exponential attractors and finite dimensional reduction for non autonomous dynamical systems we suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly holder continuous with respect to the variation of the dynamical system in some natural large class moreover we extend this construction to non autonomous dynamical systems dynamical processes treating in that case the exponential attractor as a uniformly exponentially attracting finite dimensional and time dependent set in the phase space in particular this result shows that for a wide class of non autonomous equations of mathematical physics the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is we illustrate the main results of this paper on the model example of a non autonomous reaction diffusion system in a bounded domain

quasi valuations extending a valuation abstract suppose f is a field with valuation v and valuation ring o v e is a finite field extension and w is a quasi valuation on e extending v we study quasi valuations on e that extend v in particular their corresponding rings and their prime spectra we prove that these ring extensions satisfy inc incomparability lo lying over and gd going down over o v in particular they have the same krull dimension we also prove that every such quasi valuation is dominated by some valuation extending v under the assumption that the value monoid of the quasi valuation is a group we prove that these ring extensions satisfy gu going up over o v and a bound on the size of the prime spectrum is given in addition a one to one correspondence is obtained between exponential quasi valuations and integrally closed quasi valuation rings given r an algebra over o v we construct a quasi valuation on r we also construct a quasi valuation on r o v f which helps us prove our main theorem the main theorem states that if r e satisfies r f o v and e is the field of fractions of r then r and v induce a quasi valuation w on e such that r o w and w extends v thus r satisfies the properties of a quasi valuation ring

noise robust estimates of correlation dimension and k2 entropy using gaussian kernels to define the correlation sum we derive simple formulas that correct the noise bias in estimates of the correlation dimension and k2 entropy of chaotic time series the corrections are only based on the difference of correlation dimensions for adjacent embedding dimensions and hence preserve the full functional dependencies on both the scale parameter and embedding dimension it is shown theoretically that the estimates which are derived for additive white gaussian noise are also robust for moderately colored noise simulations underline the usefulness of the proposed correction schemes it is demonstrated that the method gives satisfactory results also for non gaussian and dynamical noise

bulk and surface critical behavior of the three dimensional ising model and conformal invariance using a continuous cluster monte carlo algorithm we investigate the critical three dimensional ising model in its anisotropic limit from the ratio of the magnetic correlations in the strong and the weak coupling directions we determine the length ratio relating the isotropic ising model and the anisotropic limit on this basis we simulate the critical ising model on a spherocylinder s2 x r1 i e a curved geometry obtained from a conformal mapping of the infinite space r3 from correlation lengths along the spherocylinder combined with the prediction of conformal invariance we estimate the magnetic and thermal scaling dimensions as x h 0 5182 6 and x t 1 419 7 respectively the behavior of the binder cumulant is also determined in the limit of an infinitely long spherocylinder next free boundary conditions are imposed on the equators of the spherocylinder and thus the geometry s1 x s x r1 is obtained the surface magnetic scaling dimension is estimated as x s h 1 263 5 the consistency of the aforementioned estimations and existing results confirms that the three dimensional ising model is conformally invariant further the precision of these results reveals that as in two dimensions conformal mappings provide a powerful tool to investigate critical phenomena with the continuous cluster algorithm we also perform simulations of systems inside a conventional solid cylinder the surface magnetic correlation length differs within the estimated error margin by a factor pi 2 from that along a half spherocylinder s1 x s x r1 with the same radius

chaos or noise difficulties of a distinction in experiments the dynamical behavior of systems is reflected in time series due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension these restrictions limit our ability to distinguish between signals generated by different systems such as regular chaotic or stochastic ones when analyzed from a time series point of view we propose to classify the signal behavior without referring to any specific model as stochastic or deterministic on a certain scale of the resolution ensuremath epsilon according to the dependence of the ensuremath epsilon ensuremath tau entropy h ensuremath epsilon ensuremath tau and the finite size lyapunov exponent ensuremath lambda ensuremath epsilon on ensuremath epsilon

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

recovering smooth dynamics from time series with the aid of recurrence plots a graphical method based on recurrence plots is used in the reconstruction of the phase space from a time series of measurements it is demonstrated that if the embedding delay and dimension are correctly chosen the recurrence plot of a smooth dynamical system has a particularly simple form it is shown how to use recurrence plots to determine the correct embedding parameters so that reliable quantitative information can be drawn about the system generating the time series the average line length in the plot is shown to be directly related to the prediction horizon furthermore it is a numerical characteristic of the embedded series independent of the threshold used in the plot

