Tangencies between holomorphic maps and holomorphic laminations
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.
math.CV↗
arXiv subjects
Andrei Gabrielov
Publications and source records attributed to Andrei Gabrielov.
Analytic continuation of eigenvalues of a quartic oscillator
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 properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the plane.
math-ph↗
Aftershock identification
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 statistical procedure for clustering analysis of seismicity that can be used, in particular, for aftershock detection. The proposed approach expands the analysis of Baiesi and Paczuski [PRE, 69, 066106 (2004)] based on the space-time-magnitude nearest-neighbor distance $η$ between earthquakes. We show that for a homogeneous Poisson marked point field with exponential marks, the distance $η$ has Weibull distribution, which bridges our results with classical correlation analysis for unmarked point fields. We introduce a 2D distribution of spatial and temporal components of $η$, which allows us to identify the clustered part of a point field. The proposed technique is applied to several synthetic seismicity models and to the observed seismicity of Southern California.
physics.geo-ph↗
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
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 numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.
math.AG↗
Predictability of extreme events in a branching diffusion model
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 combined in a classical age-dependent multi-type branching diffusion process with immigration. A complete analytic description of the size- and space-dependent distributions of particles is derived. We then formulate an extreme event prediction problem and determine characteristic patterns of the system behavior as an extreme event approaches. In particlular, our results imply specific premonitory deviations from self-similarity, which have been heuristically observed in real-world and modeled complex systems. Our results suggest a simple universal mechanism of such premonitory patterns and natural framework for their analytic study.
nlin.AO↗
High energy eigenfunctions of one-dimensional Schrodinger operators with polynomial potentials
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 disctibution in the complex plane as the eigenvalues tend to infinity. This limit distribution depends only on the degree of potential and on the boundary conditions.
math-ph↗
Zeros of eigenfunctions of some anharmonic oscillators
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 potentials we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. Similar result holds for sextic potentials and their eigenfunctions with finitely many complex zeros. As a byproduct we obtain a complete classification of such eigenfunctions of sextic potentials.
math-ph↗
Elementary proof of the B. and M. Shapiro conjecture for rational functions
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) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.
math.AG↗
Inverse Cascade in Percolation Model: Hierarchical Description of Time-dependent Scaling
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 approach are the tree representation of clusters and their coalescence, and the Horton-Strahler scheme for cluster ranking. Accordingly, the evolution of percolation model is considered as a hierarchical inverse cascade of cluster aggregation. A three-exponent time-dependent scaling for the cluster rank distribution is derived using the Tokunaga branching constraint and classical results on percolation in terms of cluster masses. Deviations from the pure scaling are described. An empirical constraint on the dynamics of a rank population is reported based on numerical simulations.
cond-mat.stat-mech↗
Mystery of point charges
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 fewnomial theory we show that for a given number of charges there exists an upper bound independent on the dimension, and show it to be 12 for three charges. We conjecture the exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.
math-ph↗
Multiplicity of a zero of an analytic function on a trajectory of a vector field
Let P(x) be a germ at the origin of an analytic function in C^n, where x = (x_1,..., x_n), and let ξ= ξ_1(x) d/dx_1 + ... + ξ_n(x) d/dx_n be a germ at the origin of an analytic vector field. Suppose that ξ(0) != 0, and let γbe a trajectory of ξthrough the origin. Suppose that P|_γ/\equiv 0, and let μ(P|_γ) be the multiplicity of a zero of P|_γat the origin. Let ξP = ξ_1 dP/dx_1 + ... + ξ_n dP/dx_n be derivative of P in the direction of ξ, and let ξ^kP be the kth iteration of this derivative. We give a formula (Theorem 1) for μ(P|_γ) in terms of the Euler characteristic of the Milnor fibers defined by a deformation of P, ξP, ..., ξ^{n-1}P . For a polynomial P of degree p and a vector field ξwith polynomial coefficients of degree q, this allows one to compute μ(P|_γ) in purely algebraic terms (Theorem 2), and to give an estimate (Theorem 3) for μ(P|_γ) in terms of n, p, q, single exponential in n and polynomial in p and q. This estimate improves previous results which were doubly exponential in n.
math.CV↗