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 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. 短句来源 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. 短句来源 更多
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 Results It is get that PQ(respectively,P-Q) is Drazin invertible if and only if Q0(respectively,I-Q0) is invertible. 结果得出PQ(resp.P-Q)是Drazin可逆的充要条件是Q0(resp.I-Q0)是可逆的。 短句来源 The orientation distance graph D0(G) of G has the set O(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D' of D0(G) are adjacent if and only if d0(D,D')=1.This paper obtains characteristics of orientation distance graphs of cycles. G的定向距离图D0(G)的顶点是互不同构的定向,如果d0(D,D')=1,则D与D'在D0(G)中相邻,并获得定向距离图D0(Cn)的性质. 短句来源 G~(++-) and G~(--+) are regular if and only if G≌C_n or K_(2,n-2) or K_4; G~(++-)和G~(--+)为正则图的充要条件是G为C_n、K_(2,n-2)或K_4; 短句来源 It is proven that a product arrangement(A1×…×Ak,V1…Vk) is a supersolvable arrangement if,and only if,each arrangement(Ai,Vi),1≤i≤k is also a supersolvable arrangement. 证明了乘积构形格L中元素是模元的充要条件,并利用该结论证明了乘积构形(A1×…×Ak,V1…Vk)是超可解构形的充要条件是每个因子构形(Ai,Vi),1≤i≤k都是超可解构形。 短句来源 In this paper,we have discussed the matching uniqueness of graphs and proved that T(3,4,n)∪(∪si=0Cpi)(n4) and its complement are matching unique if and only if n≠4,9. 研究了图的匹配唯一性,给出了T(3,4,n)∪(s∪i=0Cpi)(n 5)及补图匹配唯一的充要条件. 短句来源 更多

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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. 更多
 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。 We give some useful critirion of semicontinuity for convex functional in abstract space. Let φ(x) be lower convex functional in convex set D with an interior point Xo, then 1°φ(x) in Xo weak lower semicontinuous if and only if φ(x) in Xo lower semicontinuous 2°If δ~+[φ(Xo)h]≤ M‖h‖, then φ(x) in Xo weak lower continuous. 3°If φ(x) in neighborhood of some interior point x∈D, then φ(x)in x_o weak lower continuous. 4°If φ(X_o) in X_o upper semicontinuous, then it is weak lower semicontinuous in x_o. Let... We give some useful critirion of semicontinuity for convex functional in abstract space. Let φ(x) be lower convex functional in convex set D with an interior point Xo, then 1°φ(x) in Xo weak lower semicontinuous if and only if φ(x) in Xo lower semicontinuous 2°If δ~+[φ(Xo)h]≤ M‖h‖, then φ(x) in Xo weak lower continuous. 3°If φ(x) in neighborhood of some interior point x∈D, then φ(x)in x_o weak lower continuous. 4°If φ(X_o) in X_o upper semicontinuous, then it is weak lower semicontinuous in x_o. Let φ(v,w)=integral from G [F(s,v(s),w(s))ds], F(s, v, w)lower convex with respect of W. then either °If F(s,v,w)on s∈G and|v|≤M,|w|<∞be an(H)function F(s,v,w)≥0. or 2°F(s,v,w) Continuous in G×[-M, M: ] ×(-∞, ∞), functional φ(v, w) lower semicontinuous in the sense that Vn(s) converge in measure to V_o(s):|Vn|≤M and Wn(S) weak converge to W_o(S) in L~p (p>). 本文给出了一些凸泛函的性质,特别是与极值原理有紧密連系的下半連續性,推广了以前的結果或簡化了以前的証明,也得到了一些新的事实。 << 更多相关文摘
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