Two site-specific recombinases are primarily involved in determining whether the bacteria are Phase-OFF or Phase-ON by influencing the position of a fimS

invertible element that contains the promoter for the structural gene, fimA.

Now since f is bounded below and bounded above on [OMEGA] it is an

invertible element o [H.sup.[infinity]]([OMEGA]) and so the operator [M.sub.[psi]] is invertible on H because M(H) = [H.sup.[infinity]]([OMEGA]).

Site I is present downstream of FimE gene while site II is present within

invertible element. IHF can introduce DNA bending and before recombination event is responsible for aligning inverted repeats present at the terminal sites of fim element.

This family consists of powers of the S-fixed central

invertible element [z.sub.2],

If 2e is

invertible element in R with the inverse ([2e).sup.-1], then a finite n-symmetric set A [subset or equal to] R is of the above form or of the form

They also proved that if f is harmonious labeling of a graph G of size q, then so is af + b labeling, where a is an

invertible element of [Z.sub.q] and b is any element of [Z.sub.q].

If [E.sub.N](p) is not an

invertible element, then we assume that there exist positive scalars [[lambda].sub.i], with [SIGMA] [[lambda].sub.i] = 1, and unitaries [u.sub.i] [member of] M identified with [u.sub.i] [cross product] 1 such that

Then I contains an

invertible element of R, and so I = R = [M.sub.t](E) and c[[r.sub.1], [r.sub.2]].sup.m] [member of] Z(R), for all [r.sub.1], [r.sub.2] [member of] R.

To determine the orientation of the fimS

invertible element, previously described PCR techniques were used and products visualized with FOTO/Analyst PC Image Software [41, 45].

More generally, if B is a unital complex Banach algebra, A is a closed semisimple commutative subalgebra of B containing the unit of B and [x.sub.0] is an

invertible element of B with [parallel][x.sub.0][parallel] = [parallel][x.sup.-1.sub.0] [parallel] = 1 (for example unitary elements in C*-algebras satisfy these conditions), then for the complex normed space X = A[x.sub.0], the element [x.sub.0] [member of] X has the mentioned property.

Every graded connected bialgebra B is in fact a Hopf algebra [5]--this means, by definition, that the identity map id : B [right arrow] B is an

invertible element in the convolution algebra L(B, B).

Where [PHI] : A([G.sub.1]) [right arrow] A([G.sub.2]) is a homomorphism and [tau] is an

invertible element in the Fourier-Stieltjes algebra of [G.sub.2].