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Register a new userFields with several commuting derivations
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Authors
David Pierce
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For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.
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Several authors have conjectured that Conways field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.
We consider all pure or mixed states of a quantum many-body system which exhibit the same, arbitrary but fixed measurement outcome statistics for several commuting observables. Taking those states as initial conditions, which are then propagated by the pertinent Schrodinger or von Neumann equation up to some later time point, and invoking a few additional, fairly weak and realistic assumptions, we show that most of them still entail very similar expectation values for any given observable. This so-called dynamical typicality property thus corroborates the widespread observation that a few macroscopic features are sufficient to ensure the reproducibility of experimental measurements despite many unknown and uncontrollable microscopic details of the system. We also discuss and exemplify the usefulness of our general analytical result as a powerful numerical tool.
DAquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $omega_1$-like models of PA.
We continue the study of a class of topological $mathcal{L}$-fields endowed with a generic derivation $delta$, focussing on describing definable groups. We show that one can associate to an $mathcal{L}_{delta}$ definable group a type $mathcal{L}$-definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.
The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent achievements on equipping them with a derivation, which is done by proving that surreal numbers can be seen as transseries and by finding the `simplest structure of H-field, the abstract version of a Hardy field. All the latter notions and their context are also addressed, as well as the universality of the resulting structure for surreal numbers.
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