948 research outputs found
Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems
We consider a class of ordinary differential equations describing
one-dimensional analytic systems with a quasi-periodic forcing term and in the
presence of damping. In the limit of large damping, under some generic
non-degeneracy condition on the force, there are quasi-periodic solutions which
have the same frequency vector as the forcing term. We prove that such
solutions are Borel summable at the origin when the frequency vector is either
any one-dimensional number or a two-dimensional vector such that the ratio of
its components is an irrational number of constant type. In the first case the
proof given simplifies that provided in a previous work of ours. We also show
that in any dimension , for the existence of a quasi-periodic solution with
the same frequency vector as the forcing term, the standard Diophantine
condition can be weakened into the Bryuno condition. In all cases, under a
suitable positivity condition, the quasi-periodic solution is proved to
describe a local attractor.Comment: 10 page
Newton's method and Baker domains
We show that there exists an entire function f without zeros for which the
associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic
functions without Baker domains. We also show that there exists an entire
function f with exactly one zero for which the complement of the immediate
attracting basin has at least two components and contains no invariant Baker
domains of N. The second result answers a question of J. Rueckert and D.
Schleicher while the first one gives a partial answer to a question of X. Buff.Comment: 6 page
Borel summability and Lindstedt series
Resonant motions of integrable systems subject to perturbations may continue
to exist and to cover surfaces with parametric equations admitting a formal
power expansion in the strength of the perturbation. Such series may be,
sometimes, summed via suitable sum rules defining functions of the
perturbation strength: here we find sufficient conditions for the Borel
summability of their sums in the case of two-dimensional rotation vectors with
Diophantine exponent (e. g. with ratio of the two independent
frequencies equal to the golden mean).Comment: 17 pages, 1 figur
Y-System and Deformed Thermodynamic Bethe Ansatz
We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe
Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed
TBA is a system of five coupled nonlinear integral equations, which in a
particular case reduces to the Zamolodchikov TBA equation for the 3-state Potts
model. Our method generalizes the Dorey-Tateo analysis of the (monomial) cubic
oscillator. We introduce a Y-system corresponding to the Deformed TBA and give
it an elegant geometric interpretation.Comment: 12 pages. Minor corrections in Section
The Schwarzian derivative and the Wiman-Valiron property
Consider a transcendental meromorphic function in the plane with finitely many critical values, such that the multiple points have bounded multiplicities and the inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that if the Schwarzian derivative is transcendental then the function has infinitely many multiple points, the inverse function does not have a direct transcendental singularity over infinity, and infinity is not a Borel exceptional value. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method
On a Watson-like Uniqueness Theorem and Gevrey Expansions
We present a maximal class of analytic functions, elements of which are in
one-to-one correspondence with their asymptotic expansions. In recent decades
it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.),
that the formal power series solutions of a wide range of systems of ordinary
(even non-linear) analytic differential equations are in fact the Gevrey
expansions for the regular solutions. Watson's uniqueness theorem belongs to
the foundations of this new theory. This paper contains a discussion of an
extension of Watson's uniqueness theorem for classes of functions which admit a
Gevrey expansion in angular regions of the complex plane with opening less than
or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We
present conditions which ensure uniqueness and which suggest an extension of
Watson's representation theorem. These results may be applied for solutions of
certain classes of differential equations to obtain the best accuracy estimate
for the deviation of a solution from a finite sum of the corresponding Gevrey
expansion.Comment: 18 pages, 4 figure
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
This paper concerns the dynamics of polynomial automorphisms of .
One can associate to such an automorphism two currents and the
equilibrium measure . In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
as the unique measure of maximal entropy. Then we show that the measure
has a local product structure and that the currents have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems
Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I
Classical Schur analysis is intimately connected to the theory of orthogonal
polynomials on the circle [Simon, 2005]. We investigate here the connection
between multipoint Schur analysis and orthogonal rational functions.
Specifically, we study the convergence of the Wall rational functions via the
development of a rational analogue to the Szeg\H o theory, in the case where
the interpolation points may accumulate on the unit circle. This leads us to
generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields
asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction,
Section 5 (Szeg\H o type asymptotics) is extende
Recommended from our members
Combined effects of single nucleotide polymorphisms TP53 R72P and MDM2 SNP309, and p53 expression on survival of breast cancer patients.
INTRODUCTION: Somatic inactivation of the TP53 gene in breast tumors is a marker for poor outcome, and breast cancer outcome might also be affected by germ-line variation in the TP53 gene or its regulators. We investigated the effects of the germ-line single nucleotide polymorphisms TP53 R72P (215G>C) and MDM2 SNP309 (-410T>G), and p53 protein expression in breast tumors on survival. METHODS: We pooled data from four breast cancer cohorts within the Breast Cancer Association Consortium for which both TP53 R72P and MDM2 SNP309 were genotyped and follow-up was available (n = 3,749). Overall and breast cancer-specific survival analyses were performed using Kaplan-Meier analysis and multivariate Cox's proportional hazards regression models. RESULTS: Survival of patients did not differ by carriership of either germ-line variant, R72P (215G>C) or SNP309 (-410G>T) alone. Immunohistochemical p53 staining of the tumor was available for two cohorts (n = 1,109 patients). Survival was worse in patients with p53-positive tumors (n = 301) compared to patients with p53-negative tumors (n = 808); breast cancer-specific survival: HR 1.6 (95% CI 1.2 to 2.1), P = 0.001. Within the patient group with p53-negative tumors, TP53 rare homozygous (CC) carriers had a worse survival than G-allele (GG/GC) carriers; actuarial breast cancer-specific survival 71% versus 80%, P = 0.07; HR 1.8 (1.1 to 3.1), P = 0.03. We also found a differential effect of combinations of the two germ-line variants on overall survival; homozygous carriers of the G-allele in MDM2 had worse survival only within the group of TP53 C-allele carriers; actuarial overall survival (GG versus TT/TG) 64% versus 75%, P = 0.001; HR (GG versus TT) 1.5 (1.1 to 2.0), P = 0.01. We found no evidence for a differential effect of MDM2 SNP309 by p53 protein expression on survival. CONCLUSIONS: The TP53 R72P variant may be an independent predictor for survival of patients with p53-negative tumors. The combined effect of TP53 R72P and MDM2 SNP309 on survival is in line with our a priori biologically-supported hypothesis, that is, the role of enhanced DNA repair function of the TP53 Pro-variant, combined with increased expression of the Mdm2 protein, and thus overall attenuation of the p53 pathway in the tumor cells.RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are
- …