Evolution in complex systems

What features characterize complex system dynamics? Power laws and scale invariance of fluctuations are often taken as the hallmarks of complexity, drawing on analogies with equilibrium critical phenomena. Here we argue that slow, directed dynamics, during which the system's properties change significantly, is fundamental. The underlying dynamics is related to a slow, decelerating but spasmodic release of an intrinsic strain or tension. Time series of a number of appropriate observables can be analyzed to confirm this effect. The strain arises from local frustration. As the strain is released through “quakes,” some system variable undergoes record statistics with accompanying log‐Poisson statistics for the quake event times. We demonstrate these phenomena via two very different systems: a model of magnetic relaxation in type II superconductors and the Tangled Nature model of evolutionary ecology and show how quantitative indications of aging can be found. © 2004 Wiley Periodicals, Inc. Complexity 10: 49–56, 2004     

Zeros of functions with finite Dirichlet integral

In this paper, we refine a result of Nagel, Rudin, and Shapiro (1982) concerning the zeros of holomorphic functions on the unit disk with finite Dirichlet integral.

     

Homogeneous quaternionic Kähler structures of linear type

A classification of homogeneous quaternionic Kähler structures by real tensors is given and related to Fino's representation theoretic decomposition. A relationship between the modules whose dimension grows linearly and quaternionic hyperbolic space is found. To cite this article: M. Castrillón López et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Symmetries of flat rank two distributions and sub-Riemannian structures

Flat sub-Riemannian structures are local approximations -- nilpotentizations -- of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5.

     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Anisotropic giant magnetoresistance originating from the π-d interaction in a molecule

We synthesized TPP[Fe^III(Pc)(CN)₂]₂, PTMA_x[Fe^III(Pc)(CN)₂]·y(MeCN), and PXX [Fe^III(Pc)(CN)₂], a new series of charge-transfer salts containing the axially-substituted phthalocyanine (Pc), [Fe^III(Pc)(CN)₂]⁻. In this molecular unit, the π conduction electron derived from the Pc-ring coexists with the d electron which is a potential source of a local magnetic moment. Therefore various phenomena associated with the interplay between local magnetic moments and conduction electrons are expected. We observed the giant negative magnetoresistance (GNMR) in all the three salts. The GNMR is highly anisotropic for the magnetic-field direction, and reflects the g-tensor anisotropy of the local magnetic moment in the [Fe^III(Pc)(CN)₂]⁻ unit. This indicates that the GNMR in these salts originates from the strong π-d interaction in the [Fe^III(Pc)(CN)₂]⁻ unit.     

Phase Transitions in Traffic Models

It is suggested that the question of existence of a jamming phase transition in a broad class of single-lane cellular-automaton traffic models may be studied using a correspondence to the asymmetric chipping model. In models where such correspondence is applicable, jamming phase transition does not take place. Rather, the system exhibits a smooth crossover between free-flow and jammed states, as the car density is increased.     

Equilibrium States in the Quadruple–Kerr Solution

The extended quadruple–Kerr metric is used to consider equilibrium states of four collinear Kerr particles. We explain our previous failure to solve numerically the full set of the balance equations, and we derive a self–consistent system of the axis conditions leading to the equilibrium of all four constituents which can be black holes or hyperextreme objects. The equilibrium configurations obtained in this paper exhibit similar features with those occurring in the systems of two Kerr particles, for instance, the balance of four Kerr black holes with positive masses does not seem possible. Equilibrium states of two identical compound Kerr objects are also discussed.