a new method to detect nonlinearity in a time series synthesizing surrogate data using a kolmogorov smirnoff tested hidden markov model abstract a way of statistically testing for nonlinearity in a time series is to employ the method of surrogate data this method often makes use of the fourier transform ft in order to generate the surrogate as various authors have shown this can lead to artefacts in the surrogates and spurious detection of nonlinearity can result this paper documents a new method to synthesize surrogate data using a 1st order hidden markov model hmm combined with a kolmogorov smirnoff test ks test to determine the required resolution of the hmm significance test results for a sinewave henon map and gaussian noise time series are presented it is demonstrated that ks tested hmm surrogates can be successfully used to distinguish between a deterministic and stochastic time series then by applying a simple test for linearity using linear and nonlinear predictors it is possible to determine the nature of the deterministic class and hence conclude whether the system is linear deterministic or nonlinear deterministic furthermore it is demonstrated that the method works for periodic functions too where ft surrogates break down

semidiscretization in time of nonlinear parabolic equations with blowup of the solution in this paper the author considers the first boundary value problem for the nonlinear equation u t delta u m alpha u m in omega a smooth bounded domain in mathbb r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be larger than one and alpha positive a scheme for the discretization in time of that problem is proposed it is proven that if the exact solution blows up in a finite time it is the same for the numerical solution estimates of the blow up time are obtained the stability of the method and the convergence for a class of initial conditions is proved

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

noisy dynamic simulations in the presence of symmetry data alignment and model reduction abstract we process snapshots of trajectories of evolution equations with intrinsic symmetries and demonstrate the use of recently developed eigenvector based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise our illustrative examples include a one dimensional evolutionary partial differential the kuramoto sivashinsky equation with periodic boundary conditions as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear smoluchowski equation on the surface of a sphere this is a useful first step towards data mining the symmetry adjusted ensemble of snapshots in search of an accurate low dimensional parametrization and the associated reduction of the original dynamical system we also demonstrate a technique vector diffusion maps that combines in a single formulation the symmetry removal step and the dimensionality reduction step

programming cellular permutation networks through decomposition of symmetric groups a fundamental problem in interconnection network theory is to design permutation networks with as few cells as possible and a small programming or setup time the well known networks of benes and waksman have asymptotically optimal cell counts but the best setup algorithm available for such networks with n inputs requires o n log2 n sequential time as an alternative this paper considers another class of permutation networks which are collectively referred to as cellular permutation arrays using a group theoretic formulation a natural correspondence is established between such permutation networks and iterative decompositions of symmetric groups through cosets based on this correspondence the setup problem is reduced to iteratively determining the leaders of the cosets to which the permutation to be realized belongs this in turn leads to linear time setup algorithms for cellular permutation arrays the paper describes these algorithms in detail for two families of cellular permutation arrays reported in the literature

a nine dimensional lorenz system to study high dimensional chaos we examine the dynamics of three dimensional cells with square planform in dissipative rayleigh benard convection by applying a triple fourier series ansatz up to second order we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations depending on two control parameters namely the rayleigh number and the prandtl number the asymptotic behaviour can be stationary periodic quasiperiodic or chaotic a period doubling cascade is identified as a route to chaos hereafter the asymptotic behaviour progressively evolves towards a hyperchaotic attractor for given values of control parameters beyond the accumulation point we observe a low dimensional chaotic attractor as is currently done for dissipative systems although the correlation dimension strongly suggests that this attractor could be embedded in a three dimensional space a topological characterization reveals that a higher dimensional space must be used thus we reconstruct a four dimensional model which is found to be in agreement with the properties of the original dynamics the nine dimensional lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3

on a nonlinear kalman filter with simplified divided difference approximation abstract we present a new ensemble based approach that handles nonlinearity based on a simplified divided difference approximation through stirling s interpolation formula which is hence called the simplified divided difference filter sddf the sddf uses stirling s interpolation formula to evaluate the statistics of the background ensemble during the prediction step while at the filtering step the sddf employs the formulae in an ensemble square root filter ensrf to update the background to the analysis in this sense the sddf is a hybrid of stirling s interpolation formula and the ensrf method while the computational cost of the sddf is less than that of the ensrf numerical comparison between the sddf and the ensrf with the ensemble transform kalman filter etkf as the representative is conducted the experiment results suggest that the sddf outperforms the etkf with a relatively large ensemble size and thus is a good candidate for data assimilation in systems with moderate dimensions

