Тиморин Владлен Анатольевич

Аскольд Георгиевич Хованский: Идеальный математик

2013 · CHAPTER · ru

Osculating Curves: Around the Tait-Kneser Theorem

2013 · ARTICLE · en

Теорема Тейта-Кнезера утверждает, что соприкасающиеся окружности плоской кривой с монотонной кривизной не пересекаются и вложены друг в друга. Мы обсуждаем эту теорему и несколько вариаций.

Dynamical cores of topological polynomials

2013 в печати · CHAPTER · en

We define the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects.

Counting vertices in Gelfand-Zetlin polytopes

2013 · ARTICLE · en

We discuss the problem of counting vertices in Gelfand--Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas.

Disquisitiones 235

2013 · PREPRINT · en

Section 235 of Gauss' fundamental treatise "Disquisitiones Arithmeticae" establishes basic properties that compositions of binary quadratic forms must satisfy. Although this section is very technical, it contains truly important results. We review section 235 using a more invariant language and simplifying the arguments. We also make the statements slightly stronger by removing unnecessary assumptions.

Quadratic-like dynamics of cubic polynomials

2013 · PREPRINT · en

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.

The Main Cubioid

2013 · PREPRINT · en

We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them.

Laminations from the Main Cubioid

2013 · PREPRINT · en

According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.

Aleksandrov-Fenchel inequality for coconvex bodies

2013 · PREPRINT · en

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals.

Исчисление Шуберта и многогранники Гельфанда-Цетлина

2012 · ARTICLE · ru

Мы описываем новый подход к исчислению Шуберта на многообразиях полных флагов, используя многочлен объема, связанный с многогранниками Гельфанда-Цетлина. Этот подход позволяет вычислять произведения (пересечения) циклов Шуберта, пересекая грани многогранника.