representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

strong connections and chern connes pairing in the hopf galois theory we reformulate the concept of connection on a hopf galois extension b p in order to apply it in computing the chern connes pairing between the cyclic cohomology hc 2 n b and k 0 b this reformulation allows us to show that a hopf galois extension admitting a strong connection is projective and left faithfully flat it also enables us to conclude that a strong connection is a cuntz quillen type bimodule connection to exemplify the theory we construct a strong connection super dirac monopole to find out the chern connes pairing for the super line bundles associated to a super hopf fibration

an analytic approach to practical state space reconstruction we study the three standard methods for reconstructing a state space from a time series delays derivatives and principal components we derive a closed form solution to principal component analysis in the limit of small window widths this solution explains the relationship between delays derivatives and principal components it shows how the singular spectrum scales with dimension and delay time and it explains why the eigenvectors resemble the legendre polynomials most importantly the solution allows us to derive a guideline for choosing a good window width unlike previous suggestions this guideline is based on first principles and simple quantities we argue that discrete legendre polynomials provide a quick and not so dirty substitute for principal component analysis and that they are a good practical method for state space reconstruction

comparative study of embedding methods embedding experimental data is a common first step in many forms of dynamical analysis the choice of appropriate embedding parameters dimension and lag is crucial to the success of the subsequent analysis we argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself therefore we base our analysis on an examination of systems that have explicit analytic representations a comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria the assessment also includes measures of robustness to noise the limitations of this study are explicitly delineated while bearing these limitations in mind we conclude that for the examples considered here the best identification of the embedding dimension was achieved with a global false nearest neighbors argument and the best value of lag was identified by the mutual information function

characterization of a two parameter mixed mode electrochemical behavior regime using neural networks abstract we use nonlinear signal processing techniques based on artificial neural networks to construct a continuous time model set of ordinary differential equations odes from experimental observations of mixed mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions the data was in the form of time series of the potential for different values of the applied current and chloride ion concentration we use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase space using numerical bifurcation techniques we locate stable and unstable periodic solutions calculate eigenvalues and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings for example the fitted model suggests the existence of a sub critical hopf bifurcation near the range of parameters probed in the experiments this might explain the experimental difficulty in locating small amplitude simple oscillations

dimension measurement on high dimensional systems abstract using experimental data from a variety of laboratory electrical systems the upper limits on dimension measurement are examined and it is shown that contrary to standard scaling estimates it can be possible to reliably measure correlation dimensions larger than 10 the practical implications of the constraints on such a measurement are examined including the role of statistical errors geometrical artifacts dimension algorithm metric choice and the process power spectrum these ideas are applied to electrical 1 f noise from both macroscopic and microscopic sources to show that it cannot be explained by a deterministic model with fewer than roughly 20 degrees of freedom

distortions of reconstruction for chaotic attractors abstract we consider the problem of quality evaluation for delay reconstruction of chaotic attractors from scalar time series the proposed approach is based upon local linear analysis of the reconstruction mapping it is shown that some linear distortions in principle could be corrected but usually nonlinear distortions make it dificult to efficiently apply this technique the new algorithms of estimating the scale of nonlinear distortion and irrelevance time upper limit for the reconstruction window from a time series are proposed both of them do not require pricise knowledge of minimal embedding dimension for the attractor studied

improvement of k 2 entropy calculations by means of dimension scaled distances abstract the k2 entropy is evaluated for experimental time series as well as for numerically simulated ones the conventional calculation procedure leads to results that can be extrapolated to obtain a k2 value for the time series but in the case of experimental systems the outcome is hardly significant for a classification of the states the reason for this is a distortion of distances due to the phase space embedding procedure by time delay coordinates therefore a k2 calculation based on dimension scaled distances dsd is suggested a comparison to the often suggested use of the maximum norm is discussed by the application of dsd the convergence problems are avoided and clear results were obtained in all investigated situations

