By David Yang Gao

ISBN-10: 0792361458

ISBN-13: 9780792361459

Inspired by way of useful difficulties in engineering and physics, drawing on quite a lot of utilized mathematical disciplines, this e-book is the 1st to supply, inside a unified framework, a self-contained accomplished mathematical thought of duality for basic non-convex, non-smooth platforms, with emphasis on equipment and purposes in engineering mechanics. issues coated comprise the classical (minimax) mono-duality of convex static equilibria, the attractive bi-duality in dynamical structures, the fascinating tri-duality in non-convex difficulties and the complex multi-duality normally canonical platforms. A almost certainly strong sequential canonical twin transformation process for fixing absolutely nonlinear difficulties is constructed heuristically and illustrated by means of use of many fascinating examples in addition to vast functions in a large choice of nonlinear structures, together with differential equations, variational difficulties and inequalities, restricted worldwide optimization, multi-well section transitions, non-smooth post-bifurcation, huge deformation mechanics, structural restrict research, differential geometry and non-convex dynamical platforms. With awfully coherent and lucid exposition, the paintings fills a tremendous hole among the mathematical and engineering sciences. It indicates how one can use formal language and duality ways to version usual phenomena, to build intrinsic frameworks in numerous fields and to supply rules, ideas and strong equipment for fixing non-convex, non-smooth difficulties bobbing up clearly in engineering and technological know-how. a lot of the ebook comprises fabric that's new, either in its demeanour of presentation and in its study improvement. A self-contained appendix presents a few worthwhile history from straight forward practical research. viewers: The publication might be a useful source for college students and researchers in utilized arithmetic, physics, mechanics and engineering. the entire quantity or chosen chapters is usually suggested as a textual content for either senior undergraduate and graduate classes in utilized arithmetic, mechanics, normal engineering technological know-how and different parts during which the notions of optimization and variational equipment are hired.

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**Sample text**

I=O Pi (x - y) 2' 1 + Pi (x - y) . 6) In the Exercises we show that this function is indeed a metric on X which defines the given topology by using as open balls with centre x and radius r > 0 the sets Bd,r(X) = {y EX: d(y, x) < r}. A HLCTVS X[P] is called metrizable if, and only if, its topology IP can be defined in terms of a metric. Some other special cases are addressed in the Exercises as well. We conclude this section with an example of a complete metrizable HLCTVS which will play an important role in the definition of the basic test function spaces.

Such characterizations will be given in this subsection. Recall: A sequence (Xi)iEN of points xi = (xl, ... , there is an index io, depending on r > 0, such that xi E Br (x) for all i 2: io, or expressed directly in terms of the Euclidean norm, Ix i - xl < r for all i 2: io. Similarly one proceeds in a general HLCTVS V [P] where now however instead of the Euclidean norm I . I all the semi -norms pEP have to be taken into account. 4 Let V [P] be a HLCTVS and (Xi)i EN a sequence in V [Pl Then one says: 1.

6) In the Exercises we show that this function is indeed a metric on X which defines the given topology by using as open balls with centre x and radius r > 0 the sets Bd,r(X) = {y EX: d(y, x) < r}. A HLCTVS X[P] is called metrizable if, and only if, its topology IP can be defined in terms of a metric. Some other special cases are addressed in the Exercises as well. We conclude this section with an example of a complete metrizable HLCTVS which will play an important role in the definition of the basic test function spaces.

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