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

dependent variables in broad band continuous time series in this paper we continue our development of new methods for the analysis of broad band time series by deriving quantities which are able to indicate deterministic dependence of an element in one time series on elements in other time series these methods are very broadly applicable and are particularly well suited to the study of continuous time series in which the value of the function may depend on derivatives of the function itself or on other quantities we apply our methods to a number of mathematical examples including the lorentz equation the h6non heiles equations the forced brusselator and the mackey glass equation we show that our methods are very successful at indicating deterministic dependencies in these systems even if the time series are highly chaotic statistical aspects of our procedure are discussed as are a number of interesting and surprising epistomological implications

the universality of a class of modified single stage shuffle exchange networks a class of modified single state shuffle exchange s e networks with reconfigurable connections is described the capability of these modified networks to simulate every arbitrary permutation and other multistage s e networks is studied in particular the upper bounds on simulating six multistage s e networks are given it is also shown that c l wu and t y feng s 1981 modified single stage s e network is optimal in terms of its universality and cost for the networks with one reconfigurable connection however some of the modified networks with two reconfigurable connections have better capability although they are slightly more expensive than wu and feng s network also proposed is a link folding scheme that can be used to reduce the connection complexity of implementing the connection selection mechanism

an efficient algorithm for generating linear transformations in a shuffle exchange network this paper presents an algorithm for generating all the permutations defined by linear transformations on a shuffle exchange network of 2 n processors in 2n 1 passes the proposed algorithm generates any such permutation in o n log 2 n elementary steps the subclass of bit permutations is generated in o n steps

intrinsic dimension estimation of manifolds by incising balls dimensionality reduction is a very important tool in data mining intrinsic dimension of data sets is a key parameter for dimensionality reduction however finding the correct intrinsic dimension is a challenging task in this paper a new intrinsic dimension estimation method is presented the estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball the method is compared with the previous dimension estimation methods experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the santa fe time series competition and the results show that the new method is accurate and robust

self routing technique in perfect shuffle networks using control tags the self routing technique using control tags on multiple pass perfect shuffle networks is generalized in particular they show that bit permute complement permutations can be realized and unscrambled in 2n 1 passes or less where n log sub 2 n n being the number of terminals on either side they also show that most of the frequently used permutations are in the intersection of omega realizing and inverse omega realizing sets and can be realized and unscrambled in n passes

the universality of the shuffle exchange network this paper has focused on the realization of every arbitrary permutation with the shuffle exchange network permutation properties of shuffle exchange networks are studied and are used to demonstrate several universal networks it is concluded that 3 log2 n 1 passes through a single stage regular shuffle exchange network are sufficient to realize every arbitrary permutation where n is network size a routing algorithm is also developed to calculate control settings of the shuffle exchange switches for the permutation realization three optimal universal networks namely expanded direct connection shuffle exchange network multiple pass omega network and modified shuffle exchange network are then exploited for better interconnection purposes in addition this work specifies the inherent relationship between the shuffle exchange network and the benes binary network so that designers can have a broad prospect

