linearization and krein like functionals of hypergeometric orthogonal polynomials the krein like r functionals of the hypergeometric orthogonal polynomials pn x with the kernel of the form xs x pm1 x pmr x being x the weight function on the interval r are determined by means of the srivastava linearization method the particular 2 functionals which are particularly relevant in quantum physics are explicitly given in terms of the degrees and the characteristic parameters of the polynomials they include the well known power moments and the novel krein like moments moreover various related types of exponential and logarithmic functionals are also investigated

green s operator for hamiltonians with coulomb plus polynomial potentials the hamiltonian of a coulomb plus polynomial potential in the coulomb sturmian basis has an infinite symmetric band matrix structure a band matrix can always be considered as a block tridiagonal matrix so the corresponding green s operator can be given as a matrix valued continued fraction as examples we calculate green s operator for the coulomb plus linear and quadratic confinement potential problems and determine the energy levels

on the n back and forth types of boolean algebras the objective of this paper is to uncover the structure of the back and forth equivalence classes at the nite levels for the class of boolean algebras as an application we obtain bounds on the computational complexity of determining the back and forth equivalence classes of a boolean algebra for nite levels this result has implications for characterizing the relatively intrinsically 0 relations of boolean algebras as existential formulas over a nite set of relations

weakly convergent expansions of a plane wave and their use in fourier integrals the fourier transform of an irreducible spherical tensor is normally computed with the help of the rayleigh expansion of a plane wave in terms of spherical bessel functions and spherical harmonics the angular integrations are then trivial however the remaining radial integral containing a spherical bessel function may be so complicated that the applicability of fourier transformation is severely restricted as an alternative the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested the weakly convergent expansions of a plane wave are constructed in such a way that their application in fourier integrals leads to expansions of the fourier or inverse fourier transform that converge with respect to the norm of either the hilbert space l2 r3 or the sobolev space w 1 2 r3 accordingly these weakly convergent expansions may be viewed as distributions that are defined on either l2 r3 or w 1 2 r3 the properties of some complete orthonormal se

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

quantitative theory of driven nonlinear brain dynamics abstract strong periodic stimuli such as bright flashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity but the underlying mechanisms for these phenomena are poorly understood at present the dominant features of these experimentally observed dynamics are reproduced by the dynamics of a quantitative neural field model subject to periodic drive model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f subharmonic entrainment near 2 f and harmonic generation further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude including period doubling bistable phase locking hysteresis wave mixing and chaos indicated by positive lyapunov exponents moreover photosensitive seizures are predicted for physiologically realistic model parameters yielding bistability between healthy and seizure dynamics these results demonstrate the applicability of neural field models to the new regime of periodically driven nonlinear dynamics enabling interpretation of experimental data in terms of specific generating mechanisms and providing new tests of the theory

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

entropic nonextensivity as a measure of time series complexity abstract information entropy is applied to the analysis of time series generated by dynamical systems complexity of a temporal or spatio temporal signal is defined as the difference between the sum of entropies of the local linear regions of the trajectory manifold and the entropy of the globally linearized manifold when the entropies are tsallis entropies the complexity is characterized by the value of q

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

bound and resonant states in coulomb like potentials the potential separable expansion method has been generalised to calculating bound and resonant states in coulomb like potentials the complete set of coulomb sturmian functions was taken as the basis to expand the short range potential on this basis the matrix elements of the coulomb green functions have been given in closed form as functions of the complex energy the feasibility of the method is demonstrated by a numerical example

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

boolean algebra approximations knight and stob proved that every low4 boolean algebra is 0 6 isomorphic to a computable one furthermore for n 1 2 3 4 every lown boolean algebra is 0 n 2 isomorphic to a computable one we show that this is not true for n 5 there is a low5 boolean algebra that is not 0 7 isomorphic to any computable boolean algebra it is worth remarking that because of the machinery developed the proof uses at most a 0 00 priority argument the technique used to construct this boolean algebra is new and might be useful in other applications such as to solve the lown boolean algebra problem either positively or negatively

