Duality transformations for general abelian systems

We describe the general structure of duality transformations for a very broad set of abelian statistical and field theoretic systems. This includes theories with many different types of fields and a large variety of kinds of interactions including, but not limited to nearest neighbor, next nearest neighbor, multi-spin interactions, etc. We find that the dual form of a theory does not depend directly on the dimensionality of the theory, but rather on the number of fields and number of different kinds of interactions. The dual forms we find have a generalized gauge symmetry and possess the usual property of having a temperature (or coupling constant) which is inverted from that of the original theory. Our results reduce to the well-known results in those particular cases that have heretofore been studied. Our procedure also suggests variations capable of generating other forms of the dual theory which may be useful in various specific cases.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24071/1/0000323.pd

Dynamics of infinite-range ballistic aggregation

The authors calculate the width of the growing interface of ballistic aggregation in the limit in which the range of the sticking interaction between the particles becomes infinite. They derive a scaling form for the width, and they compute the short- and long-time exponents finding nu =3/4 and alpha =1/2. Furthermore, they find that the crossover exponent defining the argument of the scaling function is gamma =1/2. They compare these exact results with computer simulations, finding excellent agreement. They also discuss the relation of these results to those of ordinary finite-range ballistic aggregation. Finally, they present a simple expression for the density of all ballistic aggregation clusters, regardless of the range of the interaction, which agrees with known results and interpolates between the infinite- and finite-range cases.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48814/2/jav20i18p6391.pd

Growth oscillations

The authors describe an up till now unrecognised phenomenon in kinetic growth models which leads to observable oscillations in such quantities as the density and velocity of growth. These oscillations, which can occur on length scales of many lattice spacings, arise because of an induced incommensuration in the growth mechanism. To illustrate the phenomenon, they present results for a particularly simple model, but the phenomenon is expected to be quite general and appear in a wide range of growth processes. The essential ingredients for the existence of the oscillations are that the growth take place at a reasonably well defined interface and that the growth process be discrete (e.g. that the cluster grows by the addition of discrete particles of finite size). The growth process is related to a functional stochastic iterative map so that the growth oscillations play the role of limit cycles. They suggest that the fixed point of this map is related to critical fractal kinetic growth.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48806/2/jav19i16pL973.pd

Time series and dependent variables

We present a new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series. The method is particularly suited to series with broad-band spectra such as chaotic series with or without noise. We derive quantities, [delta]j([var epsilon]), based on conditional probabilities, whose magnitude, roughly speaking, is an indicator of the extent to which the kth element in the series is a deterministic function of the (k - j)th element to within a measurement uncertainty, [var epsilon]. We apply our method to a number of deterministic time series generated by chaotic processes such as the tent, logistic and Henon maps, as well as to sequences of quasi-random numbers. In all cases the [delta]j correctly indicate the expected dependencies. We also show that the [delta]j are robust to the addition of substantial noise in a deterministic process. In addition, we derive a predictability index which is a measure of the extent to which a time series is predictable given some tolerance, [var epsilon]. Finally, we discuss the behavior of the [delta]i as [var epsilon] approaches zero.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29345/1/0000413.pd

Dependent variables in broad band continuous time series

In this paper we continue our development of new methods for the analysis of broad band time series by deriving quantities which are able to indicate deterministic dependence of an element in one time series on elements in other time series. These methods are very broadly applicable and are particularly well suited to the study of continuous time series, in which the value of the function may depend on derivatives of the function itself, or on other quantities. We apply our methods to a number of mathematical examples including the Lorentz equation, the Henon-Heiles equations, the forced Brusselator and the Mackey-Glass equation. We show that our methods are very successful at indicating deterministic dependencies in these systems, even if the time series are highly chaotic. Statistical aspects of our procedure are discussed, as are a number of interesting and surprising epistomological implications.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29262/1/0000319.pd

Statistical tests for deterministic effects in broad band time series

We derive a normalized version of the indicators of Savit and Green, and prove that these normalized statistics have, asymptotically, a normal distribution with a mean of zero and standard deviation of one if the time series is random in the sense of being IID (independent and identically distributed). We verify this result numerically, and study the magnitude of the finite size effects. We also show that these statistics are very sensitive to the existence of deterministic effects in the series, even if the underlying deterministic structure is complex, such as those generated by a chaotic system. We show that with moderate amounts of data, the statistics can easily indicate the presence of an underlying attractor even in the presence of IID noise which is as large as, or greater than the signal. Finally, we discuss the generalization of our approach to include (1) other null hypotheses besides IID which express assumptions of specific dependencies and (2) the study of deterministic effects between more than one time series.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30448/1/0000072.pd

Quasiperiodic behaviour in growth oscillations

A very large class of kinetic growth processes manifest oscillatory behaviour in a variety of physical quantities such as the propagation of the growing interface and the density of the resulting cluster. The authors demonstrate that these oscillations should generically display quasiperiodic behaviour. Using a formalism based on the projection method for quasicrystalline spectra, they elucidate general features such as the dominant frequency and amplitude of the oscillations. They also briefly discuss the effects of interfacial roughening on the spectra of the oscillations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48812/2/jav20i15pL987.pd

A physical picture for the phase transitions in ZN symmetric models

We show how the phase transitions in ZN symmetric spin and gauge theories can be understood as being caused by condensations of topological excitations. For the two-dimensional ZN periodic gaussian and vector Potts models, there are two phase transitions, the first caused by a condensation of strings (domain boundaries) and the second by an unbinding of vortices. The relationship between our picture and the double Coulomb gas representation of Kadanoff is discussed. Using our representation, we also explain the correspondence between these models and the recent theory of two-dimensional melting of Halperin and Nelson. Finally, we describe generalizations of our picture to higher dimensions and to gauge theories.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23169/1/0000094.pd

Classification scheme for statistical theories with unusual symmetries

A large class of theories with unusual symmetries is constructed and discussed from a unified viewpoint. These are statistical models with symmetries which correspond to an interpolation between standard globally-invariant theories and theories with a local symmetry. The models describe the dynamics of U(1) (or Z(N)) variables, ei[pi], located at the sited of a lattice of arbitrary dimension and geometry. We describe a classification of these theories based on what we call their symmetry index: the value of n for which the hamiltonian of a d-dimensional system is invariant under the transformation [pi](x1,...,xd) --> [pi] (x1,...,xd) + [Lambda] (x1,...,xd), where [Lambda] satisfies a set of n linearly-independent constraints. We discuss how the critical properties and the structure of topological defects of a model are determined by d and n.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25890/1/0000453.pd

Linear and Nonlinear Measures and Seizure Anticipation in Temporal Lobe Epilepsy

In a recent paper, we showed that the value of a nonlinear quantity computed from scalp electrode data was correlated with the time to a seizure in patients with temporal lobe epilepsy. In this paper we study the relationship between the linear and nonlinear content and analyses of the scalp data. We do this in two ways. First, using surrogate data methods, we show that there is important nonlinear structure in the scalp electrode data to which our methods are sensitive. Second, we study the behavior of some simple linear metrics on the same set of scalp data to see whether the nonlinear metrics contain additional information not carried by the linear measures. We find that, while the nonlinear measures are correlated with time to seizure, the linear measures are not, over the time scales we have defined. The linear and nonlinear measures are themselves apparently linearly correlated, but that correlation can be ascribed to the influence of a small set of outliers, associated with muscle artifact. A remaining, more subtle relation between the variance of the values of a nonlinear measure and the expectation value of a linear measure persists. Implications of our observations are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46310/1/10827_2004_Article_5252207.pd