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

characterization of local complex structures in a recurrence plot to improve nonlinear dynamic discriminant analysis structures in recurrence plots rps preserving the rich information of nonlinear invariants and trajectory characteristics have been increasingly analyzed in dynamic discrimination studies the conventional analysis of rps is mainly focused on quantifying the overall diagonal and vertical line structures through a method called recurrence quantification analysis rqa this study extensively explores the information in rps by quantifying local complex rp structures to do this an approach was developed to analyze the combination of three major rqa variables determinism laminarity and recurrence rate dlr in a metawindow moving over a rp it was then evaluated in two experiments discriminating 1 ideal nonlinear dynamic series emulated from the lorenz system with different control parameters and 2 data sets of human heart rate regulations with normal sinus rhythms n 18 and congestive heart failure n 29 finally the dlr was compared with seven major rqa variables in terms of discriminatory power measured by standardized mean difference d smd in the two experiments dlr resulted in the highest discriminatory power with d smd 2 53 and 0 98 respectively which were 7 41 and 2 09 times the best performance from rqa the study also revealed that the optimal rp structures for the discriminations were neither typical diagonal structures nor vertical structures these findings indicate that local complex rp structures contain some rich information unexploited by rqa therefore future research to extensively analyze complex rp structures would potentially improve the effectiveness of the rp analysis in dynamic discrimination studies

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

data dimensionality estimation methods a survey in this paper data dimensionality estimation methods are reviewed the estimation of the dimensionality of a data set is a classical problem of pattern recognition there are some good reviews algorithms for clustering data prentice hall englewood cliffs nj 1988 in literature but they do not include more recent developments based on fractal techniques and neural autoassociators the aim of this paper is to provide an up to date survey of the dimensionality estimation methods of a data set paying special attention to the fractal based methods

divide and conquer algorithms on the hypercube abstract we show how to implement divide and conquer algorithms without undue overhead on a wide class of networks we give an optimal generic divide and conquer implementation on hypercubes for the class of divide and conquer algorithms for which the total size of the subproblems on any level of the recursion does not exceed the parent problem size for this implementation appropriately sized subcubes have to be allocated to the subproblems generated by the dividesteps we take care that these allocation steps do not cause any unbalanced distribution of work and that asymptotically they do not increase the running time variants of our generic algorithm also work for the butterfly network and by a general simulation for the class of hypercubic networks including the shuffle exchange and the cube connected cycles network our results can also be applied to optimally solve various types of routing problems

an optimal o nlgn algorithm for permutation admissibility to extra stage cube type networks a k emctn is obtained by adding k more stages in front of a multistage cube type network mctn it is shown that a permutation is admissible to a k emctn if and only if the conflict graph is 2 sup k colorable for the case k 1 an o nlgn algorithm is given for constructing the conflict graph which leads to an o nlgn admissibility algorithm furthermore it is shown that spl omega nlgn bits must be checked in the binary representation of a permutation for determining its admissibility

parallel matrix and graph algorithms matrix multiplication algorithms for cube connected and perfect shuffle computers are presented it is shown that in both these models two n times n matrices can be multiplied in o n m log m time when n 2 m 1 leqq m leqq n processing elements pes are available when only m 2 1 leqq m leqq n pes are available two n times n matrices can be multiplied in o n 2 m m n m 2 61 time it is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms

measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

self organizing dynamics of human erythrocytes under shear stress abstract the characterization of the erythrocytes viscoelastic properties is studied from the perspective of bounded correlated random walk brownian motion based on the assumption that diffractometric data involves both deterministic and stochastic components the photometric readings are obtained by ektacytometry over several millions of shear elongated cells using a home made device called erythrodeformeter the results suggest that the samples from healthy donors are intrinsically unpredictable ordinary brownian motion while when studying beta thalassemic samples these exhibit not only a great sensitivity to initial conditions fractional brownian motion but also chaotic behavior these results could allow us to claim that we have linked nonlinear tools with clinical aspects of the erythrocytes rheological properties

invariant properties of the shuffle exchange and a simplified cost effective version of the omega network let denote the set of permutations performable by a pass through the omega network an n stage shuffle exchange network and let denote an arbitrary permutation of n 2n elements in this paper we characterize the sets called left invariants and called right invariants this characterization provides a useful criterion for determining whether or not the composition of two permutations belongs to it also allows a number of known results to be generalized and their derivations simplified

