if and only if 
A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or .


The desired equivalence of the choice models was shown to be feasible if and only if the binary relation in the pairdominant choice model is a simple semiorder.


An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely.


It is proven that all left Rmodules can be decomposed into a direct sum of chain modules if and only if the ring R is generalized uniserial.


We shall show that a complete sequence of finitedimensional subspaces {Nj}1∞ is a Bari basis if and only if each sequence {ψj{1∞(ψj€Nj, ‖ψj‖=1) is a Bari basis of its own closed linear hull.


It is shown that in the linearregression scheme, the estimates of the squares of parametric vectorfunctions are admissible in the class of unbiased estimates if and only if the observations obey a normal law.


It is shown that the ring of endomorphisms of an arbitrary free Rmodule is right selfinjective if and only if R is quasiFrobenius.


It is proven that if K is a commutative ring of characteristic pm while group G contains pelements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G' is a finite pgroup.


It is shown that the ring R is a right cogenerator if and only if in the ring of endomorphisms of any free Rmodule, r(ι(J))=J for all finitely generated right ideals J.


A space is strongly symmetrizable if and only if it is a pseudoopen IIimage of a metric space.


For functions of certain quasianalytic classes C{mn} on (∞, ∞) we determine a function ξ (x), depending on {mn}, which is such that a sequence {xk} is a sequence of the roots off(x) ε C{mn} if and only if for somea


In particular,G(M) is a torsion if and only if M is a pseudoinjective module.


It is established that the spectral measure of an infinitely divisible distribution F in a Hilbert space H is concentrated in a sphere of finite radius if and only if the integral ∫Hexp (α∥x∥ In (∥x∥+1))dF is finite for some numberα>amp;gt;0.


A set M in uniformly convex space X is shown to be approximatively compact if and only if M is Pcompact and the metric projection of X onto M is upper semicontinuous.


The main result: there exists a freefmodule over an omoduleRM with cone pm if and only if pM is halfclosed (this generalizes Vainberg's theorem for ordered Abelian groups).


It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented.


Our main result: the semigroup algebra K[H] of H over a field K of characteristic 0 satisfies an identity if and only if G has an Abelian subgroup F of finite index and, for any X, the matrix P(F, X) has finite determinant rank.


We show that a numberλ is an eigenvalue of the operator T+C for an arbitrary compact perturbation C if and only if the operator T λI is semiFredholm and ind (TλI)>amp;gt;0.


If N is a closed geodesic, then the equality is attained in the estimate if and only if M is a generalized lens space.


It is shown that each left R module is isomorphic to a direct sum of left ideals of the ring R if and only if R is quasiFrobenius and generalized uniserial.

