence to different expressions for the amplitudes. Next, it is pointed out that the non-Landau singularities fall into two classes, of which one, say the first, is given by Cutkosky. For singularities in the first class, it is shown that they may be independent of the internal masses if and only if all the internal masses are equal or if the diagram is a loop. Their positions for the latter case are given. For singularities in the second class, it is shown that they are always independent of the internal masses and that they occur only for a very restricted set of diagrams.

本文讨论了二个问题:一、当振幅须对二个以上的外质量进行开拓时,说明了开拓的次序对最后结果有关;二、指出了非朗道奇异点有二种:一种是Cutkosky所指出的,对于这一种,讨论了它与内质量无关的条件,并求出了它对于一圈的费曼图的位置;对于另一种,指出了它的来源,证明了它与内质量无关,并说明了它只对极少数图出现。

Let Cp(x) denote the cyclic code generated by p(X), and Cpn(x) the principal ideal generated by p(X) in the algebra of polynomials modulo Xn - 1. The following theorems may make clear the idea behind the design of Fire codes.Theorem Ⅰ. If n is the length of Cp(x)q(x), and p(X) is relatively prime to q(X), then Cp(X)q(x) is equal to the intersection Cpn(x)Cqn(x).Theorem Ⅱ. Consider three cyclic codes with the intersection of the first two equal to the third. The syndromes of the two distinct bursts with respect...

Let Cp(x) denote the cyclic code generated by p(X), and Cpn(x) the principal ideal generated by p(X) in the algebra of polynomials modulo Xn - 1. The following theorems may make clear the idea behind the design of Fire codes.Theorem Ⅰ. If n is the length of Cp(x)q(x), and p(X) is relatively prime to q(X), then Cp(X)q(x) is equal to the intersection Cpn(x)Cqn(x).Theorem Ⅱ. Consider three cyclic codes with the intersection of the first two equal to the third. The syndromes of the two distinct bursts with respect to the first or second code are not the same if and only if the syndromes with respect to the third are not the same.

文中运用循环码求交运算的方法,对纠突发错误循环码的设计思想作了一些探讨,给出了Fire码的一种纠突发错误算法。

 
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