discretization of the cauchy problem for a fast diffusion equation in this paper we propose and study a fully discretization for computing the positive and possibly blowing up solution of the cauchy problem u email protected x 2u email protected p 1 in r where email protected 0 1 p 1 1 a 0 with an initial condition u 0 assumed to be a nonnegative and continuous function with compact support the convergence of the numerical method is proved

notes on shuffle exchange type permutation sets properties of some shuffle exchange type permutation sets are studied from operational points of view the permutation sets studied are omega omega sup 1 psi l and u omega and omega sup 1 are respectively the omega and inverse omega permutation sets omega identical to omega intersection omega sup 1 and l and u are respectively the admissible lower and upper triangular permutation sets several intuitive operations are introduced based on these operations important known results relating to these sets are readdressed the recursive nature of the sets is also discussed

dimensionality estimation manifold learning and function approximation using tensor voting we address instance based learning from a perceptual organization standpoint and present methods for dimensionality estimation manifold learning and function approximation under our approach manifolds in high dimensional spaces are inferred by estimating geometric relationships among the input instances unlike conventional manifold learning we do not perform dimensionality reduction but instead perform all operations in the original input space for this purpose we employ a novel formulation of tensor voting which allows an n d implementation tensor voting is a perceptual organization framework that has mostly been applied to computer vision problems analyzing the estimated local structure at the inputs we are able to obtain reliable dimensionality estimates at each instance instead of a global estimate for the entire data set moreover these local dimensionality and structure estimates enable us to measure geodesic distances and perform nonlinear interpolation for data sets with varying density outliers perturbation and intersections that cannot be handled by state of the art methods quantitative results on the estimation of local manifold structure using ground truth data are presented in addition we compare our approach with several leading methods for manifold learning at the task of measuring geodesic distances finally we show competitive function approximation results on real data

geodesic entropic graphs for dimension and entropy estimation in manifold learning in the manifold learning problem one seeks to discover a smooth low dimensional surface i e a manifold embedded in a higher dimensional linear vector space based on a set of measured sample points on the surface in this paper we consider the closely related problem of estimating the manifold s intrinsic dimension and the intrinsic entropy of the sample points specifically we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold we introduce a novel geometric approach based on entropic graph methods although the theory presented applies to this general class of graphs we focus on the geodesic minimal spanning tree gmst to obtaining asymptotically consistent estimates of the manifold dimension and the re spl acute nyi spl alpha entropy of the sample density on the manifold the gmst approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples the gmst method simply constructs a minimal spanning tree mst sequence using a geodesic edge matrix and uses the overall lengths of the msts to simultaneously estimate manifold dimension and entropy we illustrate the gmst approach on standard synthetic manifolds as well as on real data sets consisting of images of faces

hamiltonian cycles in the shuffle exchange network the usefulness of the shuffle exchange network in parallel processing applications is well established the optimal embedding of a shuffle exchange network of a given size depends on the number of cycles of the shuffle permutation of that size the cost of one method of adding fault tolerance through reconfigurability depends upon the number of such cycles and the manner in which they can be connected to form larger cycles an exact equation for the number of cycles of a shuffle of size 2 sup w is presented that result is used to demonstrate that it is always possible to form a hamiltonian cycle on all processors in a shuffle exchange connected array from this it is apparent that there are a large number of ways of sharing spare processors among the members of many cycles thus redundancy can be supplied in any strength from adding on a spare processor to adding one for each cycle of the shuffle

danco an intrinsic dimensionality estimator exploiting angle and norm concentration abstract in the past decade the development of automatic intrinsic dimensionality estimators has gained considerable attention due to its relevance in several application fields however most of the proposed solutions prove to be not robust on noisy datasets and provide unreliable results when the intrinsic dimensionality of the input dataset is high and the manifold where the points are assumed to lie is nonlinearly embedded in a higher dimensional space in this paper we propose a novel intrinsic dimensionality estimator danco and its faster variant fastdanco which exploit the information conveyed both by the normalized nearest neighbor distances and by the angles computed on couples of neighboring points the effectiveness and robustness of the proposed algorithms are assessed by experiments on synthetic and real datasets by the comparative evaluation with state of the art methodologies and by significance tests