noncommutative circle bundles and new dirac operators we study spectral triples over noncommutative principal u 1 bundles basing on the classical situation and the abstract algebraic approach we propose an operatorial definition for a connection and compatibility between the connection and the dirac operator on the total space and on the base space of the bundle we analyze in details the example of the noncommutative three torus viewed as a u 1 bundle over the noncommutative two torus and find all connections compatible with an admissible dirac operator conversely we find a family of new dirac operators on the noncommutative tori which arise from the base space dirac operator and a suitable connection

learning driver response relationships from synchronization patterns we test recent claims that causal driver response relationships can be deduced from interdependencies between simultaneously measured time series we apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors the systems that we study are asymmetrically coupled simple models lorenz roessler and henon models the couplings being such that they lead to generalized synchronization if the data were perfect noise free infinitely long we should be able to detect at least in some cases which of the coupled systems is the driver and which the response this might no longer be true if the time series has finite length instead estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them and causal relationships are more difficult to detect we also show that slightly different variants of the interdependence measure can have quite different sensitivities

estimation of interrelation between chaotic observables abstract the method for the estimation of interrelation between chaotic observables from time series data is proposed relation between the suggested interrelation parameter and coupling parameter for a system of two coupled henon maps is tested numerically as an example of a spatially extended system the interrelation between oscillators linked in a one dimensional chain is studied

noncommutative instantons on the 4 sphere from quantum groups we describe an approach to the noncommutative instantons on the 4 sphere based on quantum group theory we quantize the hopf bundle 7 4 making use of the concept of quantum coisotropic subgroups the analysis of the semiclassical poisson lie structure of u 4 shows that the diagonal su 2 must be conjugated to be properly quantized the quantum coisotropic subgroup we obtain is the standard su q 2 it determines a new deformation of the 4 sphere 4 q as the algebra of coinvariants in q 7 we show that the quantum vector bundle associated to the fundamental corepresentation of su q 2 is finitely generated and projective and we compute the explicit projector we give the unitary representations of 4 q we define two 0 summable fredholm modules and we compute the chern connes pairing between the projector and their characters it comes out that even the zero class in cyclic homology is non trivial

bivariate surrogate techniques necessity strengths and caveats the concept of surrogates allows testing results from time series analysis against specified null hypotheses in application to bivariate model dynamics we here compare different types of surrogates each designed to test against a different null hypothesis e g an underlying bivariate linear stochastic process two measures that aim at a characterization of interdependence between nonlinear deterministic dynamics were used as discriminating statistics we analyze eight different stochastic and deterministic models not only to demonstrate the power of the surrogates but also to reveal some pitfalls and limitations

discrete supergroups and super riemann surfaces the properties of discrete supergroups that represent the homotopy groups of super riemann surfaces are emphasized after the study of the superforms invariant with respect to these supergroups the set of superconformal differentials are defined and an example is given of those by defining poincare s super theta functions some remarks on a possible definition of the jacobi variety of a super riemann surface are given

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

direction of coupling from phases of interacting oscillators an information theoretic approach a directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators its abilities to distinguish unidirectional from bidirectional coupling as well as to reveal and quantify asymmetry in bidirectional coupling are demonstrated using numerical examples of quasiperiodic chaotic and noisy oscillators as well as real human cardiorespiratory data

detection of weak directional coupling phase dynamics approach versus state space approach we compare two conceptually different approaches to the detection of weak directional couplings between two oscillatory systems from bivariate time series the first approach is based on the analysis of the systems phase dynamics whereas the other one tests for interdependencies in the reconstructed state spaces of the systems we analyze the sensitivity of both techniques to weak couplings in numerical experiments by considering couplings between almost identical as well as between significantly different nonlinear systems we study different degrees of phase diffusion test the robustness of the two techniques against observational noise and investigate the influence of the time series length our results show that none of the two approaches is generally superior to the other and we conclude that it is probably the combination of both techniques that would allow the most comprehensive and reliable characterization of coupled systems

