Submanifolds

This e qnession for the propagators is still exact, as long as, the principal sub-manifold h and its complement sub-manifold h arc complete, and the characteristics of the propagator is reflected in the construction of these submanifolds (47,48). It should be noted that a different (asymmetric) metric for the superoperator space, Eq. (2.5), could be invoked so that another decoupling of the equations of motion is obtained (62,63,82-84). Such a metric will not be explored here, but it just shows the versatility of the propagator methods. 

Figure 11.3 Two microchannel layers which form a complete flow circuit including a micromanifold region and connecting channels. The bottom layer is placed on the first to create a common plenum that connects to six submanifolds. Each submanifold connects to 12 parallel microchannels.

Figure 11.8 Micromanifold thermal map from a 0.5-m wide microchannei piate which contains six submanifoids, where each submanifold feeds 12 parallel channels.

Figure 11.19 Manufacturing scale-up device with submanifolds for passive flow distribution to feed thousands of parallel microchannels. Flow is mapped using ranges to show variations from a normalized value.

If space X is an H-dimensional differentiable manifold and if 7 is a subset ofX, then 7 is called an m-dimensional submanifold of X if the following additional conditions hold for 7 ... 

Obviously, the pairs (j4o,Bo) and (A, G) must be stabilizable and detectable, respectively. As we can see, controller (22) has the form of (5) and does not contain the mappings U (/x) and F (/x) thus, although the initial condition for 2 t) is not exactly known, the immersion observer (second expression in (22)) estimates the correct steady-state input and as a result, the controller is capable to drive the system towards the correct zero-error submanifold in spite of parametric variations. It can be seen from the first equation in (22) that as e t) approaches asymptotically zero, so does z. Notice also that the dynamics of Z2 is similar to immersion (21). It is important to point out that this design procedure does not require the exact calculation of mappings II (/x) and F (/x), but it suffices only to know the dimension of matrix S. 

Remark 3. Equation (32) is known as the Francis-Isidori-Byrnes equation (FIB) [8] and is the nonlinear version of equation (10) used to find the subset Z on the Cartesian product R x called, so far, the zero tracking error submanifold. 

The mapping Xss = tt to, p) represents the steady state zero output submanifold and Uss = 7 eo,p) is the steady state input which makes invariant this steady state zero output submanifold. Condition (48) expresses the fact that this steady state input can be generated, independently of the values of the parameter vector p, by the linear dynamic system... 

Equations (52)-(54) represent the steady state zero output submanifold. Once the mapping Xgs = tt u>, fi) is obtained, we need to find the mapping Wss = 7 (t, A ) in order to apply the control strategy. These mappings are... 

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

The exponential map gives a T-equivariant isomorphism between a neighborhood of 0 G T X and a neighborhood oi x G X. This shows that is a submanifold of X and that the tangent space at x is given by = F(0). Moreover, / is approximated around X by the map... 

This shows that the Hessian of / is positive definite (resp. negative definite) on (resp. N ). Therefore / is non-degenerate in the sense of Bott, i.e. the set of critical points is a disjoint union of submanifolds of X, and the Hessian is non-degenerate in the normal direction at any critical point. We put = dim N = 2 dime N which is the index of / at the critical manifold Cj,. Note that the index is always even in this case. 

The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. 

Let y be an arbitrary point in Since there exists a limit limt >oo 0t(y) G ipt maps C -equivariantly a neighborhood U of y into a neighborhood U considered above for sufficiently large t G E. This shows that fl U is also a smooth submanifold of dimension I dime X. Moreover, since it follows that is a Lagrangian submanifold... 

Remark 7.2. In the above argument, we assumed that the C -action on a holomorphic symplectic Kahler manifold X satisfies iploJc = tujc for t E C. This is possible only when X is non-compact. The reason is as follows. If X is compact there exists a critical manifold corresponding to the maximum of the Morse function for which the unstable manifold is open submanifold of X, but this contradicts the propostion which asserts that every unstable manifold is Lagrangian. 

Proof. By the argument above, we only need to calculate the index of the fixed point set Since S T, is a Lagrangian submanifold, we have... 

Let Ml, M2 be oriented smooth (possibly non-compact) manifolds of dimensions di, 2 respectively. Take a submanifold Z in Mi x M2 such that... 

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