intrinsic dimension estimation the paper reviews state of the art of the methods of intrinsic dimension id estimation the paper defines the properties that an ideal id estimator should have the paper reviews under the above mentioned framework the major id estimation methods underlining their advances and the open problems dimensionality reduction methods are preprocessing techniques used for coping with high dimensionality they have the aim of projecting the original data set of dimensionality n without information loss onto a lower m dimensional submanifold since the value of m is unknown techniques that allow knowing in advance the value of m called intrinsic dimension id are quite useful the aim of the paper is to review state of the art of the methods of id estimation underlining the recent advances and the open problems

optimal realization of any bpc permutation on k extra stage omega networks an n spl times n k omega network is obtained by adding k more stages in front of an omega network an n permutation defines a bijection between the set of n sources and the set of n destinations such a permutation is said to be admissible to a k omega if n conflict free paths one for each source destination pair defined by the permutation can be established simultaneously when an n permutation is not admissible it is desirable to divide the n pairs into a minimum number of groups passes such that the conflict free paths can be established for the pairs id each group raghavendra and varma solved this problem for bpc bit permutation complement permutations on an omega without extra stage this paper generalizes their result to a k omega where k can be any integer between 0 and n 1 an o nlgn algorithm is given which realizes any bpc permutation in a minimum number of passes on a k omega 0 spl les k spl les n 1

optimally routing lc permutations on k extra stage cube type networks it is difficult to partition an arbitrary permutation into a minimum number of groups such that conflict free paths for all source destination pairs in each group can be established on an omega network based on linear algebra theory this paper presents an optimal algorithm which solves this problem for the lc class of permutations on a large class of multi stage networks this algorithm extends the previous result which deals with the bpc class of permutations on the omega network

a self routing benes network and parallel permutation algorithms a benes permutation network capable of setting its own switches dynamically is presented the total switch setting and delay time for the n input utput self routing network is o log n it is shown that the network is capable of performing a rich class of permutations the self routing scheme leads to efficient o log n parallel algorithms to perform the same class of permutations on cube connected and perfect shuffle computers

numerical solution of a cauchy problem for nonlinear reaction diffusion processes in this paper the authors propose a numerical method to compute the solution of the cauchy problem w t w mw x x w p the initial condition is a nonnegative function with compact support m 0 p m 1 the problem is split into two parts a hyperbolic term solved by using the hopf and lax formula and a parabolic term solved by a backward linearized euler method in time and a finite element method in space the convergence of the scheme is obtained further it is proved that if m 1 m 3 if the initial condition is sufficiently small a global numerical solution exists and if p m 3 for large initial condition the solution is unbounded

approximate confidence intervals for the mean direction of a von mises distribution summary simple formulae are suggested that provide accurate confidence bounds for the mean direction of a sample from a von mises distribution these formulae obviate the need for calculating an estimate of the second trigonometric moment and dispense with nomograms

estimating the intrinsic dimension of data with a fractal based method in this paper the problem of estimating the intrinsic dimension of a data set is investigated a fractal based approach using the grassberger procaccia algorithm is proposed since the grassberger procaccia algorithm 1983 performs badly on sets of high dimensionality an empirical procedure that improves the original algorithm has been developed the procedure has been tested on data sets of known dimensionality and on time series of santa fe competition

an evaluation of intrinsic dimensionality estimators the intrinsic dimensionality of a data set may be useful for understanding the properties of classifiers applied to it and thereby for the selection of an optimal classifier in this paper the authors compare the algorithms for two estimators of the intrinsic dimensionality of a given data set and extend their capabilities one algorithm is based on the local eigenvalues of the covariance matrix in several small regions in the feature space the other estimates the intrinsic dimensionality from the distribution of the distances from an arbitrary data vector to a selection of its neighbors the characteristics of the two estimators are investigated and the results are compared it is found that both can be applied successfully but that they might fail in certain cases the estimators are compared and illustrated using data generated from chromosome banding profiles