a non subjective approach to the gp algorithm for analysing noisy time series we present an adaptation of the standard grassberger proccacia gp algorithm for estimating the correlation dimension of a time series in a non subjective manner the validity and accuracy of this approach are tested using different types of time series such as those from standard chaotic systems pure white and colored noise and chaotic systems with added noise the effectiveness of the scheme in analysing noisy time series particularly those involving colored noise is investigated one interesting result we have obtained is that for the same percentage of noise addition data with colored noise is more distinguishable from the corresponding surrogates than data with white noise as examples of real life applications analyses of data from an astrophysical x ray object and a human brain eeg are presented

red bond exponents of the critical and the tricritical ising model in three dimensions using the wolff and geometric cluster algorithms and finite size scaling analysis we investigate the critical ising and the tricritical blume capel models with nearest neighbor interactions on the simple cubic lattice the sampling procedure involves the decomposition of the ising configuration into geometric clusters each of which consists of a set of nearest neighboring spins of the same sign connected with bond probability p these clusters include the well known kasteleyn fortuin clusters as a special case for p 1 exp 2k where k is the ising spin spin coupling along the critical line k kc the size distribution of geometric clusters is investigated as a function of p we observe that unlike in the case of two dimensional tricriticality the percolation threshold in both models lies at pc 1 exp 2kc further we determine the corresponding red bond exponents as yr 0 757 2 and 0 501 5 for the critical ising and the tricritical blume capel models respectively on this basis we conjecture yr 1 2 for the latter model

robust exponential attractors for singularly perturbed hodgkin huxley equations abstract here we consider a singular perturbation of the hodgkin huxley system which is derived from the lieberstein s model we study the associated dynamical system on a suitable bounded phase space when the perturbation parameter e i e the axon specific inductance is sufficiently small we prove the existence of bounded absorbing sets as well as of smooth attracting sets we deduce the existence of a smooth global attractor a e finally we prove the main result that is the existence of a family of exponential attractors e e which is holder continuous with respect to e

an envelope solitary wave solution for a generalized nonlinear schrodinger equation we derive an envelope solitary wave solution for the following generalized nonlinear schrodinger equation ut ia2uxx a3uxxx ib mod u mod 2u b1 mod u mod 2u x by gauge transformation

universal order parameter profiles in confined critical systems with boundary fields the authors consider systems at the bulk critical temperature with an ordering field applied at the boundary for various boundary geometries universal expressions for the spatial dependence of the order parameter are obtained by conformal mapping of the known results for the half space analytical results for strip wedge and other two dimensional geometries and for systems with spherical boundaries in arbitrary dimension are given

cluster monte carlo algorithms abstract the swendsen wang and wolff monte carlo algorithms are described in some detail using the potts model as an example various generalizations are then reviewed and some applications are discussed two complete fortran programs for the algorithms are provided

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

simple riesz groups of rank one having wild intervals abstract we prove that every partially ordered simple group of rank one which is not riesz embeds into a simple riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals using this result we construct examples of simple riesz groups of rank one g containing unbounded intervals d n n 1 and d that satisfy a for each n 1 t d n g for every t q n but q n d n g where q n is a sequence of relatively prime integers b for every n 1 n d g we sketch some potential applications of these results in the context of k theory

universal amplitudes in finite size scaling generalisation to arbitrary dimensionality the relationship between the correlation length and critical exponents in finite width strips in two dimensions is generalised to cylindrical geometries of arbitrary dimensionality d for d not 2 these correspond however to curved spaces the result is verified for the spherical model

conformal invariance and the ising model on a spheroid we formulate conformal mappings between an infinite plane and a spheroid and one between a semi infinite plane and a half spheroid special cases of the spheroid include the surface of an infinitely long cylinder of a sphere and of a flat disc these mappings are applied to the critical ising model for the case of the sphere and the flat disc we derive analytical expressions for the second and the fourth moments of the magnetization density and thus for the binder cumulant next we investigate ising models on spheroids and half spheroids by means of a continuous cluster monte carlo method for simulations in curved geometries fixed and free boundary conditions are imposed for half spheroids the monte carlo data are analyzed by finite size scaling critical values of the binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above at criticality we also sample two and one point correlation functions on spheroids on half spheroids respectively the magnetic and temperature scaling dimensions as determined from the monte carlo data and the theory of conformal invariance are in good agreement with exact results

gas liquid two phase flow structure in the multi scale weighted complexity entropy causality plane abstract the multi scale weighted complexity entropy causality plane ms wcecp is proposed for characterizing the physical structure of complex system firstly we use the method to investigate typical nonlinear time series compared with the multi scale complexity entropy causality plane ms cecp the ms wcecp can not only uncover the dynamic information loss of complex system with the increase of scale but also can characterize the complexity of nonlinear dynamic system in particular the algorithm of ms wcecp performs strong anti noise ability then we calculate the ms wcecp for the conductance fluctuating signals measured from vertical upward gas liquid two phase flow experiments in a small diameter pipe the results demonstrate that the ms wcecp is a useful approach for exploring the stability and complexity in gas liquid two phase flows

