# non-invertible

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4 shows a bandpass signals and its reconstructed versions from uniform samples at the Nyquist rate (which are non-invertible) for different iteration numbers.
Naimzada, "On a rent-seeking game described by a non-invertible iterated map with denominator," Nonlinear Analysis: Theory, Methods & Applications, vol.
Lemma 1.2 The non-invertible elements of A are those elements of the form X = [[summation].sup.n-1.sub.i=1] [x.sub.i][[epsilon].sup.i].
In this case the algorithm is called non-invertible.
The logistic map is a very common one-dimensional non-invertible model for generating chaotic sequences (for n=1,2,...; and 0<[lambda][less than or equal to]4):
We show that a frame homomorphism h: L [right arrow] M is coz-codense if and only if the induced ring homomorphism Rh: RL [right arrow] RM takes non-invertible elements to non-invertible elements (Lemma 3.3), and that pseudocompact frames are precisely those L for which every finitely generated proper ideal of R x L is fixed (Lemma 3.1).
A natural question, initially discussed in , is whether or not a non-invertible element of Z[[[chi]]] can be factored over Z[[[chi]]].
It should also be noted that there is some freedom in choosing pivots (which variables to substitute or which columns to 'wipe out') and they should be chosen in order that they not be zero (for then there is a division by zero or a non-invertible row replacement, in the substitution and elimination views, respectively).
The third state generates a non-invertible transformation, in some regions of the state space, due to the appearance in its derivatives of the exponential nonlinearities which induce the chaotic behavior.
It was proved earlier that operators (1.1) and (1.2) for cases D = [R.sup.m], D = [R.sup.m.sub.t] are invertible or non-invertible in spaces [L.sub.2]([R.sup.m]), [L.sub.2]([R.sup.m.sub.+]) and [L.sub.2](h[Z.sup.m]), [L.sub.2](h[Z.sup.m.sub.+]) simultaneously [18,22].
The second chapter, on higher-dimensional and complex maps, presents exercises for both invertible and non-invertible maps.
The fingerprint template is transformed into non-invertible forms, called cancellable templates, to provide revocability as well as privacy to the fingerprint data.

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