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The inverse problem of the calculus of variations was first studied by Helmholtz in 1887 and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations. This book looks at second-order differential equations and asks if they can be written as Euler-Lagrangian equations. If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi-Civita for some Riemann metric.

To solve the inverse problem, the authors use the formal integrability theory of overdetermined partial...

A collection of lectures on functional analysis and applications. Areas addressed include: sets, spaces and functions; integrals; topological spaces; spaces of operators and functionals; linear operators; spectral theory of linear operators; and nonlinear problems of functional analysis.

This book is intended for those having only a moderate background in mathematics, who need to increase their mathematical knowledge for development in their areas of work and to read the related mathematical literature. The material covered, which includes practically all the information on functional analysis that may be necessary for those working in various areas of applications of mathematics, as well as the simplicity of presentation, differentiates this book from others. About 300 examples and more than 500 problems are provided to help readers understand and master the theories...

A presentation of the general theory and basic methods of linear and nonlinear stochastic systems (StS) - in other words, dynamical systems described by stochastic finite - and infinite-dimensional differential, integral, integrodifferential and difference equations. The general StS theory is based on the equations for characteristic functions and functionals. The text outlines StS structural theory, including direct numerical methods, methods of normalization, equivalent linearization and parametrization of one- and multi-dimensional distributions, based on moments, quasimoments,...