Maximum Principles and Principal Eigenvalues Essay

40264 Words Apr 24th, 2016 162 Pages

During the past century, the impact of mathematics on humanity has been more tremendous than ever since Galileo's agonizing fight against the old establishment and the revolution which physics experienced after Newton's subsequent synthesis.
At the beginning of the last century, mathematical ideas and techniques were spread to theoretical and applied physics by the influence of two of the greatest mathematicians of all times, D. Hilbert and H. Poincar6, being then at the zenith of their careers. Their ability to establish very deep at first glance often hidden connections between a priori separated branches of science convinced physicists to adopt and work with the most powerful existing mathematical tools. Whereas the
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In the present era of over-active industrialization of science, propaganda, and the explosive manipulation of the social and personal basis of science, I believe that we find ourselves in such a period of danger. In our time of mass media, the call for reform, as a result of propaganda, can just as easily lead to a narrowing and choking as to a liberating of i We follow the translation of C. Reid in Hilbert (Springer, New York, 1996, p. 220).


J. Ferrera, J. L@ez-G6mez, E R. Ruiz del Portal

mathematical knowledge. That applies, not only to research in the universities, but also to the instruction in the schools. The danger is that the combined forces so press in the direction of abstraction that only that side of the great Hilbertian tradition is carried on.
Living mathematics rests on the fluctuation between the antithetical powers of intuition and logic, the individuality of 'grounded' problems and the generality of farreaching abstractions. We ourselves must prevent the development being forced to only one pole of the life-giving antithesis.
Mathematics must be cherished and strengthened as a unified, vital branch in the broad river of science; it dares not trickle away in the sand.
Hilbert has shown us through his impressive example that such dangers are easily preventable, that there is no gap between pure and applied mathematics, and that between

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