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
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
The fingerprint template is transformed into non-invertible
forms, called cancellable templates, to provide revocability as well as privacy to the fingerprint data.