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Alexander Brudnyi

Publications and source records attributed to Alexander Brudnyi.

At least 37 records · Page 2Linked to original sources
arXiv2014

Universal Curves in the Center Problem for Abel Differential Equations

We study the center problem for the class $\mathcal E_Γ$ of Abel differential equations $\frac{dv}{dt}=a_1 v^2+a_2 v^3$, $a_1,a_2\in L^\infty ([0,T])$, such that images of Lipschitz paths $\tilde A:=\bigl(\int_0^\cdot a_1(s)ds, \int_0^\cdot a_2(s)ds\bigr): [0,T]\rightarrow\mathbb R^2$ belong to a fixed compact rectifiable curve $Γ$. Such a curve is called universal if whenever an equation in $\mathcal E_Γ$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_1, a_2$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with product being the composition of series, explicit examples of universal curves and approximation of Lipschitz triangulable curves by universal ones.

math.CA↗
arXiv2013

A Rolle type theorem for cyclicity of zeros of families of analytic functions

Let $\{f_{λ; j}\}_{λ\in V; 1\le j\le k}$ be families of holomorphic functions in the open unit disk $\Di\subset\Co$ depending holomorphically on a parameter $λ\in V\subset \Co^n$. We establish a Rolle type theorem for the generalized multiplicity (called {\em cyclicity}) of zero of the family of univariate holomorphic functions ${\sum_{j=1}^k f_{λ;j}}_{λ\in V}$ at $0\in\Di$. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $\sum_{k=1}^m P_k(z)e^{Q_k(z)}$, $z\in\Co$, where $P_k$ and $Q_k$ are holomorphic polynomials of degrees $p$ and $q$, respectively, parameterized by vectors of coefficients of $P_k$ and $Q_k$.

math.CV↗
arXiv2013

Moments Finiteness Problem and Center Problem for Ordinary Differential Equations

We study the moments finiteness problem for the class of Lipschitz maps $F: [a,b]\rightarrow\mathbb R^n$ with images in a compact Lipschitz triangulable curve $Γ$. We apply the obtained results to the center problem for ODEs describing in some cases (including equations with analytic coefficients) the set of universal centers of such equations by vanishing of finitely many moments from their coefficients.

math.DS↗
arXiv2011

Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^\infty

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H^\infty of bounded holomorphic functions on the unit disk D\subset C with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H^\infty, prove that the maximal ideal space of the algebra H_{\rm comp}^\infty (A) of holomorphic functions on $\Di$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H^\infty and A.

math.CV↗
arXiv2008

Center Problem for the Group of Rectangular Paths

We solve the center problem for ODEs \frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x) v^{i+1} such that the first integrals of vectors of their coefficients determine rectangular paths in finite dimensional complex vector spaces.

math.CA↗
arXiv2007

On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of $\Co^{N}$. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in $\Co^{2}$.

math.CV↗
arXiv2007

On Center Sets of ODEs Determined by Moments of their Coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a {\em center}. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.

math.DS↗