Calculus of variations

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  • This book introduces a variety of problem statements in classical optimal control, in optimal estimation and filtering, and in optimal control problems with non-scalar-valued performance criteria. Many example problems are solved completely in the body of the text. All chapter-end exercises are sketched in the appendix. The theoretical part of the book is based on the calculus of variations, so the exposition is very transparent and requires little mathematical rigor.

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  • Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief, in constructing and exploiting natural differentiable structures on abstract probability spaces; in other words, Stochastic Calculus of Variations proceeds from a merging of differential calculus and probability theory. As optimization under a random environment is at the heart of mathematical finance, and as differential calculus is of paramount importance for the search of extrema, it is not surprising that Stochastic Calculus of Variations appears in mathematical finance.

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  • The mathematical modeling of microstructures in solids is a fascinating topic that combines ideas from different fields such as analysis, numerical simulation, and materials science. Beginning in the 80s, variational methods have been playing a prominent rˆole in modern theories for microstructures, and surprising developments in the calculus of variations were stimulated by questions arising in this context.

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  • In the quest for knowledge, it is not uncommon for researchers to push the limits of simulation techniques to the point where they have to be adapted or totally new techniques or approaches become necessary. True multiscale modeling techniques are becoming increasingly necessary given the growing interest in materials and processes on which large-scale properties are dependent or that can be tuned by their low-scale properties. An example would be nanocomposites, where embedded nanostructures completely change the matrix properties due to effects occurring at the atomic level.

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  • This new text, intended for the senior undergraduate finite element course in mechanical, civil and aerospace engineering departments, gives students a solid, practical understanding of the principles of the finite element method within a variety of engineering applications. Hutton discusses basic theory of the finite element method while avoiding variational calculus, instead focusing upon the engineering mechanics and mathematical background that may be expected of senior engineering students.

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  • Poincar´ made the first attempt in 1896 on applying variational calculus e to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions. Little progress had been made on resolving this difficulty until a recent breakthrough by Chenciner and Montgomery. Afterward, variational methods were successfully applied to the N -body problem to construct new classes of solutions.

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  • This book collects the lecture notes of two courses and one mini-course held in a winter school in Bologna in January 2005. The aim of this school was to popularize techniques of geometric measure theory among researchers and PhD students in hyperbolic differential equations. Though initially developed in the context of the calculus of variations, many of these techniques have proved to be quite powerful for the treatment of some hyperbolic problems.

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  • It is the space of ordinary human experience and the starting point of our geometric intuition. Studied for two-and-a-half millenia, it has been the object of celebrated controversies, the most famous concerning the minimum number of properties necessary to define it completely.

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  • We have attempted to explain the concepts which have been used and developed to model the stochastic dynamics of natural and biological systems. While the theory of stochastic differential equations and stochastic processes provide an attractive framework with an intuitive appeal to many problems with naturally induced variations, the solutions to such models are an active area of research, which is in its infancy. Therefore, this book should provide a large number of areas to research further.

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  • This paper introduces to the calculus of regular expressions a replace operator and defines a set of replacement expressions that concisely encode alternate variations of the operation. Replace expressions denote regular relations, defined in terms of other regular expression operators. The basic case is unconditional obligatory replacement. We develop several versions of conditional replacement that allow the operation to be constrained by context O.

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