Andrei Gabrielov

Irreducibility of some spectral determinants (2009)

Abstract. We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on...

Approximation of definable sets by compact families, and upper bounds on homotopy and homology (2009)

This is a complement to our paper arXiv:0802.1461. We study irreducibility of spectral determinants of some one-parametric eigenvalue problems in dimension one with polynomial potentials.

Gabrielov, Andrei, Vorobjov, Nicolai

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti...

Tangencies between holomorphic maps and holomorphic laminations (2008)

Analytic continuation of eigenvalues of a quartic oscillator (2008)

We prove that the set of leaves of a holomorphic lamination of codimension one that are non-transversal to a germ of a holomorphic map is discrete.

Andrei Gabrielov, Nicolai Vorobjov, Bath Ba Ay, Of A. Gabrielov, N. Vorobjov

Abstract. Let X be a semialgebraic set in R n defined by a Boolean combination of atomic formulae of the kind h ∗ 0 where ∗∈{>, ≥, =}, deg(h) <d, and the number of distinct polynomials...

DIMACS Series in Discrete Mathematics and Theoretical Computer Science On the Number of Connected Components of the Relative Closure of a Semi-Pfaffian Family (2008)

We consider the Schrodinger operator on the real line with even quartic potential and study analytic continuation of eigenvalues, as functions of the coefficient of the potential. We prove several...

Andrei Gabrielov, Thierry Zell

Abstract. The notion of relative closure (X, Y)0 of a semi-Pfaffian couple (X, Y) was introduced by Gabrielov to give a description of the o-minimal structure generated by Pfaffian functions. In this...

Zeros of eigenfunctions of some anharmonic oscillators (2008)

Re Eremenko, Andrei Gabrielov, Boris Shapiro

We consider eigenvalue problems of the form −y ′ ′ + P(z)y = λy, y(−∞)=y(∞)=0, (1) where P is a real even polynomial with positive leading coefficient, which is called a potential. The...

Contents (2008)

Andrei Gabrielov, Dmitry Novikov, Boris Shapiro

To Vladimir Igorevich Arnold who taught us to study classics Abstract. We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point...

Analytic continuation of eigenvalues of a quartic oscillator (2008)

Re Eremenko, Andrei Gabrielov

We consider the Schrödinger operator on the real line with even quartic potential x 4 +αx 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties...

Analytic continuation of eigenvalues of a quartic oscillator (2008)

Re Eremenko, Andrei Gabrielov

We consider the Schrödinger operator on the real line with even quartic potential x 4 +αx 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties...

Zeros of eigenfunctions of some anharmonic oscillators (2008)

Eremenko, Alexandre, Gabrielov, Andrei, Shapiro, Boris

Aftershock identification (2007)

Zaliapin, Ilya, Gabrielov, Andrei, Keilis-Borok, Vladimir, Wong, Henry

Earthquake aftershock identification is closely related to the question ``Are aftershocks different from the rest of earthquakes?'' We give a positive answer to this question and introduce a general...

Approximation of definable sets by compact families, and upper bounds on homotopy and homology (2007)

Gabrielov, Andrei, Vorobjov, Nicolai

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti...

Predictability of extreme events in a branching diffusion model (2007)

Gabrielov, Andrei, Keilis-Borok, Vladimir, Zaliapin, Ilya

We propose a framework for studying predictability of extreme events in complex systems. Major conceptual elements -- direct cascading or fragmentation, spatial dynamics, and external driving -- are...

High energy eigenfunctions of one-dimensional Schrodinger operators with polynomial potentials (2007)

Eremenko, Alexandre, Gabrielov, Andrei, Shapiro, Boris

For a class of one-dimensional Schrodinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit...

Zeros of eigenfunctions of some anharmonic oscillators (2006)

Eremenko, Alexandre, Gabrielov, Andrei, Shapiro, Boris

We study eigenfunctions of Schrodinger operators -y"+Py on the real line with zero boundary conditions, whose potentials P are real even polynomials with positive leading coefficients. For quartic...

Elementary proof of the B. and M. Shapiro conjecture for rational functions (2005)

Rational functions and real Schubert calculus (2005)

We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g)...

Eremenko, Alex, Gabrielov, Andrei, Shapiro, Michael, Vainshtein, Alek

Vorobjov Betti numbers of semi-algebraic sets defined by quantifier-free formulae, Discrete and Computational Geometry (2005)

Andrei Gabrielov, Andrei Gabrielov, Nicolai Vorobjov, Nicolai Vorobjov

Abstract. Let X be a semialgebraic set in R n defined by a Boolean combination of atomic formulae of the kind h ∗ 0where∗∈{>, ≥,=}, deg(h) <d, and the number of distinct polynomials h...

