Every P-convex subset of R2\R^2 is already strongly P-convex

A classical result of Malgrange says that for a polynomial P and an open subset $\Omega$ of $\R^d$ the differential operator $P(D)$ is surjective on $C^\infty(\Omega)$ if and only if $\Omega$ is P-convex. H\"ormander showed that $P(D)$ is surjective as an operator on $\mathscr{D}'(\Omega)$ if and only if $\Omega$ is strongly P-convex. It is well known that the natural question whether these two notions coincide has to be answered in the negative in general. However, Tr\`eves conjectured that in the case of d=2 P-convexity and strong P-convexity are equivalent. A proof of this conjecture is given in this note

Chaos for the Hyperbolic Bioheat Equation

The Hyperbolic Heat Transfer Equation describes heat processes in which extremely short periods of time or extreme temperature gradients are involved. It is already known that there are solutions of this equation which exhibit a chaotic behaviour, in the sense of Devaney, on certain spaces of analytic functions with certain growth control. We show that this chaotic behaviour still appears when we add a source term to this equation, i.e. in the Hyperbolic Bioheat Equation. These results can also be applied for the Wave Equation and for a higher order version of the Hyperbolic Bioheat Equation.The authors are supported in part by MEC and FEDER, Projects MTM2010-14909 and MTM2013-47093-P.Conejero, JA.; Ródenas Escribá, FDA.; Trujillo Guillen, M. (2015). Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems - Series A. 35(2):653-668. doi:10.3934/dcds.2015.35.653S65366835

Distributionally chaotic families of operators on Fréchet spaces

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915

R497K polymorphism in epidermal growth factor receptor gene is associated with the risk of acute coronary syndrome

<p>Abstract</p> <p>Background</p> <p>Previous studies suggested that genetic polymorphisms in the epidermal growth factor receptor (EGFR) gene had been implicated in the susceptibility to some tumors and inflammatory diseases. EGFR has been recently implicated in vascular pathophysiological processes associated with excessive remodeling and atherosclerosis. Acute coronary syndrome (ACS) is a clinical manifestation of preceding atherosclerosis. Our purpose was to investigate the association of the EGFR polymorphism with the risk of ACS. In this context, we analyzed the HER-1 R497K and EGFR intron 1 (CA)_nrepeat polymorphisms in 191 patients with ACS and 210 age- and sex-matched controls in a Chinese population, using a polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP) strategy and direct sequencing.</p> <p>Results</p> <p>There were significant differences in the genotype and allele distribution of R497K polymorphism of the EGFR gene between cases and controls. The <it>Lys </it>allele had a significantly increased risk of ACS compared with the <it>Arg </it>allele (adjusted OR = 1.49, 95% CI: 1.12–1.98, adjusted <it>P </it>= 0.006). However, no significant relationship between the number of (CA)_nrepeats of EGFR intron 1 (both alleles < 20 or any allele ≥ 20) and the risk of ACS was observed (adjusted OR = 0.97, 95% CI: 0.58–1.64, adjusted <it>P </it>= 0.911). Considering these two polymorphisms together, there was no statistically significant difference between the two groups.</p> <p>Conclusion</p> <p>R497K polymorphism of the EGFR gene is significantly associated with the risk of ACS. Our data suggests that R497K polymorphism may be used as a genetic susceptibility marker of the ACS.</p

Chaotic C0C_0-semigroups induced by semiflows in Lebesgue and Sobolev spaces

We give characterizations of chaos for $C_0$-semigroups induced by semiflows on $L^p_\rho(\Omega)$ for open $\Omega\subseteq\R$ similar to the characterizations of hypercyclicity and mixing of such $C_0$-semigroups proved in \cite{kalmes2009hypercyclic}. Moreover, we characterize hypercyclicity, mixing, and chaos for these classes of $C_0$-semigroups on $W^{1,p}_*(I)$ for a bounded interval $I\subset\R$ and prove that these $C_0$-semigroups are never hypercyclic on $W^{1,p}(I)$. We apply our results to concrete first order partial differential equations, such as the von Foerster-Lasota equation