model misspecification tests model building and predictability in complex systems abstract in this paper we show that significant advantages can be realized by using functions of simple variables such as time lags as directions in reconstruction spaces for complicated time series first we show that more sensitive model misspecification tests can be constructed when the candidate model is used as one of the directions in the reconstruction space this naturally results in misspecification tests based on the construction of conditional probabilities which can reveal misspecified models even when other powerful methods such as the bds test applied to a sequence of residuals fails second we show that model building and predictability of the time series can be substantially improved by using an informational criterion to determine the functions with which to associate the directions of the reconstruction space this criterion is of the form of a conditional probability and is related to a measure of the short term predictability of the time series we consider specifically an example in which the search space of functions is linear combinations of time lags and we compare the resulting model with a linear least squares fit to the data we also show that this optimized reconstruction space can improve longer term predictability we demonstrate this by comparing longer term predictions made using the optimized space with predictions made using a simple time lagged reconstruction space

measure of predictability many techniques have been developed to measure the difficulty of forecasting data from an observed time series this paper introduces a measure which we call the forecast entropy designed to measure the predictability of a time series we use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor we then consider these distributions on different scales we present a formula for calculating the forecast entropy to provide a standard of predictability we define an idealized random system whose forecast entropy will be maximal we then use this measure to rescale the forecast entropy to lie in the range 0 1 the time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure we present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors we also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one

braided affine geometry and q analogs of wave operators the main goal of this review is to compare different approaches to constructing the geometry associated with a hecke type braiding in particular with that related to the quantum group uq sl n we place emphasis on the affine braided geometry related to the so called reflection equation algebra rea all objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators we begin by comparing the poisson counterparts of quantum varieties and describe different approaches to their quantization also we exhibit two approaches to introducing q analogs of vector bundles and defining the chern connes index for them on quantum spheres in accordance with the serre swan approach the q vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras besides we describe the basic properties of the rea used in this construction and compare different ways of defining q analogs of partial derivatives and differentials on the rea and algebras close to them in particular we present a way of introducing a q differential calculus via koszul type complexes the elements of the q calculus are applied to defining q analogs of some relativistic wave operators

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

deformation quantization of surjective submersions and principal fibre bundles in this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold in case of a principal fibre bundle we require in addition invariance under the principal action we prove existence and uniqueness of such deformations the commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators applications to noncommutative gauge field theories and phase space reduction of star products are discussed

mutual information functions versus correlation functions this paper studies one application of mutual information to symbolic sequences the mutual information functionm d this function is compared with the more frequently used correlation function d an exact relation betweenm d and d is derived for binary sequences for sequences with more than two symbols no such general relation exists in particular d 0 may or may not lead tom d 0 this linear but not general independence between symbols separated by a distance is studied for ternary sequences also included is the estimation of the finite size effect on calculating mutual information finally the concept of symbolic noise is discussed

topologically nontrivial field configurations in noncommutative geometry in the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated fuzzy sphere the corresponding field theory actions conserve all basic symmetries of the standard commutative version space isometries and global chiral symmetry but due to the noncommutativity of the space the fields are regularized and they contain only a finite number of modes

strong connections on quantum principal bundles a gauge invariant notion of a strong connection is presented and characterized it is then used to justify the way in which a global curvature form is defined strong connections are interpreted as those that are induced from the base space of a quantum bundle examples of both strong and non strong connections are provided in particular such connections are constructed on a quantum deformation of the two sphere fibrations2 rp2 a certain class of stronguq 2 connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq dependent hermitian metric a particular form of the yang mills action on a trivialuq 2 bundle is investigated it is proved to coincide with the yang mills action constructed by a connes and m rieffel furthermore it is shown that the moduli space of critical points of this action functional is independent ofq

generalized redundancies for time series analysis abstract extensions to various information theoretic quantities such as entropy redundancy and mutual information are discussed in the context of their role in nonlinear time series analysis we also discuss linearized versions of these quantities and their use as benchmarks in tests for nonlinearity many of these quantities can be expressed in terms of the generalized correlation integral and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics such as the bds and green savit statistics further numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation finally we consider several local versions of these quantities including a local kolmogorov sinai entropy which gives an estimate of variability of the short term predictability

singular value decomposition in attractor reconstruction pitfalls and precautions applicability of singular value decomposition for reconstructing the strange attractor from one dimensional chaotic time series proposed by broomhead and king is extensively tested and discussed previously published doubts about its reliability are confirmed singular value decomposition by nature a linear method can bring distorted and misleading results when nonlinear structures are studied