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

coping with nonstationarity by overembedding we discuss how nonstationarity in observed time series data due to pronounced fluctuations of system parameters can be resolved by making use of embedding techniques for scalar data if a d dimensional deterministic system is driven by p slowly time dependent parameters a d p dimensional manifold has to be reconstructed from the scalar time series which is done by an mg2 d p dimensional time delay embedding we show that in this space essential aspects of determinism are restored we demonstrate the validity of the idea heuristically for numerical examples and for human speech data

estimating invariants of noisy attractors we propose a method for estimating the correlation dimension and correlation entropy of a time series it is based on a generalization of the correlation integral that is specifically useful when the time series is corrupted with gaussian measurement noise from computational experiments we conclude that reasonable estimates for the noise level correlation dimension and correlation entropy can be found for time series with up to 20 noise the method appears to be fairly robust with respect to the noise distribution

noise level estimation of time series using coarse grained entropy we present a method of noise level estimation that is valid even for high noise levels the method makes use of the functional dependence of coarse grained correlation entropy k 2 ensuremath varepsilon on the threshold parameter ensuremath varepsilon we show that the function k 2 ensuremath varepsilon depends in a characteristic way on the noise standard deviation ensuremath sigma it follows that observing k 2 ensuremath varepsilon one can estimate the noise level ensuremath sigma although the theory has been developed for the gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of langevin equation corresponding to the dynamical noise we have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the chua electronic circuit contaminated by noise

determination of the noise level of chaotic time series we propose a method to determine the amount of measurement noise present in a chaotic time series if the data are embedded in a space of higher dimension than that strictly required to reconstruct the dynamics the extra dimensions are dominated by the noise which results in a certain shape of the correlation integral for the case in which only gaussian noise is present this shape can be calculated analytically as a function of the noise level thus the noise level can be obtained from a simple function fit the analytical result also shows that a noise level of more than 2 will obscure any possible scaling of the correlation integral and thus makes it impossible to estimate the correlation dimension

monoids of intervals of ordered abelian groups abstract for any partially ordered abelian group g we relate the structure of the ordered monoid g of intervals of g i e nonempty upward directed lower subsets of g to various properties of g as for example interpolation properties or topological properties of the state space when g has an order unit this allows us to solve a problem by k r goodearl by proving that even in most natural cases multiplier groups of dimension groups often fail to be interpolation groups furthermore the study of monoids of intervals in the totally ordered case yields a characterization of hahn powers of the real line by a first order sentence on the positive interval monoid

on a model stationary nonlinear wave in an active medium a lienard type model equation describing stationary nonlinear wave motion in an active medium is investigated this model equation has been derived from the evolution equation governing pulse transmission in a nerve fibre two models the classical third order fitzhugh nagumo equation and the model second order equation are compared and analysed the possibility of calculating a single pulse in the whole time range without any convergence problems offers a certain advantage to a lienard type model equation in physical terms this equation is able to describe both a threshold effect and the possibility of amplification or attenuation a qualitative analysis of the phase portrait of this model is presented it is proved that limit cycles in the phase plane cannot exist under the conditions arising from the physics of the nerve fibre the threshold problem is analysed in detail and an algorithm is presented to find the threshold distinguishing the processes of amplification and attenuation the results obtained in this work permit a full map of the solutions to be given for a general lienard type equation the model equation under consideration describes a single pulse but the full map of the solutions also contains periodic solutions corresponding to the well known van der pol equation

existence and continuity of uniform exponential attractors of the singularity perturbed hodgkin huxley system abstract in order to show that the standard partially dissipative hodgkin huxley system approximates the hyperbolic hodgkin huxley system for large time we are interested in establishing the existence of exponential attractors for the hyperbolic system with estimates of the fractal dimension and the rate of attraction of trajectories which are uniform with respect to the perturbation furthermore we obtain the continuity of exponential attractors

surface critical phenomena in three dimensional percolation using monte carlo methods and finite size scaling we investigate surface critical phenomena in the bondpercolation model on the simple cubic lattice with two open surfaces in one direction we decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios q1 and qb defined on the basis of the moments of the cluster size distribution these ratios are used to determine critical points at the bulk percolation threshold pbc we determine the surface bond occupation probability at the special transition as p 1c ssd 0 418 17s2d and further obtain the corresponding surface thermal and magnetic exponents as y t1 ssd 0 5387s2d and y h1 ssd 1 8014s6d respectively next from the pair correlation function on the surfaces we determine y h1 sod 1 0246s4d and y h1 sed 1 25s6d for the ordinary and the extraordinary transition respectively of which the former is consistent with the existing result y h1 sod 1 024s4d we also numerically derive the line of surface phase transitions occurring at pb pbc and determine the pertinent asymptotic values of the universal ratios q1 and qb

