To calculate the optimality conditions of the following problem [mathematical expression not reproducible], where C ([[OMEGA].sub.0]) is the set of admissible forms obtained by

diffeomorphism, the cost function J([OMEGA]) is the compliance defined by

An optimized framework with nonrigid registration and the

diffeomorphism strategy was used, a similarity energy function based on grey value was designed, and termination conditions were set to stop iterations adaptively.

Further, we assume that there exists a

diffeomorphism h from [K.sub.1] to [K.sub.2] = h([K.sub.1]), defined by

And this turns out to be true, since black hole entropy can be shown to be the Noether charge of the

diffeomorphism symmetry [78].

Then there is a

diffeomorphism h : [[summation].sub.s] [right arrow] M, having the form h(0, y) = (a(y),y), where [alpha]'(u) > 0 at u = a(y), that maps the flow of the sliding vector field [Z.sub.s] to the flow of the reduced problem on M.

It is clear that det D[phi](u, v, [tau]) = (1/q)(a + [K.sup.2][u.sup.2]) > 0, and [phi] is a

diffeomorphism [14].

A solution to the Ricci flow is called Ricci soliton if it moves only by a one parameter group of

diffeomorphism and scaling.

Technically, the Smale's construction involves a planar

diffeomorphism, acting on a square, whose image has the shape of a horseshoe that crosses the square in a suitable manner (see [43, 54, 55] for the mathematical details).

Lim, "Feedback linearisation of time-varying nonlinear systems via time-varying

diffeomorphism," IEE Proceedings: Control Theory and Applications, vol.

If the map is a

diffeomorphism on R, the sets [W.sup.s]([bar.x]) and [W.sup.u]([bar.x]) are the stable and unstable manifolds of [bar.x].

If [rho] < n, then a

diffeomorphism restricted to a neighborhood N can be built as