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247
Nonlinear dynamics of networks: the groupoid formalism
 Bull. Amer. Math. Soc
, 2006
"... Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which ..."
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Cited by 76 (13 self)
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Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the grouptheoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend grouptheoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a highdimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.
Dynamic Semiotics
"... this paper I shall make a case for a dynamic semiotics. I list a set of phenomena that are difficult to understand in standard theories, and suggest a model borrowed from theories of complex dynamic systems. Since such theories rely on processes of selforganization that often defy analytical treatm ..."
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Cited by 59 (3 self)
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this paper I shall make a case for a dynamic semiotics. I list a set of phenomena that are difficult to understand in standard theories, and suggest a model borrowed from theories of complex dynamic systems. Since such theories rely on processes of selforganization that often defy analytical treatment, I use small computational models for assessing the empirical consequences of the theories.
Chromatic roots are dense in the whole complex plane
 In preparation
, 2000
"... to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic ..."
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Cited by 42 (15 self)
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to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Pottsmodel partition functions) ZG(q,v) outside the disc q + v  < v. An immediate corollary is that the chromatic roots of notnecessarilyplanar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,
Image Coding By Block Prediction Of Multiresolution Subimages
 IEEE Transactions on Image Processing
"... The redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images, but it has not been fully recognized and exploited for general images. Recently, fractal block coders have exploited the selfsimilarity among blocks in images. In this work we devise ..."
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Cited by 35 (2 self)
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The redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images, but it has not been fully recognized and exploited for general images. Recently, fractal block coders have exploited the selfsimilarity among blocks in images. In this work we devise an image coder in which the causal similarity among blocks of different subbands in a multiresolution decomposition of the image is exploited. In a pyramid subband decomposition, the image is decomposed into a set of subbands which are localized in scale, orientation and space. The proposed coding scheme consists of predicting blocks in one subimage from blocks in lower resolution subbands with the same orientation. Although our prediction maps are of the same kind of those used in fractal block coders, which are based on an iterative mapping scheme, our coding technique does not impose any contractivity constraint on the block maps. This makes the decoding procedure very simple and...
Observed structure of addresses in IP traffic
 In IMC
, 2002
"... This paper investigates the structure of addresses contained in IP traffic. Specifically, we analyze the structural characteristics of destination IP addresses seen on Internet links, considered as a subset of the address space. These characteristics may have implications for algorithms that deal wi ..."
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Cited by 28 (0 self)
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This paper investigates the structure of addresses contained in IP traffic. Specifically, we analyze the structural characteristics of destination IP addresses seen on Internet links, considered as a subset of the address space. These characteristics may have implications for algorithms that deal with IP address aggregates, such as routing lookups and aggregatebased congestion control. We find that address structures are well modeled by a multifractal Cantor dust with two parameters. The model may be useful for simulations where realistic IP addresses are preferred. We also develop concise characterizations of address structures, including active aggregate counts and discriminating prefixes. Our structural characterizations are stable over short time scales at a given site, and different sites have visibly different characterizations, so that the characterizations make useful “fingerprints ” of the traffic seen at a site. Also, changing traffic conditions, such as worm propagation, significantly alter these “fingerprints”. 1
Canonical heights, transfinite diameters, and polynomial dynamics
 J. Reine Angew. Math
"... Abstract. Let φ(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating φ gives rise to a dynamical system and a corresponding canonical height function ˆ hφ, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of th ..."
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Cited by 28 (6 self)
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Abstract. Let φ(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating φ gives rise to a dynamical system and a corresponding canonical height function ˆ hφ, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of φ over various completions of K, and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero. 1.
L.: 2002, Complex earthquake rupture and local tsunamis
 J. Geophys. Res. 107, ESE
"... [1] In contrast to farfield tsunami amplitudes that are fairly well predicted by the seismic moment of subduction zone earthquakes, there exists significant variation in the scaling of local tsunami amplitude with respect to seismic moment. From a global catalog of tsunami runup observations this v ..."
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Cited by 23 (8 self)
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[1] In contrast to farfield tsunami amplitudes that are fairly well predicted by the seismic moment of subduction zone earthquakes, there exists significant variation in the scaling of local tsunami amplitude with respect to seismic moment. From a global catalog of tsunami runup observations this variability is greatest for the most frequently occurring tsunamigenic subduction zone earthquakes in the magnitude range of 7 < Mw < 8.5. Variability in local tsunami runup scaling can be ascribed to tsunami source parameters that are independent of seismic moment: variations in the water depth in the source region, the combination of higher slip and lower shear modulus at shallow depth, and rupture complexity in the form of heterogeneous slip distribution patterns. The focus of this study is on the effect that rupture complexity has on the local tsunami wave field. Awide range of slip distribution patterns are generated using a stochastic, selfaffine source model that is consistent with the falloff of farfield seismic displacement spectra at high frequencies. The synthetic slip distributions generated by the stochastic source model are discretized and the vertical displacement fields from point source elastic dislocation expressions are superimposed to compute the coseismic vertical displacement field. For shallow subduction zone earthquakes it is demonstrated that selfaffine irregularities of the slip distribution result in significant variations in
Ray tracing deterministic 3D fractals
 In: SIGGRAPH Comput. Graph
, 1989
"... As shown in 1982, Julia sets of quadratic functions as well as many other deterministic fractals exist in spaces of higher dimensionality han the complex plane. Originally a boundarytracking al orithm was used to view these structures but required a large amount of storage space to operate. By ray ..."
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Cited by 19 (6 self)
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As shown in 1982, Julia sets of quadratic functions as well as many other deterministic fractals exist in spaces of higher dimensionality han the complex plane. Originally a boundarytracking al orithm was used to view these structures but required a large amount of storage space to operate. By ray tracing these objects, the storage facilities of a graphics workstation frame buffer are sufficient. A short discussion of a specific set of 3D deterministic fractals precedes a full description of a raytracing algorithm applied to these objects. A comparison with the boundarytracking method and applications to other 3D deterministic fractals are also included. CR Categor ies and Subject Descriptors:
Taming Chaotic Circuits
, 1992
"... Control algorithms that exploit chaotic behavior and its precursors can vastly improve the performance of many practical and useful systems. Phaselocked loops, for example, are normally designed using linearization. This approximation hides the global dynamics that lead to lock and capture range li ..."
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Cited by 16 (3 self)
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Control algorithms that exploit chaotic behavior and its precursors can vastly improve the performance of many practical and useful systems. Phaselocked loops, for example, are normally designed using linearization. This approximation hides the global dynamics that lead to lock and capture range limits. Design techniques that are equipped to exploit the real nonlinear and chaotic nature of the device can loosen these limitations. The program Perfect Moment is built around a collection of such techniques. Given a differential equation, a control parameter, and two statespace points, the program explores the system's behavior, automatically choosing interesting and useful parameter values and constructing statespace portraits at each one. It then chooses a set of trajectory segments from those portraits, uses them to construct a composite path between the objectives, and finally causes the system to follow that path by switching the parameter value at the segment junctions. Rules embo...