numerical solution of nonlinear reaction diffusion processes in this paper the author considers a nonlinear reaction diffusion equation i e a nonlinear heat equation with a source term the solution of this equation may blow up in a finite time a scheme for the discretization in time of that problem is proposed and the same sufficient condition of a blowup is obtained for the numerical solution as for the exact solution the stability and the convergence of the scheme is proved

an enhanced universal n x n fully nonblocking quantum switch this study develops a quantum switching device with fully nonblocking properties although previous studies have also presented quantum based solutions for the blocking problem the proposed schemes are characterized by an increased packet loss a large number of quantum swap gates and an increased propagation delay time complexity the current study overcomes these drawbacks by designing an n x n fully nonblocking quantum switch in which the packet payload is passed through quantum swap gates while the packet header is passed through quantum control gates designed by applying a modified quantum karnaugh mapping method the allocation of quantum swap gates to the different layers within the switch is solved using a perfect matching in complete graph pmicg algorithm with a time complexity of o n 2n 2 n 2 a symmetry based heuristic method is proposed to reduce the time complexity of the search process for all the perfect matching pairs to a time complexity of o n2 the performance of the proposed quantum switch is compared with that of a quantum self routing packet switch and a quantum switching quantum merge sorting scheme respectively in terms of the hardware complexity the propagation delay time complexity the auxiliary qubit complexity and the packet loss probability

an optimal algorithm for permutation admissibility to multistage interconnection networks this paper introduces a simple o nlogn sequential algorithm that determines the admissibility of an arbitrary permutation to an n spl times n multistage cube type network mctn implemented by 2 spl times 2 switching elements ses in contrast to previous o nlog sup 2 n algorithms it is proven that the new algorithm is optimal in the sense that any algorithm based on bit operations has to examine at least n 4 logn different bits among the total nlogn bits in the binary representations of the destinations numbered from 0 through n 1

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

rearrangeability of 2 protectn 1 stage shuffle exchange networks rearrangeable networks can realize each and every permutation in one pass through the network shuffle exchange networks provide an efficient interconnection scheme for implementing various types of parallel processes whether 2n stage shuffle exchange networks with n 2n inputs outputs are rearrangeable has remained an open question for approximately three decades this question has been answered affirmatively in this paper an important corollary of the main result is the proof that two passes through an omega network are sufficient and necessary to implement any permutation in obtaining the main results of this paper frames that look like grids with horizontal links of different lengths are shown to be remarkable tools for identifying and characterizing the binary matrix representations of permutations

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

parallel permutation and sorting algorithms and a new generalized connection network o k log n algorithms are obtained to permute and sort n data items on cube and perfect shuffle computers with n sup 1 1 k processing elements 1 k log n these algorithms lead directly to a generalized connection network construction having o k log n delay and o kn sup 1 1 k log n contact pairs this network has the advantage that the switches can be set in o k log n time by either a cube or perfect shuffle computer with n sup 1 1 k processing elements 23 references

convergence of the k 2 entropy for random noises with power law spectra abstract we discuss a counter example to the traditional expectation that stochastic processes lead to a non convergence of the k 2 entropy in computed or measured time series we consider a simple class of colored random noises with power law spectra which have already been shown to generate finite and predictable values of the correlation dimension we show that the k 2 entropy of these random noises converges as well these results imply that the observation of a finite or null value of the k 2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low dimensional chaos

hierarchical classification of permutation classes in multistage interconnection networks this paper explores a new hierarchy among different permutation classes that has many applications in multistage interconnection networks the well known lc linear complement class is shown to be merely a subset of the closure set of the bp bit permute class known as the bpcl bit permute closure class the closure is obtained by applying certain group transformation rules on the bp permutations it indicates that for every permutation p of the lc class there exists a permutation pi in the bp class such that the conflict graphs of p and p are isomorphic for n stage min s this obviates the practice of treating the lc class as a special case the existing algorithm for optimal routing of bpc class in an n stage min can take care of optimal routing of the lc class as well finally the relationships of bpcl with other classes of permutations e g lie linear input equivalence bpie bit permute input equivalence bpoe bit permute output equivalence are also exposed apart from lending better understanding and an integral view of the universe of permutations these results are found to be useful in accelerating routability in n stage min s as well as in 2n 1 stage benes and shuffle exchange networks

