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.

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.Lisa Goldberg [11] addressed the following question: how many equivalence classes of rational functions of degree d with a given critical set of 2d − 2 points may exist? She reduced this to the following problem of enumerative geometry:Problem P. Given 2d − 2 lines in general position in projective space CP d , how many projective subspaces of codimension 2 intersect all of them?

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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. Fractals 1997.05:603-614. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/17/15. For personal use only.

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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.

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 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 α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.

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