# Introduction to Microlocal Analysis (L'Enseignement by Masaki Kashiwara

By Masaki Kashiwara

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Extra resources for Introduction to Microlocal Analysis (L'Enseignement Mathématique, # 32)

Sample text

E A C G is a mapping (operator): H X A ~ K, where K is a Banach space, too. The Banach spaces are function spaces, where H contains only functions that satisfy the boundary conditions. 32) H = {u E C2Iu'(0) = u'(L) = O} contains twice differentiable functions and K = {u E CO} continuous ones (Appendix A). ]RI. 3 Definitions of stability As already mentioned in the Introduction, we must distinguish between two basically different concepts. On the one hand, one can investigate how a specific solution of a fixed system behaves under small perturbations of its initial conditions.

4). 5) < 7r 2 t > 0. n7rX, n 8u 8t{x,0) = n O~x~l. 2 Time discrete system If the time variable takes discrete values instead of continuous ones, a discrete flow or, as it is also called, a point mapping P : IRn ---+ IRn is obtained. 6) Point mappings occur quite naturally for impulsively loaded systems. A simple example is the plane rigid pendulum with impulsive loading illustrated in Fig. 2 ([61 D. The equation of motion is d21jJ I dt 2 d1jJ + bdj + c1jJ + [00 PoL m~oo 8{t - 1. mr) sm 1jJ = 0.

In Fig. 6a the deformation from the position 1/;1 = 1/;2 = 0 to 1/;1 :j:. 0 and 1/;2 :j:. 0 is done in two steps. First only rod 1 is rotated about the angle 1/;1 and 1/;2 is kept to zero. This results in the work Al = PC(l - cos 1/;1). Then 1/;1 is kept fixed and rod 2 is rotated about the angle 1/;2. 2 Statical systems contribute to the work because the load is displaced perpendicularly to its direction. 6b no work is done at all by P since P is always displaced perpendicularly to its current direction.