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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 pair-dominant 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 R-modules 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 finite-dimensional 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 linear-regression scheme, the estimates of the squares of parametric vector-functions 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 R-module is right self-injective if and only if R is quasi-Frobenius.
      
It is proven that if K is a commutative ring of characteristic pm while group G contains p-elements, 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 p-group.
      
It is shown that the ring R is a right cogenerator if and only if in the ring of endomorphisms of any free R-module, r(ι(J))=J for all finitely generated right ideals J.
      
A space is strongly symmetrizable if and only if it is a pseudo-open II-image 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 pseudo-injective 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 P-compact and the metric projection of X onto M is upper semicontinuous.
      
The main result: there exists a freef-module over an o-moduleRM with cone pm if and only if pM is half-closed (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 semi-Fredholm and ind (T-λI)>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 quasi-Frobenius and generalized uniserial.
      
 

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