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

statistical analysis of the extreme values of stress time series from the portevin le châtelier effect in an effort to understand the deterministic vs stochastic character of the portevin char21 le ch atelier plc phenomenon we investigate the structure of the underlying mechanism that generates the stick slip patterns of stress over time the stress time series is reduced to a series of successive pairs of minimum and maximum values representing the stick slip patterns and a statistical analysis by means of hypothesis testing is applied to it the null hypothesis of least deterministic structure is that the time series of extreme values is a bounded random walk of alternating direction brwad that is besides the constraint of succession of minima to maxima bounded at a predefined range there are no other correlations in the data to implement the test we use surrogate data generated by a model consistent with a brwad type process which also uses the statistics of the original data to best mimic them the proposed hypothesis testing is found to perform properly on simulated data from stochastic and deterministic systems for the plc time series the null hypothesis is rejected at a high level of confidence giving evidence for some deterministic structure in the succession of the extreme stress values this result allows for further statistical analysis including also the time aspect of the stick slip patterns

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

a schneider type theorem for hopf algebroids abstract comodule algebras of a hopf algebroid h with a bijective antipode i e algebra extensions b a by h are studied assuming that a lifted canonical map is a split epimorphism of modules of the non commutative base algebra of h relative injectivity of the h comodule algebra a is related to the galois property of the extension b a and also to the equivalence of the category of relative hopf modules to the category of b modules this extends a classical theorem by h j schneider on galois extensions by a hopf algebra our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor a notion introduced and analysed hereby

forecasting a chaotic time series using an improved metric for embedding space abstract a method is presented for improving the accuracy of forecasts of future values of a time series of one variable of a chaotic dynamical system the metric for the embedding space of time delay vectors of time series values is obtained from a metric tensor whose components are varied to minimize root mean squared error of forecasts the improvement of forecasts is demonstrated for time series of the henon and ikeda maps the lorenz system of coupled differential equations and the mackey glass delay differential equation

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

testing for nonlinearity using redundancies quantitative and qualitative aspects abstract a method for testing nonlinearity in time series is described based on information theoretic functionals redundancies linear and nonlinear forms of which allow either qualitative or after incorporating the surrogate data technique quantitative evaluation of dynamical properties of scrutinized data an interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated evaluation of redundancies and redundancy based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

addition theorems for spherical wave solutions of the vector helmholtz equation addition theorems for spherical wave solutions of the vector helmholtz equation are discussed the theorems allow one to expand a vector spherical wave about a given origin into spherical waves about a shifted origin a simplified derivation of the results obtained earlier by cruzan o r cruzan q appl math 20 33 1962 is presented

multiple scattering by multiple spheres a new proof of the lloyd berry formula for the effective wavenumber we provide the first classical derivation of the lloyd berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers our approach clarifies the assumptions under which the lloyd berry formula is valid more precisely we derive an expression for the effective wavenumber which assumes the validity of lax s quasi crystalline approximation but makes no further assumptions about scatterer size and then we show that the lloyd berry formula is obtained in the limit as the scatterer size tends to zero

effective scaling regime for compution the correlating dimension from chaotic time series abstract in the analysis of chaotic time series a standard technique is to reconstruct an image of the original dynamical system using delay coordinates if the original dynamical system has an attractor then the correlation dimension d2 of its image in the reconstruction can be estimated using the grassberger procaccia algorithm the quality of the reconstruction can be probed by measuring the length of the linear scaling region used in this estimation in this paper we show that the quality is constrained by both the embedding dimension m and more importantly by the delay time for a given embedding dimension and a finite time series there exists a maximum allowed delay time beyond which the size of the scaling region is no longer reliably discernible we derive an upper bound for this maximum delay time numerical experiments on several model chaotic time series support the theoretical argument they also clearly indicate the different roles played by the embedding dimension and the delay time in the reconstruction as the embedding dimension is increased it is necessary to reduce the delay time substantially to guarantee a reliable estimate of d2 our results imply that it is the delay time itself rather than the total observation time m 1 which plays the most critical role in the determination of the correlation dimension

