Chaos and fractals in fish school motion

The once abstract notions of fractal patterns and processes now appear naturally and inevitably in various chaotic dynamical systems. The examples range from Brownian motion [1], [2], [3], [4], [5] to the dynamics of social relations [6]. In this paper, after introducing a certain hybrid mathematical model of the plankton–fish school interplay, we study the fractal properties of the model fish school walks. We show that the complex planktivorous fish school motion is dependent on the fish predation rate. A decrease in the rate is followed by a transition from low-persistent to high-persistent fish school walks, i.e., from a motion with frequent to a motion with few changes of direction. The low-persistent motion shows fractal properties for all time scales, whereas the high-persistent motion has pronounced multifractal properties for large-scale displacements.     

Chaos, solitons and fractals in (2 + 1)-dimensional KdV system derived from a periodic wave solution

With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2 + 1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

Quantum solitons with cylindrical symmetry

Soliton solutions with cylindrical symmetry are investigated within the nonlinear -model disregarding the Skyrme stabilization term. The solitons are stabilized by quantization of the collective breathing mode and collapse in the 0 limit. It is shown that for such a stabilization mechanism the model, apart from solitons with integer topological numberB, admits solitons with half-oddB. The solitons with integerB have standard spin-isospin classification, but theB=1/2 solitons are shown to be characterized by spin, isospin, and some additional momentum.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 248–259, August, 1995.     

Topological symmetry,background independence and matrix models

We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that the space-time uncertainty principle of string may be understood as a manifestation of the breakdown of the topological symmetry in the large N matrix model. Next, we construct a new type of matrix models which is a matrix model analog of the topological Chern-Simons and BF theories. It is of interest that these topological matrix models are not only completely independent of the background metric but also have nontrivial “p-brane” solutions as well as commuting classical space-time as the classical solutions. In this paper, we would like to point out some elementary and unsolved problems associated to the matrix models, whose resolution would lead to the more satisfying matrix model in future.     

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.     

On mirror symmetry in Wess-Zumino-Novikov-Witten models

We examine the problem of constructing N=2 superconformal algebras out of N=1 non-semi-simple affine Lie algebras. These N=2 superconformal theories share the property that the super Virasoro central charge depends only on the dimension of the Lie algebra. We find, in particular, a construction having a central charge c=9. This provides a possible internal space for string compactification and where mirror symmetry might be explored.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 55–63, July, 1995.     

Nonstandard analysis,parastatistics, and fractals

To estimate divergent integrals, it is convenient, on one hand, to use ideas of nonstandard analysis and, on the other hand, to approximate the integral with a special lattice model that can be interpreted as space quantization. We apply these methods in the case of noninteger (fractal) and negative (hole) dimensions and present some refined formulas, in particular, for the spectrum of flicker noise. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 262–270, November, 2007.     

On quasi-static models hidden in Maxwell's equations

In this paper we introduce the electromagnetic quasi-static models in a simple but meaningful way, relying on the dimensional analysis of Maxwell's equations. This analysis puts in evidence the three characteristic times of an electromagnetic phenomenon. It allows to define the range of validity of well-known models, such as the eddy-current (MQS) or the electroquasistatic (EQS) ones, and thus their pertinence to describe a given phenomenon. The role of the so-called “small parameters” of a model is explained in detail for two classical examples, namely a capacitor and a solenoid. We show how the MQS and EQS models result from having replaced fields by their first order truncations of Taylor expansions with respect to these small parameters. We finally investigate the connection between quasi-static models and circuit theory, clarifying the role of the fields with respect to classical circuit elements, and provide an example of application to study the electromagnetic fields in a simple case.     

Efficient sensitivity analysis in hidden markov models

Sensitivity analysis in hidden Markov models (HMMs) is usually performed by means of a perturbation analysis where a small change is applied to the model parameters, upon which the output of interest is re-computed. Recently it was shown that a simple mathematical function describes the relation between HMM parameters and an output probability of interest; this result was established by representing the HMM as a (dynamic) Bayesian network. To determine this sensitivity function, it was suggested to employ existing Bayesian network algorithms. Up till now, however, no special purpose algorithms for establishing sensitivity functions for HMMs existed. In this paper we discuss the drawbacks of computing HMM sensitivity functions, building only upon existing algorithms. We then present a new and efficient algorithm, which is specially tailored for determining sensitivity functions in HMMs.     

Interactive hidden Markov models and their applications

^** Email: wching{at}hkusua.hku.hk In this paper, we propose an Interactive hidden Markov model(IHMM). In a traditional HMM, the observable states are affecteddirectly by the hidden states, but not vice versa. In the proposedIHMM, the transitions of hidden states depend on the observablestates. We also develop an efficient estimation method for themodel parameters. Numerical examples on the sales demand dataand economic data are given to demonstrate the applicabilityof the model.     

Chaos in periodically forced discrete-time ecosystem models

Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran–Ricker, and Hassell, and also a Maynard Smith predator–prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.     

Master partial differential equations for a Type II hidden symmetry

An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).     

Temperature inversion symmetry in gauge-Higgs unification models

We study the temperature inversion symmetry R → 1/T for the finite-temperature effective potential of the N=1, d=5 supersymmetric SU(3) _c ×SU(3) _w model on the orbifold S ¹ /Z ₂ . For the value of the Wilson line parameter α = 1 (SU(2) _L breaks to U′(1)), we show that the effective potential contains a symmetric part and an antisymmetric part under ξ → 1/ξ, ξ = RT. For α = 0 (SU(2) _L is preserved in this case), we find that the only contribution to the effective potential that breaks the temperature inversion symmetry comes from the fermions in the fundamental representation of the gauge group with the Z ₂ parities (+, +) or (−,−). This is interesting because it implies that the bulk effective potential corresponding to models with fundamental fermions localized at a fixed point in the orbifold (and models with no bulk fundamental fermions) has the temperature inversion symmetry. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 109–130, April, 2009.     

Ergodic and adaptive control of hidden Markov models

A partially observed stochastic system is described by a discrete time pair of Markov processes. The observed state process has a transition probability that is controlled and depends on a hidden Markov process that also can be controlled. The hidden Markov process is completely observed in a closed set, which in particular can be the empty set and only observed through the other process in the complement of this closed set. An ergodic control problem is solved by a vanishing discount approach. In the case when the transition operators for the observed state process and the hidden Markov process depend on a parameter and the closed set, where the hidden Markov process is completely observed, is nonempty and recurrent an adaptive control is constructed based on this family of estimates that is almost optimal.