This paper is a survey of the upper bounds on the complexity of basic algebraic and geometric operations with Pfaffian and Noetherian functions, and with sets definable by these functions. Among… (More)

We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation… (More)

We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same… (More)

Let X be a semialgebraic set inRn defined by a Boolean combination of atomic formulae of the kind h ∗ 0 where ∗ ∈ {>,≥,=}, deg(h) < d, and the number of distinct polynomials h is k. We prove that the… (More)

Let X be a subset in [−1, 1]n0 ⊂ Rn0 defined by a formula X = {x0|Q1x1Q2x2 . . . Qνxν((x0,x1, . . . ,xν) ∈ Xν)}, where Qi ∈ {∃,∀}, Qi 6= Qi+1, xi ∈ Rni , and Xν be either an open or a closed set in… (More)

We consider two classes of threshold failure models, Abelian avalanches and sandpiles, with the redistribution matrices satisfying natural conditions guaranteeing absence of infinite avalanches. We… (More)

We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. Our arguments explore the… (More)

This paper considers fractal trees with self-similar side branching. The Tokunaga classification system for side branching is introduced, along with the Tokunaga self-similarity condition. Area… (More)

A differential ring of analytic functions in several complex variables is called a ring of Noetherian functions if it is finitely generated as a ring and contains the ring of all polynomials. In this… (More)

We introduce the “relative closure” operation on one-parametric families of semi-Pfaffian sets. We show that finite unions of sets obtained with this operation (“limit sets”) constitute a structure,… (More)