Consequently, system (14) is right

invertible by Definition 7.

However, as parameters of each layer of Circulant Rainbow should satisfy the parameter constraints of Circulant UOV to prevent HighRank attack and make L easily

invertible, this makes the public key size of Circulant Rainbow 1.5 times larger than regular Rainbow for 80 bit security over GF(31).

(2) If A [member of] A', B, C [member of] A with B [greater than or equal to] C [greater than or equal to] [theta] and I - A [member of] A' is

invertible, then [(I - A).sup.-1] B [greater than or equal to] [(I - A).sup.-1]C.

coli cells expressing these plasmids were mixed with mannosecoated Sepharose and the orientation of the

invertible element followed after 0 h, 4 h, and 24 h postmixing.

([s.sub.j], [s.sub.j+1], [i.sub.j]/[o.sub.j]) is an IDOS if and only if every transition ([s.sub.k], [s.sub.k+1], [i.sub.k]/[o.sub.k]) (2 [less than or equal to] k [less than or equal to] j) is

invertible.

Now we show that if the IRK is such that its coefficient matrix A is

invertible, then the system (3.6a), (3.6b), (3.6c) is reduced by explicitly solving for [K.sub.i], i = 1,2,..., s.

Let [mathematical expression not reproducible] and [mathematical expression not reproducible] are a full rank [n.sub.y] < [n.sub.u] where M is right

invertible and [psi] is selected such that [[psi].sub.opt] = arg min [[summa].sup.N.sub.i=0] [[u.sup.T.sub.j+1] (i)[u.sub.j+1](i)].

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].

(i) a function [x.sub.j] = [[PSI].sub.j](t) is

invertible near [x.sub.j] = [x.sup.0.sub.j] = [[PSI].sup.0.sub.j]([[theta].sub.l]) for [x.sub.j] [less than or equal to] [x.sup.0.sub.j] or/and [x.sub.j] [greater than or equal to] [x.sup.0.sub.j], and the one sided derivative [mathematical expression not reproducible] or/and [mathematical expression not reproducible] is equal to zero, respectively,

Site-specific recombining sequences are short and recombination requires single site of reaction within recombining sequences.10 In a number of instances, site-specific inversion is a simple ON to OFF switching of genes positioned within or neighbouring

invertible region.

Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of

invertible bimodule categories.

Conversely, let U [member of] GL(m, R) and V [member of] GL(n, R) be

invertible matrices such that