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

a search for chaotic behavior in large and mesoscale motions in the pacific ocean the lagrangian trajectories of three satellite tracked bouys deployed in the pacific ocean for a one year period are studied from the standpoint of dynamical systems theory time series of the bouy positions are analyzed to estimate the correlation dimension and kolmogorov entropy we find a small finite value of correlation dimension near 1 4 and a finite kolmogorov entropy of about 0 2 day 1 these results at first suggest that the bouy motions may be dominated by a low dimensional chaotic attractor using standard ideas from the mathematics of fractals and a new method which we call multivariate scaling analysis we are able to demonstrate however that the bouy trajectories are instead fractal curves in the physical plane x y these results may be viewed from the standpoint of multivariate nongaussian convergence of the scales which occur when the fourier spectrum is described by a power law such systems can have a small finite value for the correlation dimension and are therefore fractal even when no underlying attractor is present systems of this type are more easily related to turbulent behavior stochastic many degrees of freedom rather than to strange attractor deterministic few degrees of freedom

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

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

constant geometry fast fourier transforms on array processors matrix algebra is used to design and validate parallel algorithms for large constant geometry fast fourier transforms ffts on fixed size array processors the n point radix 2 case for a linear array processor with n 2 cells is identical to the usual procedure corresponding to the matrix factorization of m c pease 1968 the algorithms are engendered by matrix factorizations which themselves depend on a basic factorization of the perfect shuffle the resulting data movement is realized in parallel as relatively small perfect shuffles inside each local memory and along each row and column of the array processor without requiring that the complete array itself have the shuffle exchange network

on chaotic graphs a different approach for characterizing aperiodic dynamics fractal worlds with limited connectivity are the topological result of growing graphs from chaotic series we show how this model presents original characteristics which cannot be detected by means of the standard network descriptors in detail intrinsic inaccessibility to the fully connected configuration is demonstrated to be a universal feature associated with this family of graphs and strictly related to the fractality of a specific chaotic source here we discuss the potential of our model to be a generator of fractal graphs and also a self consistent tool for differentiating chaotic dynamics from stochastic processes

reconfiguration algorithms for interconnection networks the correspondence examines the functional relations within a class of multistage interconnection networks it is known that these networks are not rearrangeable this fact has led to some research on interconnection network relations the correspondence deals with one aspect of this research namely that of constructing an equivalence map between two interconnection networks procedures are given to test whether two such networks are equivalent whenever they are these procedures also produce a map to conjugate one network onto the other

an easily controlled network for frequently used permutations a network which is a concatenation of 2 networks 2 along with a simple control algorithm is proposed this network is capable of performing all network realizable permutations and the bit permute complement bpc class of permutations 5 in 0 log n time the control algorithm is actually a multiple pass control algorithm on the network which is more general than pease s lu decomposition method 6 and lenfant s decomposition method 4

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

a permutation entropy based test for causality the volume stock price relation abstract the purpose of this paper is to propose a newly developed non parametric test for linear and nonlinear causality based on permutation entropy and to show its usefulness in analyzing the potential causal relationship between trading volume and security prices most of the empirical applications and tests for causality rely on using granger causality based test for linear models although these tests have high power in uncovering linear causal relations their power against nonlinear causal relations can be low our test is designed to deal with the detection of linear and non linear causality we also compare our permutation entropy based test with other granger causality tests monte carlo simulations show excellent performance in terms of size and power of the new test for detecting linear and non linear causality under different scenarios our conclusions point that there is a bidirectional causal relation from volume to price returns not only in the mean but also in the variance