quantum geometry of algebra factorisations and coalgebra bundles we develop the noncommutative geometry bundles connections etc associated to algebras that factorise into two subalgebras an example is the factorisation of matrices m2 2 2 we also further extend the coalgebra version of theory introduced previously to include frame resolutions and corresponding covariant derivatives and torsions as an example we construct q monopoles on all the podle quantum spheres s2q s

indications of nonlinearity intraindividual specificity and stability of human eeg the unfolding dimension abstract with the help of a parameter which we call the unfolding dimension ww provide a bivariate representation of dynamical eeg analysis applied to human eeg our approach successfully discriminates surrogate data from raw eeg and similarly shows human eeg to be both intraindividually specific and stable over time the heart of our approach is the grassberger procaccia algorithm for the determination of the correlation dimension d 2 within the context of the well known method of time delays to guarantee the reproducibility of results this algorithm as well as an estimate for the k 2 entropy and the determination of f spectra have been integrated into a computer program which encompasses an operator user independent automatic and reproducible specification of both an optimal time delay for calculating the correlation integral as well as an optimal scalar invariant plateau region for the extraction of d 2 and k 2 our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms in particular with false nearest neighbor considerations and sauer s minimal embedding criteria particular intelligent features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological psychological data

principal fibrations from noncommutative spheres we construct noncommutative principal fibrations s 7 s 4 which are deformations of the classical su 2 hopf fibration over the four sphere we realize the noncommutative vector bundles associated to the irreducible representations of su 2 as modules of coequivariant maps and construct corresponding projections the index of dirac operators with coefficients in the associated bundles is computed with the connes moscovici local index formula the algebra inclusion is an example of a not trivial quantum principal bundle

on proper orthogonal co ordinates as indicators of modal activity proper orthogonal decomposition pod is applied for examining modal activity the extraction of proper orthogonal modal co ordinates pocs is outlined the proper orthogonal values povs are the mean squared values of the pocs the number of povs above the noise floor provides a bound on the number of significant modes based on povs above the noise floor the ideas are illustrated on a linear cantilevered beam experiment the displacement ensembles are obtained by processing six strain measurements coherent proper orthogonal modes poms and povs above the noise floor together confirm that six active modes are detected in the system the distribution of modal components in pocs is discussed the characteristics of the poms povs and pocs are then examined in the presence of added noise

a hopf bundle over a quantum four sphere from the symplectic group we construct a quantum version of the su 2 hopf bundle s7 s4 the quantum sphere s7q arises from the symplectic group spq 2 and a quantum 4 sphere s4q is obtained via a suitable self adjoint idempotent p whose entries generate the algebra a s4q of polynomial functions over it this projection determines a deformation of an anti instanton bundle over the classical sphere s4 we compute the fundamental k homology class of s4q and pair it with the class of p in the k theory getting the value 1 for the topological charge there is a right coaction of suq 2 on s7q such that the algebra a s7q is a non trivial quantum principal bundle over a s4q with structure quantum group a suq 2

comparison of algorithms calculating optimal embedding parameters for delay time coordinates the success of noise reduction processes and forecasting methods as well as accurate calculations of fractal dimensions and lyapunov exponents from scalar time series depends strongly on the quality of the reconstructed phase space we present a comparison of four methods which simultaneously estimate the two embedding parameters delay time and sufficiently large embedding dimension dime for takens delay time coordinates recently we introduced a global geometrical measure the fill factor and a local dynamical measure the averaged integral local deformation we will discuss these methods briefly and compare them to the topological wavering product in addition a simple algorithm is presented which observes the spreading of trajectories at the transition from dime to dime 1 and gives the opportunity to estimate the correlation entropy k2 we applied these algorithms to experimental data series measured from a rotational taylor couette experiment and verified the quality of the obtained embedding parameters by the estimation of correlation dimensions for different reconstructed attractors

noncommutative principal bundles through twist deformation we construct noncommutative principal bundles deforming principal bundles with a drinfeld twist 2 cocycle if the twist is associated with the structure group then we have a deformation of the fibers if the twist is associated with the automorphism group of the principal bundle then we obtain noncommutative deformations of the base space as well combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space more in general the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new hopf galois extensions as twists of hopf galois extensions a sheaf approach is also considered and examples presented

