By Vladimir Pestov
The "infinite-dimensional teams" within the identify confer with unitary teams of Hilbert areas, the endless symmetric team, teams of homeomorphisms of manifolds, teams of changes of degree areas, and so on. The ebook provides an method of the examine of such teams in line with principles from geometric useful research and from exploring the interaction among dynamical houses of these teams, combinatorial Ramsey-type theorems, and the phenomenon of focus of degree. The dynamics of infinite-dimensional teams is especially a lot not like that of in the community compact teams. for example, each in the neighborhood compact crew acts freely on an appropriate compact house (Veech). in contrast, a 1983 consequence via Gromov and Milman states that each time the unitary team of a separable Hilbert area constantly acts on a compact area, it has a standard mounted element. within the e-book, this new fast-growing conception is outfitted strictly from well-understood examples up. The ebook has no shut counterpart and relies on contemporary study articles. even as, it really is prepared in an effort to be quite self-contained. the subject is basically interdisciplinary and should be of curiosity to mathematicians operating in geometric useful research, topological and ergodic dynamics, Ramsey conception, good judgment and descriptive set concept, illustration idea, topological teams, and operator algebras
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Extra info for Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon
Example text
We then ask what we should do. We are not asking what to do in all possible situations, just what to do based on our present state. If there is no memory in a model, then there is no essential difference between steady-state and transient modeling. But if there is memory, which is normally the case for important decisions, stochastic programming is primarily about transient modeling. We are where we are, be that good or bad, and we want to know what to do. Even optimal decisions might leave us worse off than we had been before we made the decisions if constraints are tough and resources limited.
If you were considering some major changes in procedures or some major investment that would change the whole portfolio, this problem might be inherently two-stage. Random variables: Market developments and success of individual projects. Objective function: Here you wish to model risk aversion probably caused by some of the projects having correlated profits. If all divisions and projects had uncorrelated profits, there would be no risk management problem to attend to. Why not deterministic? Risk management makes little sense in a deterministic world.
1 Building an Oil Platform For a concrete example, assume you are consulting on the construction of an oil platform. One of the issues relates to the size of the platform deck. 8 Alternative Approaches 25 The bigger the deck, the larger the platform legs, and the more expensive the platform. By adding extra deck space, you buy an option, however: you make it possible later on to add more equipment. Should the field produce more sand than expected, you could add an extra unit for sand removal. Should the platform produce more oil than expected, you could add a production unit, and maybe you would end up producing so much water that you would need an extra unit for that.