Andrei Gabrielov

Clustering analysis of seismicity and aftershock identification

We introduce a statistical methodology for clustering analysis of seismicity in the time-space-energy domain and use it to establish the existence of two statistically distinct populations of earthquakes: clustered and nonclustered. This result can be used, in particular, for nonparametric aftershock identification. The proposed approach expands the analysis of Baiesi and Paczuski [Phys. Rev. E 69, 066106 (2004)10.1103/PhysRevE.69.066106] based on the space-time-magnitude nearest-neighbor distance eta between earthquakes. We show that for a homogeneous Poisson marked point field with exponential marks, the distance eta has the Weibull distribution, which bridges our results with classical correlation analysis for point fields. The joint 2D distribution of spatial and temporal components of eta is used to identify the clustered part of a point field. The proposed technique is applied to several seismicity models and to the observed seismicity of southern California.

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 → (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.

Pole Placement by Static Output Feedback for Generic Linear Systems

We consider linear systems with m inputs, p outputs, and McMillan degree n such that n=mp. If both m and p are even, we show that there is a nonempty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that, for each system in U, there exists an open set of pole configurations, symmetric with respect to the real line, which cannot be assigned by any real static output feedback.

Abelian avalanches and Tutte polynomials

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 analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For an AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In the general case, statistics of avalanches is linked to an analog of a Tutte polynomial defined for any symmetric matrix.

Reverse tracing of short-term earthquake precursors

Abstract We introduce a new approach to short-term earthquake prediction named “ Reverse Tracing of Precursors ” (RTP), since it considers precursors in reverse order of their appearance. First, we detect the “candidates” for the short-term precursors; in our case, these are newly introduced chains of earthquakes reflecting the rise of an earthquake correlation range. Then we consider each chain, one by one, checking whether it was preceded by an intermediate-term precursor in its vicinity. If yes , we regard this chain as a precursor; in prediction it would start a short-term alarm. The chain indicates the narrow area of possibly complex shape, where an intermediate-term precursor should be looked for. This makes possible to detect precursors undetectable by the direct analysis. RTP can best be described on an example of its application; we describe retrospective prediction of two prominent Californian earthquakes—Landers (1992), M =7.6, and Hector Mine (1999), M =7.3, and suggest a hypothetical prediction algorithm. This paper descripes the RTP methodology, which has potentially important applications to many other data 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. Validation of the algorithm per se requires its application in different regions with a substantial number of strong earthquakes. First (and positive) retrospective results are obtained for 21 more strong earthquakes in California ( M ≥6.4), Japan ( M ≥7.0) and the Eastern Mediterranean ( M ≥6.5); these results are described elsewhere. The final validation requires, as always, prediction in advance for which this study sets up a base. We have the first case of a precursory chain reported in advance of a subsequent strong earthquake (Tokachi-oki, Japan, 25 September 2003, M =8.1). Possible mechanisms underlying RTP are outlined.

Block model of earthquake sequence

Abstract A model of interaction of lithospheric blocks is suggested. Dynamics of the lithosphere is described as an alternation of slow deformations and ruptures (‘earthquakes’). The model is used to produce an artificial catalog of seismicity, and the properties of this catalog are compared with and contrasted to the properties of real seismicity catalogs. By this means it is possible to determine which properties of earthquake sequences can be derived from simple assumptions about the processes that occur in the lithosphere. We consider both the instability of the process and possibility that large earthquakes might be predicted.

An exact renormalization model for earthquakes and material failure: statics and dynamics

Abstract Earthquake events are well-known to posses a variety of empirical scaling laws. Accordingly, renormalization methods offer some hope for understanding why earthquake statistics behave in a similar way over orders of magnitude of energy. We review the progress made in the use of renormalization methods in approaching the earthquake problem. In particular, earthquake events have been modeled by previous investigators as hierarchically organized bundles of fibers with equal load sharing. We consider by computational and analytic means the failure properties of such bundles of fibers, a problem that may be treated exactly by renormalization methods. We show, independent of the specific properties of an individual fiber, that the stress and time thresholds for failure of fiber bundles obey universal, albeit different, scaling laws with respect to the size of the bundles. The application of these results to fracture processes in earthquake events and in engineering materials helps to provide insight into some of the observed patterns and scaling - in particular, the apparent weakening of earthquake faults and composite materials with respect to size, and the apparent emergence of relatively well-defined stresses and times when failure is seemingly assured.

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 whenever the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.

Fractal Trees with Side Branching

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 filling (D = 2) and volume filling (D = 3) deterministic fractal tree constructions are introduced both with and without side branching. Applications to diffusion limited aggregation (DLA), actual drainage networks, as well as biology are considered. It is suggested that the Tokunaga taxonomy may have wide applicability in nature.

Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Let X be a subset in [-1, 1](n0) subset of R-n0 defined by the formula X = {x(0) \ Q(1)x(1) Q(2)x(2) ... Q(v)x(v) ((x(0), x(1), ...,x(v)) is an element of X-v)}, where Q(i) is an element of {There Exists, For All}, Q(i) not equal Q(i+1), x(i) is an element of [-1, 1](ni), and X-v may be either an open or a closed set in being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of X is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving X-v. In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of X-v are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.