I think p-adic mathematical physics has so far nothing to do with real phenomena. You get in such a way varieties analogous to complex varieties. The following survey articles by Ducros for french readers are excellent:. Well, you can show that an old arithmetic coding algorithm can be reformulated in p-adic terms.
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Proof of this theorem is based on 2-adic numbers. Alexey Ustinov. Oct 17 '16 at Oct 29 '13 at Maurizio Monge.
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See my answer to this MO question mathoverflow. Some problems in math like making sense of quantum field theory are so difficult that one should try to understand as much as possible on simpler toy models where these problems are present but have a clear and clean formulation. See the slides in French of my Colloquium at the University of Lyon for a pedagogical introduction to this circle of ideas. The first part is a survey of axiomatic utility under risk, beginning with expected utility and proceeding to cover the generalisations of expected utility developed over the past twenty years.
The second part is a discussion of non-Archimedean representations of preferences, incorporating a range of boundary effects. The third part is a discussion of a range of problems in insurance economics from the perspective of generalized expected utility models incorporating first-order risk-aversion. The objectives of the first part of the book are modest -- to present an updated version of the numerous surveys already available and to lay the basis for subsequent sections.
The basic approach is closest in spirit to that of Chew and Epstein, classifying the main generalizatons of expected utility as either rank-dependent or betweenness-preserving and then considering various unifying models. Discussion of the empirical evidence is brief and somewhat casual, but is used to draw the conclusion that the bulk of the observed violations of EU may be attributed to certainty effects and, more generally, boundary effects arising when the probability of some outcome falls to zero.
This leads the author to the consideration of 'non-Archimedean' models in which the assumption of continuity is dropped, but which are consistent with EU at 'interior' points roughly speaking, on the interior of any face of the unit simplex. Schmidt briefly discusses arguments for and against the imposition of a continuity argument. One sort of argument relies on appeals to introspection of the form 'Consider the worst possible consequence, and some trivial positive consequence. Would you take any risk of the worst consequence, however small, in order to receive the positive consequence with probability close to 1'.
After one looks at a couple of examples risking execution for a few dollars, risking road death to drive to the video store , it seems that not much more can be said, so Schmidt is right to be brief on this point. On the other hand, I think more attention should be paid to the argument that continuity is a mere mathematical convenience, with no observable outcomes.
To put this argument in context, consider that the original and strongest counterexamples to EU, the original Allais paradox and the coincidence of lottery gambling and insurance, involve large positive or negative outcomes occurring with probabilities of 1 per cent or less. By contrast, few recent experiments have involved such small probabilities and large payoffs, partly because protocols involving some gambles being played 'for real' cannot be credibly implemented if some payoffs exceed the experimenters' budget.
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The question of continuity relevant to the current debate may be rephrased as 'do violations of EU occur near boundaries or exactly on boundaries'. For example, would it affect the sales of lottery tickets if the set of prizes remained the same, but the odds of winning were reduced.
A model based on boundary effects would suggest not, whereas a probability weighting model would suggest that demand for lottery tickets should be quite sensitive to chances of winning. Although this is issue downplayed in his initial discussion of continuity, Schmidt's analysis of non-Archimedean models brings him back to a closely related point in the end.
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After a brief survey of the early literature on lexicographic preferences, Schmidt analyses a model of expected utility with certainty preferences EUCP , in which an expected utility representation holds for non-degenerate lotteries, while a separate valuation function is applied to degenerate lotteries.
By virtue of the expected utility representation each non-degenerate lottery has a well-defined conditional certainty equivalent that may be used in 'folding back' decision trees and other EU operations. Assuming the valuation functions are separately continuous, each non-degenerate lottery also has an unconditional certainty equivalent.