ANALYTIC HYPOELLIPTICITY FOR SUMS OF SQUARES IN THE PRESENCE OF SYMPLECTIC NON TREVES STRATA

2019 ◽  
Vol 19 (6) ◽  
pp. 1877-1888 ◽  
Author(s):  
Antonio Bove ◽  
Marco Mughetti
Keyword(s):  
Critical Points ◽  
Sums Of Squares ◽  
Characteristic Variety ◽  
The Real ◽  
Real Analytic ◽  
Funct Anal ◽  
Analytic Regularity ◽  
Analytic Hypoellipticity

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

scholarly journals On the Construction of Large Algebras Not Contained in the Image of the Borel Map

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Céline Esser ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Analytic Functions ◽  
General Setting ◽  
Smooth Functions ◽  
The Real ◽  
Cauchy Product ◽  
Real Analytic Functions ◽  
Borel Map ◽  
Real Analytic ◽  
The Stability

AbstractThe Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

The Dynamics of a Piecewise Linear Map and its Smooth Approximation

1997 ◽  
Vol 07 (02) ◽  
pp. 351-372 ◽  
Author(s):  
D. Aharonov ◽  
R. L. Devaney ◽  
U. Elias
Keyword(s):  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Invariant Curves ◽  
Smooth Approximation ◽  
Linear Map ◽  
Island Structure ◽  
The Real ◽  
Piecewise Linear Map ◽  
Real Analytic ◽  
Area Preserving

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

scholarly journals Analysis of the atmosphere behaviour in the proximities of an orographic obstacle

1995 ◽  
Vol 2 (1) ◽  
pp. 30-48 ◽  
Author(s):  
E. Hernández ◽  
J. Díaz ◽  
L. C. Cana ◽  
A. García
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Critical Points ◽  
Complex Terrain ◽  
Laboratory Experiments ◽  
Real Experiment ◽  
The Real ◽  
Linear And Nonlinear ◽  
Atmospheric Variables

Abstract. The atmospheric behaviour near an orographic obstacle has been thoroughly studied in the last decades. The first papers in this field were mainly theoretical, being more recent the laboratory experiments which represented that behaviour in ideal conditions. The numerical simulations have been addressed lately thanks to the development of computers. But the study of meteorology in complex terrain has lacked experiments in the atmosphere to understand the real influence the relief has on it. In this paper the problem has been considered from the last perspective, and so, seasons of measure of the atmospheric variables within the boundary layer have been organized with the goal of checking existing theories and bringing right conclusions from real experiment in the atmosphere. Controverted aspects of linear and nonlinear theories, as the location of critical points upwind and downwind of an orographic obstacle, will be analyzed. The results obtained show a large adequacy between the forecasted behaviour and the experimentally detected.

scholarly journals Extension sets for real analytic functions and applications to Radon transforms

1996 ◽  
Vol 19 (4) ◽  
pp. 625-632
Author(s):  
Vernor Arguedas ◽  
Ricardo Estrada
Keyword(s):  
Entire Function ◽  
Analytic Functions ◽  
Radon Transforms ◽  
Support Theorem ◽  
The Real ◽  
Real Analytic Functions ◽  
Small Set ◽  
Real Analytic

The real analytic character of a functionf(x,y)is determined from its behavior along radial directionsfθ(s)=f(scosθ,ssinθ)forθ∈E, whereEis a “small” set. A support theorem for Radon transforms in the plane is proved. In particular iffθextends to an entire function forθ∈Eandf(x,y)is real analytic inℝ2then it also extends to an entire function inℂ2.

Sign in / Sign up

Export Citation Format

Share Document