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

quantum principal bundles and corresponding gauge theories a generalization of classical gauge theory is presented in which compact quantum groups play the role of the internal symmetry groups all considerations are performed in the framework of a noncommutative geometric formalism of locally trivial quantum principal bundles over classical smooth manifolds quantum counterparts of classical gauge bundles and classical gauge transformations are introduced and investigated a natural dierential calculus on quantum gauge bundles is constructed and analyzed kinematical and dynamical properties of corresponding gauge theories are discussed a particular attention is given to the purely quantum phenomenas appearing in the formalism and their physical interpretation an example with quantum su 2 group is considered

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

identification and prediction of low dimensional dynamics this contribution focuses upon extracting information from dynamic reconstructions of experimental time series data in addition to the problem of distinguishing between deterministic dynamics and stochastic dynamics applied questions such as the detection of parametric drift are addressed nonlinear prediction and dimension algorithms are applied to geophysical laboratory data and the significance of these results is established by comparison with results from similar surrogate series generated so as not to contain the property of interest a global nonlinear predictor is introduced which attempts to correct systematic bias due to the inhomogeneous distribution of data common in strange attractors variations in the quality of predictions with location in phase space are examined in order to estimate the uncertainty in a forecast at the time it is made finally the application of these methods to truly stochastic systems is discussed and the distinction between deterministic stochastic and low dimensional dynamics is considered

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

retarded hydrodynamic interactions in suspensions abstract we extend the method to determine the grand mobility matrix presented in a previous article to the problem of two particles in an unbounded fluid satisfying the time dependent linear incompressible navier stokes equations use of this method combined with a generalization of the fluctuation dissipation theorem results in the determination of the correlation matrix we determine the effect of hydrodynamic interactions on the time dependent diffusion coefficients d 11 and d 12 finally we study the effects of the correlation functions on brownian motion

projective module description of the q monopole the dirac q monopole connection is used to compute projector matrices of quantum hopf line bundles for arbitrary winding number the chern connes pairing of cyclic cohomology and k theory is computed for the winding number 1 the non triviality of this pairing is used to conclude that the quantum principal hopf fibration is non cleft among general results we provide a left right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode we also provide for arbitrary strong connections on algebraic quantum principal bundles hopf galois extensions their associated covariant derivatives on projective modules

verification of chaotic behavior in an experimental loudspeaker the dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis delay time embedding dimension fractal dimension and other empirical quantities are determined from experimental data particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty lyapunov exponents and fractal dimension are measured using several independent techniques results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant lyapunov exponent we thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series

an alternative expression for the addition theorems of spherical wave solutions of the helmholtz equation an alternative formulation of the addition theorem for spherical wave solutions of the helmholtz equation is presented the 3 j symbols of wigner or more precisely the gaunt coefficient complete solid angle integral of a triple product of spherical harmonics which appear in the formerly introduced expressions of these addition theorems are replaced by an explicit matrix expression relating the spherical wave solutions defined with respect to the different origins the generalized gaunt coefficients which are complete solid angle integrals of a multiple product of spherical harmonics can then be written in terms of a matrix product of basic matrices representing the gaunt coefficient

reconstructing attractors from scalar time series a comparison of singular system and redundancy criteria abstract a delay vector phase space reconstruction in which the delay time satisfiesa minimum redundancy criterion is compared with a reconstruction obtained using a singular system approach minimum redundancy produces the better reconstruction the reconstructions are compared using a distortion functional d which measures how well the location of a point in the original phase space can be determined on the basis of its image under the reconstruction process the superiority of the redundancy analysis over the singular system analysis is found to arise from the former s foundation on the notion of general independence as opposed to the latter s foundation on the notion of linear independence

box counting algorithm and dimensional analysis of a pulsar abstract it is argued that the use of the box counting algorithm to calculate the correlation dimension is a better choice than the grassberger procaccia correlation integral algorithm for dealing with an experimental data set this is illuminated by treating three classical examples the logistic map the hdnon map and the lorenz equation the intensity data of a pulsar is also treated which is revealed to have a least embedding dimension of 14 and the correlation dimension of about 4 5