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.

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);

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.

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.

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.

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.

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.

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.

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>).