By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This booklet will deliver the sweetness and enjoyable of arithmetic to the school room. It bargains critical arithmetic in a full of life, reader-friendly variety. integrated are routines and plenty of figures illustrating the most thoughts.

The first bankruptcy provides the geometry and topology of surfaces. between different issues, the authors speak about the Poincaré-Hopf theorem on severe issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses quite a few features of the concept that of measurement, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of three-d manifolds. specifically, it truly is proved that the 3-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a chain of lectures given through the authors at Kyoto college (Japan).

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**Sample text**

Notice that the condition imposes no restriction on how large r can be. Many functions that arise in interesting analytic problems turn out to satisfy "growth conditions" which, as IIx II becomes large enough, prevent IIf(x) II from growing at the same rate, so that eventually IIx II ~ II f (x) II, which produces the Leray-Schauder condition in this special form . The result that replaces the requirement in Schauder 's theorem, of a convex set carried to itself, by the Leray-Schauder boundary condition is really just a corollary of the Schauder theorem.

Then K is convex since it the cartesian product of all the closed intervals {-~, ~] . An open subset of K in the metric topology on 12 is open in the product topol6g/on K. 5 to conclude that the "Hilbert cube" K has the fpp. In the sort of analytic problems we will be interested in, there is a map f that takes a normed linear space X into itself, and the solution to the analytic problem is a fixed point of f . However, the Schauder theorem turns out to be awkward to use in 4 . Schauder Fixed Point Theory 27 Figure S.

5. The Forced Pendulum 33 Now let's suppose that L was not just a linear function but actually an isomorphism with a (continuous) inverse L -I . Then we could rewrite the differential equation Ly = y " = f(t , y, l) as y = L -I f(t, y, y'). This really would be a fixed point problem because we could define a function S by setting S( y) = L -I f(t, y, y') and then a solution to the differential equation would be described by y = S(y), that is, a fixed point of S. I can't really get away with all that for two reasons.