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if and only if
It is proved that the mutual information between a, b, and c can be extracted if and only if the values of the conditional mutual informations I(a : b|c), I(a : c|b), and I(b : c|a) are negligible.
      
Each receiver receives a vector of M sets, where a set at each time instant contains a fixed frequency if and only if the corresponding elementary signal was sent by either some sender or the jammer.
      
In complex two-dimensional Euclidean space, the Hamilton-Jacobi or Schr?dinger equation with a given "nondegenerate" potential is maximally superintegrable if and only if it is separated in more than one coordinate system.
      
It is proved that, except in a few explicitly described cases, two connected tube domains in ?2 are biholomorphically equivalent if and only if they are affinely equivalent.
      
left) regular if and only if for each fuzzy subset f of S we have f ? f2 ° 1 (resp.
      
It is regular if and only if for each fuzzy subset f of S we have f ? f ° 1 ° f, and it is intra-regular if and only if f ? 1 ° f2 ° 1 for each fuzzy subset f of S (where 1 is the greatest element of the set of fuzzy subsets of S).
      
An orthogonal scaling function ?(t) can realize perfect A/D (Analogue/Digital) and D/A if and only if ?(t) is cardinal in the case of scalar wavelet.
      
X* is a KUR space if and only if X is a KUS space, X* is a KUS space if and only if X is a KUR space.
      
As a corollary, it is proved that two automorphisms of order three are conjugate if and only if their fixed point sets are isomorphic.
      
For example, supposing that R is Gsymmetric and is constant on the set of fixed points on S2under G (where Gis a subgroup of O(3)), itis proved that the equation is solvable if and only if Ris positive somewhere.
      
It is proved that a (n, k) linear block code in GF(q) are the good code for error-detection, if and only if its dual code is also.
      
The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ.
      
Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2.
      
We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary.
      
We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>amp;gt;1 and f satisfies the Wecken property.
      
We show that a congruence 3-permutable algebra is congruence meetsemidistributive if and only if it is distributive.
      
We prove M=⊕i=1n Mi is δ-coatomic if and only if each Mi (i=1,…, n) is δ-coatomic.
      
Especially, in the case when the Ricci operator is diagonalizable, a Bochner flat para-K?hlerian manifold is paraholomorphically pseudosymmetric if and only if the Ricci operator has at most two eigenvalues.
      
We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.
      
We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property.
      
 

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