series expansions for the 3d transverse ising model at t 0 both weak coupling and strong coupling series expansions are calculated for the ising model in a transverse field at zero temperature in three dimensions series are obtained for the ground state energy the magnetization the susceptibility and the energy gaps on the simple cubic the body centred cubic and the face centred cubic lattices the analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four dimensional simple ising model there is a remarkable degree of universality between all three lattices

dynamical analysis of the hindmarsh rose neuron with time delays this brief is mainly concerned with a series of dynamical analyses of the hindmarsh rose hr neuron with state dependent time delays the dynamical analyses focus on stability hopf bifurcation as well as chaos and chaos control through the stability and bifurcation analysis we determine that increasing the external current causes the excitable hr neuron to exhibit periodic or chaotic bursting spiking behaviors and emit subcritical hopf bifurcation furthermore by choosing a fixed external current and varying the time delay the stability of the hr neuron is affected we analyze the chaotic behaviors of the hr neuron under a fixed external current through time series bifurcation diagram lyapunov exponents and lyapunov dimension we also analyze the synchronization of the chaotic time delayed hr neuron through nonlinear control based on an appropriate lyapunov krasovskii functional with triple integral terms a nonlinear feedback control scheme is designed to achieve synchronization between the uncontrolled and controlled models the proposed synchronization criteria are derived in terms of linear matrix inequalities to achieve the global asymptotical stability of the considered error model under the designed control scheme finally numerical simulations pertaining to stability hopf bifurcation periodic chaotic and synchronized models are provided to demonstrate the effectiveness of the derived theoretical results

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed hodgkin huxley systems abstract in an effort to show that the standard hodgkin huxley system approximates the hyperbolic hodgkin huxley system we view the hyperbolic system as a singular perturbation of the standard system we establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and hausdorff dimensions we then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero

cluster monte carlo simulation of the transverse ising model we formulate a cluster monte carlo method for the anisotropic limit of ising models on d 1 dimensional lattices which in effect are equivalent with d dimensional quantum transverse ising models using this technique we investigate the transverse ising models on the square triangular kagome honeycomb and simple cubic lattices the monte carlo data are analyzed by finite size scaling in each case we find as expected that the critical behavior fits well in the d 1 dimensional ising universality class for the transverse ising model on the square lattice we determine the binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or classical direction and that in the other two directions matching this universal function with the case of the isotropic ising model yields the length ratio relating the isotropic ising model with the anisotropic limit the efficiency of the present algorithm is reflected by the precision of its results which improves significantly on earlier analyses

an initial boundary value problem of physiological significance for equations of nerve conduction a model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid loligo of hodgkin and huxley 5 this nonlinear parabolic partial differential equation pde when considered as part of a particular initial boundary value problem ibvp models the electrical activity of an axon under conditions found both in nature and in the laboratory this ibvp for the hodgkin huxley equations is proved to be well posed in the sense of hadamard 3 and a priori bounds on the solution are derived

on theory of pulse transmission in a nerve fibre the ray method is used to analyse pulse propagation in a nerve fibre the initial equations are considered to be hyperbolic i e with a finite velocity of propagation and the corresponding evolution equation is obtained the evolution equation that describes the dynamical side of the process is an equation of the first order being the natural result of applying the ray method this approach avoids the paradox of infinite velocity arising from the initial parabolic system used in the hodgkin huxley model but does not make the final governing equation more complicated the biochemical side of the process is described by means of the hodgkin huxley or fitzhugh nagumo models the steady state solution if it exists leads to an equation of the lienard type and the correspondence between the equations of the catastrophe theory and the usual nerve pulse equations is established

mutual information and global strange attractors in taylor couette flow abstract we apply a new method for the estimation of the mutual information m to experimental time series from different locations in weakly turbulent taylor couette flow we observe a dependence of m on the partition scale e and find m e dm log 1 e for small e and sufficient embedding dimension m our results confirm that the overall dynamics of the taylor couette flow is dominated by a global strange attractor of fractal information dimension d 1 d m at moderate reynolds numbers the dimension estimates obtained by our method are found to be more reliable than results from dimension algorithms based on single measurements alone a fact which makes our method most attractive for dimension analysis in extended systems

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

a new method to estimate the kolmogorov entropy from recurrence plots its application to neuronal signals recurrence plots are a useful graphic tool for studying dynamical systems a new variable is proposed to quantify their deterministic structure it allows an easy computation of the k2 entropy since the distance between points needs to be calculated only once instead of every time for each new embedding dimension and it bypasses some of the limitations inherent to the classical grassberger and procaccia method it is valuable for biological data as illustrated here by tests with various dynamical models and its application to experimental signals recorded from neurons