By Pierre Deligne

ISBN-10: 038708066X

ISBN-13: 9780387080666

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**Sample text**

Such a jump is also known as a "catastrophe", about which there is a whole theory, the catastrophe theo ry of Thorn, which appears to have interesting applications in widely different fields ([Tl], [Zl]). We shall give another example of this kind, somewhat more complicated, but not offering anything new in principle. now has two "folds" instead of one. would be, say, y = z 4 + cz 2 + xz). The surface F (An equation for such a surface 54 / / / / F E / ..

In the Acta eruditorum of 1696, Johann Bernoulli posed the following problem, whose solution he already k new, and published i n 16 9 7 : given t wo points from A to B A a nd B in a ve r tical p l a n e , fi nd a curve so that a point P which slides from A to B along the curve under gravity takes the shortest possible time. He called the curve which s o lves this problem the brachistoch rone (8pax ta•os xpo vos = shortest time ) . Th e proble m wa s solved by several ma the matic i a n s , some i mmed i a t ely : Ne wto n, Le i bni z , Jacob Be rnoull i and L'Hospital .

F are the For parameter values "out- side" the outline there is exactly one equilibrium state,"inside" there are three. However, the "middle" one of these three is unstable. one continuously varies the control parameters along a curve in When E, it is obvious that the system may suddenly jump from one of the two stable states to the other on crossing the outline. Thus a quantitatively small change gene rates a qualitatively large jump. Such a jump is also known as a "catastrophe", about which there is a whole theory, the catastrophe theo ry of Thorn, which appears to have interesting applications in widely different fields ([Tl], [Zl]).

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