The well-posedness of direct and inverse problems for parabolic equations with

involution was considered in [3-5].

and Torrence B.,

Involutions fixing the union of odd-dimensional projective spaces, Canadian Math.

By induction and a previous case, [[pi].sup.e.sub.1] and c([[pi].sup.o.sub.2]) are 3412-avoiding

involutions, so c([beta]) = (1 [??] [[pi].sup.e.sub.1] [??] 1) [direct sum] c([[pi].sup.o.sub.2]) is also a 3412-avoiding

involution.

Now we give a similar Lemma for fixed point free

involutions. In this case, we slightly modify the two concepts of inflation and deflation.

Delayed in uterine

involution, chronic metritis and poor conception rate are common sequelae to retention of foetal membranes (Roberts, 1971).

Such matrices define a unique

involution on [X.sub.0](N) that is called the Atkin Lehner

involution.

Note that, if [iota] is an r-admissible

involution and if [iota](b) = c > b and [iota](z) = z, then z < b or z > c (this is easily proved by induction on [absolute value of Z]).

Algorithm 3.3 (InvolutionJumper) Input: G = (X) and an

involution x [member of] G.

If A is a central simple k-algebra with a non-trivial

involution, then either A [equivalent] [M.sub.n](D), where D is a quaternion division algebra, or A is trivial on k.

Thus the composite operation of toggling the elements of P from left to right is well-defined; Striker and Williams (2012) call this operation promotion, and show that it is conjugate to rowmotion in the toggle group (the group generated by the toggle

involutions).

The topics include the isotopy of links, most knots are wild, infinite cyclic coverings, some curious

involutions of spheres, two complexes that are homeomorphic but combinatorially distinct, uses of the fundamental group, and a unique decomposition theorem for three-manifolds.

In particular, several authors examined the properties of the polynomial [I.sub.n](x) = [[summation].sup.n-1.sub.j=0] [i.sub.n,j], [x.sup.j] where [i.sub.n,j] denotes the number of

involutions on [n] = {1, 2, ..., n} with j descents.