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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A geometric approach to an equation of J. D’Alembert
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by A. Pràstaro and Th. M. Rassias PDF
Proc. Amer. Math. Soc. 123 (1995), 1597-1606 Request permission

Abstract:

By using a geometric framework of PDE’s we prove that the set of solutions of the D’Alembert equation $( \ast )(\frac {{{\partial ^2}\log f}}{{\partial x\partial y}}) = 0$ is larger than the set of smooth functions of two variables $f(x,y)$ of the form $( \ast \ast )f(x,y) = h(x) \bullet g(y)$. This agrees with a previous counterexample by Th. M. Rassias given to a statement by C. M. Stéphanos. More precisely, we have the following result. Theorem. The set of 2-dimensional integral manifolds of PDE $( \ast )$ properly contains the ones representable by graphs of 2-jet-derivatives of functions $f(x,y)$ expressed in the form $( \ast \ast )$. A generalization of this result to functions of more than two variables is sketched also by considering the equation $(\frac {{{\partial ^n}\log f}}{{\partial {x_1} \cdots \partial {x_n}}}) = 0$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1597-1606
  • MSC: Primary 58A15; Secondary 58G99
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1232143-5
  • MathSciNet review: 1232143