Mutational Analysis A Joint Framework for Cauchy Problems in and Bey
Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications, however, it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions,
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Zbl 1198.37003 Lorenz, Thomas Mutational analysis. A joint framework for Cauchy problems in and beyond vector spaces. (English) Lecture Notes in Mathematics 1996. Berlin: Springer. xiv, 509 p. EUR 89.95/net; £ 81.00; SFR 140.00 (2010). ISBN 978-3-642-12470-9/pbk; ISBN 978-3-642-124716/ebook http://dx.doi.org/10.1007/978-3-642-12471-6
0. Introduction. Diverse evolutions come together under the same roof. Some interlocutory examples: A region growing method of image segmentation; image smoothing via anisotropic diffusion; A stochastic differential game without precisely known realization of opponents. Extending the traditional horizon: evolution equation beyond vector spaces: Aubin’s initial notion: regard affine linear maps just as a special type of Elementary Deformations, mutational analysis as an adaptive black box, for initial value problems, the initial problem decomposition and the final link to more popular meanings of abstract solutions, the new steps of generalization; mutational inclusions are given in the introduction. Chapter 1. This chapter contains Extending ordinary differential equations to metric spaces: Aubin’s Suggestion: The key for avoiding linear structures: transitions; the mutation as counterpart of time derivative; feedback leads to mutational equations; proofs for existence and uniqueness of solutions without state constraints; an essential advantage of mutational equations solutions to systems; proof for existence of solutions under state constraints; some elementary properties of the contingent transition set; example: ordinary differential equations in RN ; Example: morphological equations for compact sets in RN ; the Pompein-Hausdorff distance, morphological Transitions, morphological primitives as reachable sets, some examples of morphological primitives, some examples of contingent transition sets, solution to morphological equations; Example: morphological transitions for image segmentation problem, analytical tools of the continuous segmentation problem, solving the continuous segmentation problem, the application to computer images; Example: Modified morphological equations via bounded one-sided Lipschitz maps. Chapter 2. This chapter deals with adapting mutational equations to examples in vector spaces: The topological environment, specifying transition and mutation. Solutions to mutational equations: continuity with respect to initial states and right-hand side, limits of point wise converging solutions: convergence theorem, existence of mutational equations without state constraints, convergence theorem and existence for systems, existence for mutational equations with delay, existence under state constraints for finite index set. Examples: semi linear evolution equations in reflexive Banach spaces, nonlinear transport equations for random measures, the dual matrix on random measures, linear transport equations induce transitions, conclusion about nonlinear transport equations; A structured population model with random measures, linear population model, conclusions about the full
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