# Qualitative theory of ODE

• D. Batenkov and Y. Yomdin, “Taylor domination, Turán lemma, and Poincaré-Perron sequences,” in Contemporary Mathematics, B. Mordukhovich, S. Reich, and A. Zaslavski, Eds., Providence, Rhode Island: American Mathematical Society, 2016, vol. 659, pp. 1-15.
[Bibtex]
@InCollection{mordukhovich_taylor_2016,
author = {Batenkov, Dmitry and Yomdin, Yosef},
title = {Taylor domination, {Tur{\'a}n} lemma, and {Poincar{\'e}}-{Perron} sequences},
booktitle = {Contemporary {Mathematics}},
publisher = {American Mathematical Society},
year = {2016},
editor = {Mordukhovich, Boris and Reich, Simeon and Zaslavski, Alexander},
volume = {659},
pages = {1--15},
doi = {10.1090/conm/659/13162},
file = {Batenkov_Yomdin_Taylor Domination, Tur{\'a}n lemma, and Poincar{\'e}-Perron Sequences.pdf:/Users/dima/Library/Application Support/Zotero/Profiles/pvpguanx.default/zotero/storage/Z7WU5V63/Batenkov_Yomdin_Taylor Domination, Tura?n lemma, and Poincare?-Perron Sequences.pdf:application/pdf},
isbn = {978-1-4704-1736-9 978-1-4704-2904-1},
language = {en},
owner = {dima},
timestamp = {2016.08.04},
url = {http://dx.doi.org/10.1090/conm/659/13162},
urldate = {2016-04-07}
}
• D. Batenkov and G. Binyamini, “Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation,” Journal of Differential Equations, vol. 259, iss. 11, pp. 5769-5781, 2015.
[Bibtex]
@Article{batenkov_uniform_2015,
author = {Batenkov, Dmitry and Binyamini, Gal},
title = {Uniform upper bounds for the cyclicity of the zero solution of the {Abel} differential equation},
journal = {{Journal of Differential Equations}},
year = {2015},
volume = {259},
number = {11},
pages = {5769--5781},
doi = {10.1016/j.jde.2015.07.009},
file = {Batenkov_Binyamini_2015_Uniform upper bounds for the cyclicity of the zero solution of the Abel.pdf:/Users/dima/Library/Application Support/Zotero/Profiles/pvpguanx.default/zotero/storage/9M8HT664/Batenkov_Binyamini_2015_Uniform upper bounds for the cyclicity of the zero solution of the Abel.pdf:application/pdf},
issn = {0022-0396},
mrnumber = {3397308},
owner = {dima},
timestamp = {2016.08.04},
url = {http://www.ams.org/mathscinet-getitem?mr=3397308},
urldate = {2016-08-04}
}
• D. Batenkov and Y. Yomdin, “Taylor Domination, Difference Equations, and Bautin Ideals,” To appear in Springer Proceedings in Mathematics and Statistics, 2014.
[Bibtex]
@Article{batenkov_taylor_2014,
author = {Batenkov, Dmitry and Yomdin, Yosef},
title = {{Taylor Domination, Difference Equations, and Bautin Ideals}},
journal = {{To appear in Springer Proceedings in Mathematics and Statistics}},
year = {2014},
month = nov,
note = {{arXiv}: 1411.7629},
abstract = {We compare three approaches to studying the behavior of an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$
from its Taylor coefficients. The first is Taylor domination'' property for $f(z)$ in the complex disk $D_R$, which is an
inequality of the form
$|a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1.$
The second approach is based on a possibility to generate $a_k$ via recurrence relations. Specifically, we consider linear
non-stationary recurrences of the form
$a_{k}=\sum_{j=1}^{d}c_{j}(k)\cdot a_{k-j},\ \ k=d,d+1,\dots,$
with uniformly bounded coefficients.
In the third approach we assume that $a_k=a_k(\lambda)$ are polynomials in a
finite-dimensional parameter $\lambda \in C^n.$ We study Bautin ideals'' $I_k$ generated by
$a_{1}(\lambda),\ldots,a_{k}(\lambda)$ in the ring $C[\lambda]$ of polynomials in $\lambda$.
\smallskip
These three approaches turn out to be closely related. We present some results and questions in this direction.},
annote = {Comment: {arXiv} admin note: substantial text overlap with {arXiv}:1301.6033},
bdsk-url-1 = {http://arxiv.org/abs/1411.7629},
date-modified = {2016-02-01 13:47:18 +0000},
file = {arXiv.org Snapshot:/Users/dima/Library/Application Support/Zotero/Profiles/pvpguanx.default/zotero/storage/P6BPTZMN/1411.html:text/html;Batenkov_Yomdin_2014_Taylor Domination, Difference Equations, and Bautin Ideals.pdf:/Users/dima/Library/Application Support/Zotero/Profiles/pvpguanx.default/zotero/storage/8K4JMTJQ/Batenkov_Yomdin_2014_Taylor Domination, Difference Equations, and Bautin Ideals.pdf:application/pdf},
keywords = {30B10, 39A06, Mathematics - Classical Analysis and {ODEs}},
owner = {dima},
timestamp = {2014.12.15},
url = {http://arxiv.org/abs/1411.7629},
urldate = {2014-12-15}
}
• D. Batenkov and G. Binyamini, “Moment vanishing of piecewise solutions of linear ODEs,” To appear in Springer Proceedings in Mathematics & Statistics, 2013.
[Bibtex]
@Article{batenkov2013moment,
author = {Batenkov, Dmitry and Binyamini, Gal},
title = {{Moment vanishing of piecewise solutions of linear ODEs}},
journal = {{To appear in Springer Proceedings in Mathematics \& Statistics}},
year = {2013},
abstract = {We consider the "moment vanishing problem" for a general class of piecewise-analytic functions which satisfy on each continuity interval a linear ODE with polynomial coefficients. This problem, which essentially asks how many zero first moments can such a (nonzero) function have, turns out to be related to several difficult questions in analytic theory of ODEs (Poincare's Center-Focus problem) as well as in Approximation Theory and Signal Processing ("Algebraic Sampling"). While the solution space of any particular ODE admits such a bound, it will in the most general situation depend on the coefficients of this ODE. We believe that a good understanding of this dependence may provide a clue for attacking the problems mentioned above.
In this paper we undertake an approach to the moment vanishing problem which utilizes the fact that the moment sequences under consideration satisfy a recurrence relation of fixed length, whose coefficients are polynomials in the index. For any given operator, we prove a general bound for its moment vanishing index. We also provide uniform bounds for several operator families.},
date-modified = {2016-02-01 12:48:18 +0000},
owner = {dima},
timestamp = {2013.04.15}
}