Éz fields Walsberg, Erik; Ye, Jinhe
Journal of algebra,
01/2023, Letnik:
614
Journal Article
Recenzirano
Odprti dostop
Let K be a field. The étale open topology on the K-points V(K) of a K-variety V was introduced in 23. The étale open topology is non-discrete if and only if K is large. If K is separably, real, ...p-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and p-adically closed fields. We introduce and study the class of éz fields: K is éz if K is large and every definable set is a finite union of étale open subsets of Zariski closed sets. This should be seen as a generalized notion of model completeness for large fields. Algebraically closed, real closed, p-adically closed, and bounded PAC fields are éz. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are éz.) We develop the basics of a theory of definable sets in éz fields. This gives a uniform approach to the theory of definable sets across all characteristic zero local fields and a new topological theory of definable sets in bounded PAC fields. We also show that some prominent examples of possibly non-model complete model-theoretically tame fields (characteristic zero t-Henselian fields and Frobenius fields) are éz.
This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit d'Auvergne, Aurillac, France. Articles included ...in this book feature recent developments in various areas of non-Archimedean analysis: summation of p-adic series, rational maps on the projective line over \mathbb{Q}p, non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G-modules with a convex base, non-compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean K the spaces, p-adic Nevanlinna theory and applications, and sub-coordinate representation of p-adic functions. Moreover, a paper on the history of p-adic analysis with a comparative summary of non-Archimedean fields is presented. Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area.
Suppose that R is a local domain with fraction field K. If R is Henselian, then the R‐adic topology over K refines the étale open topology. If R is regular, then the étale open topology over K ...refines the R‐adic topology. In particular, the étale open topology over L((t1,…,tn))$L((t_1,\ldots ,t_n))$ agrees with the Lt1,…,tn$Lt_1,\ldots ,t_n$‐adic topology for any field L and n≥1$n \ge 1$.
Significance Ideal fluids have a conserved quantity—helicity—which measures the degree to which a fluid flow is knotted and tangled. In real fluids (even superfluids), vortex reconnection events ...disentangle linked and knotted vortices, jeopardizing helicity conservation. By generating vortex trefoil knots and linked rings in water and simulated superfluids, we observe that helicity is remarkably conserved despite reconnections: vortex knots untie and links disconnect, but in the process they create helix-like coils with the same total helicity. This result establishes helicity as a fundamental building block, like energy or momentum, for understanding the behavior of complex knotted structures in physical fields, including plasmas, superfluids, and turbulent flows.
The conjecture that helicity (or knottedness) is a fundamental conserved quantity has a rich history in fluid mechanics, but the nature of this conservation in the presence of dissipation has proven difficult to resolve. Making use of recent advances, we create vortex knots and links in viscous fluids and simulated superfluids and track their geometry through topology-changing reconnections. We find that the reassociation of vortex lines through a reconnection enables the transfer of helicity from links and knots to helical coils. This process is remarkably efficient, owing to the antiparallel orientation spontaneously adopted by the reconnecting vortices. Using a new method for quantifying the spatial helicity spectrum, we find that the reconnection process can be viewed as transferring helicity between scales, rather than dissipating it. We also infer the presence of geometric deformations that convert helical coils into even smaller scale twist, where it may ultimately be dissipated. Our results suggest that helicity conservation plays an important role in fluids and related fields, even in the presence of dissipation.
Topological fields with a generic derivation Cubides Kovacsics, Pablo; Point, Françoise
Annals of pure and applied logic,
March 2023, 2023-03-00, Letnik:
174, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We study a class of tame L-theories T of topological fields and their Lδ-extension Tδ⁎ by a generic derivation δ. The topological fields under consideration include henselian valued fields of ...characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has L-open core (i.e., every Lδ-definable open set is L-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of T such as relative elimination of field sort quantifiers, NIP and distality also transfer to Tδ⁎. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of p-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.
This book contains the proceedings of the 14th International Conference on $p$-adic Functional Analysis, held from June 30-July 5, 2016, at the Universite d'Auvergne, Aurillac, France. Articles ...included in this book feature recent developments in various areas of non-Archimedean analysis: summation of p -adic series, rational maps on the projective line over Q p , non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G -modules with a convex base, non-compact Trace class operators and Schatten-class operators in p -adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean Köthe spaces, p -adic Nevanlinna theory and applications, and sub-coordinate representation of p -adic functions. Moreover, a paper on the history of p -adic analysis with a comparative summary of non-Archimedean fields is presented.Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area.
Fusion of defects Bartels, Arthur; Douglas, Christopher L; Henriques, Andr e G
2019., 2019, 2019-10-04, Letnik:
1237
eBook
Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal ...field theories. They introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. The authors' most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
This volume contains the proceedings of the NSF-CBMS Regional Conference on Topological and Geometric Methods in QFT, held from July 31-August 4, 2017, at Montana State University in Bozeman, ...Montana.In recent decades, there has been a movement to axiomatize quantum field theory into a mathematical structure. In a different direction, one can ask to test these axiom systems against physics. Can they be used to rederive known facts about quantum theories or, better yet, be the framework in which to solve open problems? Recently, Freed and Hopkins have provided a solution to a classification problem in condensed matter theory, which is ultimately based on the field theory axioms of Graeme Segal.Papers contained in this volume amplify various aspects of the Freed-Hopkins program, develop some category theory, which lies behind the cobordism hypothesis, the major structure theorem for topological field theories, and relate to Costello's approach to perturbative quantum field theory. Two papers on the latter use this framework to recover fundamental results about some physical theories: two-dimensional sigma-models and the bosonic string. Perhaps it is surprising that such sparse axiom systems encode enough structure to prove important results in physics. These successes can be taken as encouragement that the axiom systems are at least on the right track toward articulating what a quantum field theory is.
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