Differentiability properties of log-analytic functions

Kaiser, Tobias; Andre, Opris,

doi:10.1216/rmj.2022.52.1423

Cited by 4 publications

(13 citation statements)

References 12 publications

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“…The interaction between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots was observed in [11]. The derivative of a log-analytic function is log-analytic, as demonstrated in [12]. That study showed that log-analytic functions have strong quasianalytic properties.…”

Section: Literature Review Of Studies Of C ∞ -Rings and Vector Fieldsmentioning

confidence: 86%

“…In particular, on page 37, we see an algebraic approach of the definition of smooth manifolds. There are recent publications in 2022 and 2023 that provide further directions and constructions, such as [11][12][13]. The interaction between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots was observed in [11].…”

Section: Literature Review Of Studies Of C ∞ -Rings and Vector Fieldsmentioning

confidence: 98%

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Algebraic Structures on Smooth Vector Fields

Alkinani,

Alghamdi

2023

Symmetry

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The aim of this work is to investigate some algebraic structures of objects which are defined and related to a manifold. Consider L to be a smooth manifold and Γ∞(TL) to be the module of smooth vector fields over the ring of smooth functions C∞(L). We prove that the module Γ∞(TL) is projective and finitely generated, but it is not semisimple. Therefore, it has a proper socle and nonzero Jacobson radical. Furthermore, we prove that this module is reflexive by showing that it is isomorphic to its bidual. Additionally, we investigate the structure of the Lie algebra of smooth vector fields. We give some questions and open problems at the end of the paper. We believe that our results are important because they link two different disciplines in modern pure mathematics.

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Section: Literature Review Of Studies Of C ∞ -Rings and Vector Fieldsmentioning

confidence: 86%

Section: Literature Review Of Studies Of C ∞ -Rings and Vector Fieldsmentioning

confidence: 98%

Algebraic Structures on Smooth Vector Fields

Alkinani,

Alghamdi

2023

Symmetry

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“…, b s are log-analytic and therefore the preparation keeps log-analyticity (which cannot be obtained by using the preparation theorem from [9]). Consequently if g(t) := lim x↘0 f (t, x) exists for a log-analytic f : R n × R → R, (t, x) → f (t, x), we see that g is also log-analytic (compare with Theorem 3.1 of Kaiser-Opris [8]). In light of this consideration important differentiability properties for log-analytic functions like non-flatness or a parametric version of Tamm's theorem could be established in [8].…”

Section: Andre Oprismentioning

confidence: 89%

On preparation theorems for R<sub>an, exp</sub>-definable functions

Andre

2023

J. Log. Anal.

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In this article we give strong versions for preparation theorems for $\mathbb{R}_{\textnormal{an,exp}}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{\textnormal{an,exp}}$ is the o-minimal structure generated by all restricted analytic functions and the global exponential function). By a deep model theoretic fact of Van den Dries, Macintyre and Marker every $\mathbb{R}_{\textnormal{an,exp}}$-definable function is piecewise given by $\mathcal{L}_{\textnormal{an}}(\exp,\log)$-terms where $\mathcal{L}_{\textnormal{an}}(\exp,\log)$ denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. So our idea is to consider log-analytic functions at first, i.e. functions which are iterated compositions from either side of globally subanalytic functions and the global logarithm, and then $\mathbb{R}_{\textnormal{an,exp}}$-definable functions as compositions of log-analytic functions and the global exponential.

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“…We recall the precise definition of a log-analytic function (see Lion and Rolin [5]) and state consequences of preparation results on special sets (compare with [3]).…”

Section: Setting and Preliminariesmentioning

confidence: 99%

“…They are iterated compositions from either side of globally subanalytic functions (see [2]) and the global logarithm. In [3] it was shown that from the point of view of differentiability log-analytic functions behave similarly to globally subanalytic functions. We have strong quasianalyticity and Tamm's theorem hold.…”

Section: Introductionmentioning

confidence: 99%