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    Then T is a compact operator if and only if . (2) Let T ∈ B(H).
    (2) 若T∈B(H),则T是迹类算子的充分必要条件是是有界的。
    Then T is a trace-class operator if and only if Wk(T) is bounded. On the other hand, M. T. Chien, Shu-Hsien and Pei Yuan Wu have given the geometric properties of k- numerical ranges of quadratic operators (satisfying T2 + aT + bl = 0 for some scalars o and b in C).
    M.T.Chien,Shu-Hsien Tso和Pei Yuan Wu在文[7]中给出了二次算子(满足T~2+aT+bI=0,(a,b∈C))中两类算子(幂零算子和幂等算子)的k-数值域的几何性质。
    Secondly, we obtain necessary and sufficient conditions for positive solutions of X + A*X-t A = I and get that if X + A*X-tA =I has a solution X, then ||X|| = 1 if and only if A is not bounded below.
    其次,给出了算子方程X+A~*X~(-t)A=I(t≥1)有正算子解的充要条件,并得到了更为特殊的情况:方程的正算子解的范数为1的充要条件是A不是下有界的。
    That is, a C*- algebra A is commutative if and only if every positive linear map from C*- algebra A to C*-algebra B becomes completely positive.
    即一个C~*-代数A是可交换的充要条件是任何一个从C~*-代数A到C~*-代数B的正线性映射是完全正的;
    satisfying1) Φ |v= 02)Φ (H,V) =0Moreover, 0 is nondegenerate if and only if Φ is nondegenerate.
    2) Φ(H,V)=0 而且φ是非退化的当且仅当Φ是非退化的。
    1 Let 5,T be two semigroups,a : a(s) be a given homomorphism,then the semidirect product S xf, T is a right Clifford quasi- regular semigroup if and only if:In chapter 5,we give the necessary and sufficient conditions of a regular semigroup: LR-C semigroup.
    :s*End(劝,、*a(、)为给定的半群同态映射,半直积5 xQT是LR一C半群当且仅当 1)对任意的C任E(S),S为LR一C半群,厂为正则半群,其中Te={telt eT};
    Liu Yingming and Liang jihua proved that for all core compact space X a DCPO D is continuous L - domain if and only if [X → D] is continuous L - domain.
    刘应明,梁基华证明了对一个DCPO D,D是连续L-domain当且仅当对所有核紧空间X,[X→D]是连续L-domain。
    Now the most prevalent description of the first fundamental theorem of asset pricing is: there are the equivalent martingale measures in a market if and only if the bounded semi-martingale S(t) satisfies no free lunch with vanishing risk.
    目前对资产定价第一基本定理最普遍的描述是:市场存在等价鞅测度当且仅当有界半鞅折准价格过程S(t)满足没有无风险免费午餐(No Free Lunch with Vanishing Risk,NFLVR).
    Finally, further more, there is a main theorem in the paper, it is that the bounded semi-martingale S(t) satisfiesNFLVR if and only if V - L+∞ ∩ L+∞ {0} Thus we construct a new topology description on NFLVR.
    最后更进一步的,在深入理解和研究前人结论的基础上证明了本文的主要定理:有界半鞅S(t)满足NFLVR等价于V-L_+~∞ ∩L_+~∞={0}。 从而得到NFLVR的一种新的拓扑描述。
    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,.
    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。
    For example, H.oruc and Neciber Tuncer ([8]) proved that for fixed 9, 0 < q < 1, B_(n,q)(f, x) converges uniformly to f(x) if and only if f is linear.
    例如,H. oruc和Neciber Tuncer([18])证明了:若0
    Let Γ be a strongly connected digraph of girth g ≠ 2 and K_s be a completed graph with s vertexes. Then Γ[K_s] is a weakly distance-regular digraph if and only if Γ is a weakly distance-regular digraph.
    ※Γ是强连通有向图,围长g≠2,K_s是有s个顶点的完全图,Γ[K_s]是弱距离正则的充分且必要条件为Γ是弱距离正则的。
    ※ Let Γ is a strongly connected digraph of girth g ≠ 2 and C~*_r be an undirected graph of length r. Then Γ'=Γ × C~*_r is a weakly distance-regular digraph if and only if Γ is a weakly distance-regular digraph and the following hold .
    ※Γ是一个强连通有向图,围长g≠2,Γ′=Γ×C_r~*,C_r~*是长为r的无向圈,Γ′是弱距离正则的充分且必要条件为Γ是弱距离正则的且满足以下条件:
    Specially, we prove that left R-semimodule P is projective if and only if Hom_R(P, -) is a covariant congruence exact functor if and only if, for all congruences p over all semimodules M and homomor-phism α : P → M/ρ, there exist homomorphisms α|- : P→ M such that α = π_ρα|-, where π_ρ:M→M/ρ.
    主要结果是:若P为左R-半模,则P是投射R-半模当且仅当Hom_R(P,-)是共变的同余正合函子当且仅当对于任意左R-半模M上的任意同余ρ和同态α∶P→M/ρ,都存在同态(?) :P→M,使α=πρ(?)
    regular ring if and only if every factor module of the injective . R-module is flat(refer to theory 2.1.9).
    正则环当且仅当任意内射R-模的商模是平坦模(见定理2.1.9).
    If m denotes the number of Hamilton-path in T, The equation is true if and only if for every i e {1,2,... s}.
    等式成立当且仅当对于每个i∈{1,2,…,s},T_i≌C_3或T_i≌K_1。
    The sum graph G+(S) of a finite subset S C N is the graph (S, E) with uv ∈ E if and only if u + v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S(?)
    Harary提出了和图的概念,令N表示正整数集,N的非空有限子集S的和图G+(S)是指图(S,E),其中uv∈E当且仅当u+v∈S,一个图G称为和图,若它同构于某个S(?)
    The sum graph G+(S) of a nonempty finite subset S C N{Z) is the graph (S. E) with uv ∈ E if and only if u + v ∈ S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S (?)
    令N(Z)表示正整数(整数)集,N(Z)的非空有限子集S的和图G+(S)是图(S.E). 其中uv∈E当且仅当u+v∈S;
    We prove that R is a Krull type domain if and only if R[x] is a Krulltype domain, if and only if R[X]Nv is also a Krull type domain, if and onlyif R[X]Nv is a Pru¨fer domain and R[X]Nv has finite character, if and only ifR[X]Nv is a B′ezout domain and R[X]Nv has finite character.
    证明了R是Krull型整环, 当且仅当R[X]是Krull型整环, 当且仅当R[X]Nv是Krull型整环, 当且仅当R[X]Nv是有限特征的Prufer整环, 当且仅当R[X]Nv是有限特征的B′ezout整环;
    We provethat R is a valuation domain if and only if R is a local Krull type domain andR is a TL domain and Spec(R) is totally ordered.
    证明了R是赋值环当且仅当R是局部的Krull型整环, 且R是TL整环,Spec(R)是全序;
 

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