arXiv subjects

A. Gabrielov

Publications and source records attributed to A. Gabrielov.

6 recordsLinked to original sources

arXiv2026

From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem

In \cite{GNS} we proved that, for every $\alpha>0$, the potential of three positive point charges has at most $12$ nondegenerate equilibrium points. We also observed that the same method would give the sharper bound $6$ if a certain auxiliary polynomial system $Q=R=0$ had at least four solutions, counted with multiplicity, in each open quadrant of the $(f,g)$-plane. Here we prove this four-solution statement. The main new ingredient is a separation argument at the unique saddle point of a separated-variable first integral. Consequently, the upper bound for three charges improves from $12$ to $6$.

math.CA↗

arXiv2004

Rational functions and real Schubert calculus

We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.

math.AG↗

arXiv2004

Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry

Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z : ... : z^d) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d^{th} Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example on the real line), then the function maps this circle into a circle.

math.AG↗

arXiv2003

Reverse Detection of Short-Term Earthquake Precursors

We introduce a new approach to short-term earthquake prediction based on the concept of selforganization of seismically active fault networks. That approach is named "Reverse Detection of Precursors" (RDP), since it considers precursors in reverse order of their appearance. This makes it possible to detect precursors undetectable by direct analysis. Possible mechanisms underlying RDP are outlined. RDP is described with a concrete example: we consider as short-term precursors the newly introduced chains of earthquakes reflecting the rise of an earthquake correlation range; and detect (retrospectively) such chains a few months before two prominent Californian earthquakes - Landers, 1992, M = 7.6, and Hector Mine, 1999, M = 7.3, with one false alarm. Similar results (described elsewhere) are obtained by RDP for 21 more strong earthquakes in California (M >= 6.4), Japan (M >= 7.0) and the Eastern Mediterranean (M >= 6.5). Validation of the RDP approach requires, as always, prediction in advance for which this study sets up a base. We have the first case of advance prediction; it was reported before Tokachi-oki earthquake (near Hokkaido island, Japan), Sept. 25, 2003, M = 8.1. RDP has potentially important applications to other precursors and to prediction of other critical phenomena besides earthquakes. In particular, it might vindicate some short-term precursors, previously rejected as giving too many false alarms.

physics.geo-ph↗

arXiv2003

Multiscale Trend Analysis

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 algorithm is constructed for decomposition of a time series into a hierarchy of trends (local linear approximations) at different scales. The top of the hierarchy is the global linear approximation over the whole observational interval, the bottom is the original time series. Each internal level of the hierarchy corresponds to a piecewise linear approximation of analyzed series. Possible applications of the introduced Multiscale Trend Analysis (MTA) go far beyond the linear interpolation problem: This paper develops and illustrates methods of self-affine, hierarchical, and correlation analyses of time series.

physics.data-an↗

arXiv1999

An Exactly Soluble Hierarchical Clustering Model: Inverse Cascades, Self-Similarity, and Scaling

We show how clustering as a general hierarchical dynamical process proceeds via a sequence of inverse cascades to produce self-similar scaling, as an intermediate asymptotic, which then truncates at the largest spatial scales. We show how this model can provide a general explanation for the behavior of several models that has been described as ``self-organized critical,'' including forest-fire, sandpile, and slider-block models.

adap-org↗