Elementary proof of the B. and M. Shapiro conjecture for rational functions (2005)

Alex Eremenko, Andrei Gabrielov

We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ...

Elementary proof of the B. and M. Shapiro conjecture for rational functions (2005)

Alex Eremenko, Andrei Gabrielov

We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ...

Inverse Cascade in Percolation Model: Hierarchical Description of Time-dependent Scaling (2004)

Gabrielov, Andrei, Wong, Henry Hang Lam, Zaliapin, Ilya

The dynamics of a 2D site percolation model on a square lattice is studied using the hierarchical approach introduced by Gabrielov et al., Phys. Rev. E, 60, 5293-5300, 1999. The key elements of the...

Mystery of point charges (2004)

Gabrielov, Andrei, Novikov, Dmitry, Shapiro, Boris

We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R^n. This question goes back to J.C.Maxwell and M.Morse. Using...

An inverse cascade model for self-organized complexity and natural hazards (2004)

Gleb Yakovlev, William I. Newman, Donald L. Turcotte, Andrei Gabrielov

natural hazards

Contents (2004)

Ilya Zaliapin, Henry Wong, Andrei Gabrielov

Complexity of computations with Pfaffian and Noetherian functions (2004)

Multiscale Trend Analysis (2004)

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...

Ilya Zaliapin, Andrei Gabrielov, Vladimir Keilis-borok

This paper introduces a multiscale analysis based on optimal piecewise linear approximations of time series. An optimality criterion is formulated and on its base a computationally effective...

Relative closure and the complexity of Pfaffian elimination (2003)

Degrees of real Grassmann varieties (2001)

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...

Eremenko, Alex, Gabrielov, Andrei

The paper has been withdrawn by the authors, due to an error in the proof of Theorem 1.1. Corrected version is published in Discrete and Computational Geometry, 28 (2002) 331-347.

Complexity of cylindrical decompositions of sub-Pfaffian sets (2001)

Colliding Casding Model for Earthquake Prediction (2000)

Abstract. We construct an algorithm for a cylindrical cell decomposition of aclosedcubeI n ⊂ R n compatible with a “restricted ” sub-Pfaffian subset Y ⊂ I n, provided an oracle deciding...

Andreigaeilc Depa Gcl, Andrei Gabrielov, Ilya Zaliapin, William I. Newman

. A wide set of premonitory seismicitypaKBKjL is reproduced ona numerica model of seismicity,ag their performaK2 in prediction ofma joreaLVK:K. es is evaLjKCKL Seismicity is generaLT by the...

Colliding cascades model for earthquake prediction (2000)

Andrei Gabrielov, Ilya Zaliapin, William I. Newman

1.1 The model We explore here the process wherein seismicity undergoes a qualitative change, culminating in a major earthquake. This is done for synthetic seismicity generated by the model of...

Complexity Of Cylindrical Decompositions Of Sub-Pfaffian Sets (1999)

Frontier and Closure of a Semi-Pfaffian Set (1998)

We construct an algorithm for a cylindrical cell decomposition of a closed cube I n # R n compatible with a "restricted" sub-Pfaffian subset Y # I n , provided an oracle deciding...

Multiplicity of a zero of an analytic function on a trajectory of a vector field (1997)

For a semi-Pfaffian set, i.e., a real semi-analytic set defined by equations and inequalities between Pfaffian functions in an open domain G, the frontier and closure in G are represented as...

Gabrielov, Andrei

Let P(x) be a germ at the origin of an analytic function in C^n, where x = (x_1,..., x_n), and let \xi = \xi_1(x) d/dx_1 + ... + \xi_n(x) d/dx_n be a germ at the origin of an analytic vector field....

Multiplicity Of A Zero Of An Analytic Function On A Trajectory Of A Vector Field (1997)

Counter-Examples to Quantifier Elimination for Fewnomial and Exponential Expressions (1997)

. The multiplicity # of a zero of a restriction of an analytic function P in C n to a trajectory of a vector #eld # with analytic coe#cients is equal to the sum of the Euler characteristics of Milnor...

Multiplicity Of A Noetherian Intersection (1997)

Introduction. The famous Tarski-Seidenberg [10,11] theorem asserts that any real semialgebraic expression with quantifiers is equivalent to a real semialgebraic expression without quantifiers, i.e.,...

Andrei Gabrielov, Askold Khovanskii

. A di#erential 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...

Complements of Subanalytic Sets and Existential Formulas for Analytic Functions (1995)

Asymmetric Abelian Avalanches and Sandpiles (1994)

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...

Abelian Avalanches and Tutte Polynomials (1992)

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...

Avalanches, Sandpiles And Tutte Decomposition (1992)

: We introduce a class of deterministic lattice models of failure, Abelian avalanche #AA# models, with continuous phase variables, similar to discrete Abelian sandpile #ASP# models. We investigate...