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

algebraic evaluation of matrix elements in the laguerre function basis the laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron structure theory with applications in hadronic and nuclear theory as well while similar in form to the coulomb bound state eigenfunctions from the schrodinger eigenproblem or the coulomb sturmian functions from a related sturm liouville problem the laguerre functions unlike these former functions constitute a complete discrete orthonormal set for square integrable functions in three dimensions we construct the su 1 1 so 3 dynamical algebra for the laguerre functions and apply the ideas of factorization or supersymmetric quantum mechanics to derive shift operators for these functions we use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators involving powers of the radial coordinate and radial derivative in the laguerre function basis we illustrate how matrix elements for more general spherical tensor operators in three dimensional space such as the gradient may then be constructed from these radial matrix elements

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

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

hybrid taguchi chaos of multilevel immune and the artificial bee colony algorithm for parameter identification of chaotic systems in this paper a novel evolutionary learning algorithm is proposed by hybridizing the taguchi method chaos disturbance operation multilevel immune algorithm mia and artificial bee colony algorithm abc the algorithm is thus called htcmiabc to estimate the parameter of chaotic systems the htcmiabc comprises two main different phases first we use the mia as the recognition phase to balance local and global searches and accelerate the search speed to enhance the evolutionary phase second the evolutionary phase is built on the abc and chaos disturbance operation to have the capabilities of exploration and exploitation moreover the taguchi method and crossover operation are inserted between the recognition phase and evolutionary phase for the recombination and diversification of several antibodies to improve the searching ability finally the htcmiabc algorithm is examined by parameter identification of the nonlinear chaotic system simulation results show that the proposed algorithm is more efficient than some typical existing algorithms the effects of noise and population size are investigated as well

detecting intrinsic dynamics of traffic flow with recurrence analysis and empirical mode decomposition abstract in this paper we apply the empirical mode decomposition emd method to the recurrence plot rp and recurrence quantification analysis rqa to evaluate the frequency and time evolving dynamics of the traffic flow based on the cumulative intrinsic mode functions extracted by the emd the frequency evolving rp regarding different oscillation of modes suggests that apparent dynamics of the data considered are mainly dominated by its components of medium and low frequencies while severely affected by fast oscillated noises contained in the signal noises are then eliminated to analyze the intrinsic dynamics and consequently the denoised time evolving rqa diversely characterizes the properties of the signal and marks crucial points more accurately where white bands in the rp occur whereas a strongly qualitative agreement exists between all the non denoised rqa measures generally the emd combining with the recurrence analysis sheds more reliable abundant and inherent lights into the traffic flow which is meaningful to the empirical analysis of complex systems

a computability theoretic equivalent to vaught s conjecture we prove that for every theory t which is given by an l 1 sentence t has less than 2 0 many countable models if and only if we have that for every x 2 on a cone of turing degrees every x hyperarithmetic model of t has an x computable copy we also find a concrete description relative to some oracle of the turing degree spectra of all the models of a counterexample to vaught s conjecture

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

recursive labelling systems and stability of recursive structures in hyperarithmetical degrees we show that under certain assumptions of recursiveness in 2t the recursive structure 2t is ao stable for ca w we need to deal with limit ordinals and we frame an analogous result proposition 2 for yn systems where yn is an increasing sequence of ordinals propositions 1 and 2 are then established by the

chaotic bayesian optimal prediction method and its application in hydrological time series the embedding dimension and the number of nearest neighbors are very important parameters in the prediction of chaotic time series to reduce the prediction errors and the uncertainties in the determination of the above parameters a new chaos bayesian optimal prediction method cbopm is proposed by choosing optimal parameters in the local linear prediction method llpm and improving the prediction accuracy with bayesian theory in the new method the embedding dimension and the number of nearest neighbors are combined as a parameter set the optimal parameters are selected by mean relative error mre and correlation coefficient cc indices according to optimization criteria real hydrological time series are taken to examine the new method the prediction results indicate that cbopm can choose the optimal parameters adaptively in the prediction process compared with several llpm models the cbopm has higher prediction accuracy in predicting hydrological time series

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

useful bases for problems in nuclear and particle physics a set of exactly computable orthonormal basis functions that are useful in computations involving constituent quarks is presented these basis functions are distinguished by the property that they fall off algebraically in momentum space and can be exactly fourier bessel transformed the configuration space functions are associated laguerre polynomials multiplied by an exponential weight and their fourier bessel transforms can be expressed in terms of jacobi polynomials in 2 k2 2 a simple model of a meson containing a confined quark antiquark pair shows that this basis is much better at describing the high momentum properties of the wave function than the harmonic oscillator basis