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

a classification of cube connected networks with a simple control scheme the correspondence presents three classes of cube connected networks with individual stage control based on a group theoretic modeling of interconnection networks it is shown that these classes of networks retain nonisomorphic groups of permutations although the permutations realizable by such networks are rather limited in number the simplicity of their control scheme makes them attractive moreover the interconnection power of these networks can be enhanced by simulating the networks which belong to one class by any member of that class

a new benes network control algorithm a new benes network control algorithm is presented unlike the original looping algorithm the new algorithm is not recursive in this algorithm n x n benes network is viewed as a concatenation of two subnetworks sn1 and sn2 the first log n 1 stages of a benes network correspond to sn1 and the remaining log n stages correspond to sn2 sn1 is controlled by a full binary tree of set partitioning functions called a complete residue partition tree and sn2 is bit controlled the new control algorithm sets switches one stage at a time stage by stage

on a class of rearrangeable networks benes networks are nonblocking rearrangeable networks that can realize arbitrary permutations without conflict a class of 2 log sub b n stage networks which are equivalent to the benes networks is introduced networks in this class can be either symmetric or asymmetric in their structure regular or irregular in their interstage connections and even 2 log sub b n or 2 log sub b n 1 stages all networks in this class are nonblocking rearrangeable a switch labeling scheme is proposed to provide testing of the topological and functional equivalency for this class of networks this switch labeling scheme can also provide a novel matrix representation for network configuration this representation introduces a portability concept for the routing scheme on this class of networks with this representation a general routing scheme is also developed this routing scheme can realize arbitrary permutation for the whole class of 2 log sub b n stage networks

the kr benes network a control optimal rearrangeable permutation network the benes network has been used as a rearrangeable network for over 40 years yet the uniform n 2 log n 1 control complexity of the n spl times n benes is not optimal for many permutations in this paper we present a novel o log n depth rearrangeable network called kr benes that is permutation specific control optimal the kr benes routes every permutation with the minimal control complexity specific to that permutation and its worst case complexity for arbitrary permutations is bounded by the benes thus it replaces the benes when considering control complexity latency we design the kr benes by first constructing a restricted 2log k 2 depth rearrangeable network called k benes for routing k bounded permutations with control 2n log k 0 spl les k spl les n 4 we then show that the n spl times n benes network itself with one additional stage contains every kr benes network as a subgraph and use this property to construct the kr benes network with regard to the control optimality of the kr benes we show that any optimal network for rearrangeably routing k bounded permutations must have depth 2log k 2 and therefore the k benes and hence the kr benes is optimal

an efficient routing algorithm for realizing linear permutations on p sup t shuffle exchange networks the authors present an efficient routing algorithm for realizing any permutation in lin linear permutation class on single stage shuffle exchange networks with k k switching elements where k p is a prime number for any positive integer number n there are n k sup n processors connected by the network the proposed algorithm can realize lin in 2n 1 passes it can be implemented by using nn processors in o n time it can also be extended to the shuffle exchange networks with p sup t p sup t switching elements where t is a positive integer number in addition the routing of any arbitrary permutations on the networks with any integer k 2 is discussed further by using the techniques developed here the authors present an optimal o log n parallel algorithm for solving a set of linear equations with a nonsingular coefficient matrix when the arithmetic is over the finite field gf p sup t

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

uniform framework for the recurrence network analysis of chaotic time series we propose a general method for the construction and analysis of unweighted e recurrence networks from chaotic time series the selection of the critical threshold e c in our scheme is done empirically and we show that its value is closely linked to the embedding dimension m in fact we are able to identify a small critical range e numerically that is approximately the same for the random and several standard chaotic time series for a fixed m this provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors we explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes n we also present two practical applications of the scheme detection of transition between two dynamical regimes in a time delayed system and identification of the dimensionality of the underlying system from real world data with a limited number of points through recurrence network measures the merits limitations and the potential applications of the proposed method are also highlighted

numerical solution of fast diffusion or slow diffusion equations in this paper the author proposes a semi discretization in time of a nonlinear reaction diffusion equation fast or slow diffusion the solution of which may vanish or blow up in a finite time the approximate value at each time step is solution of a nonlinear equation which is solved by using an iterative method it is proved that if the exact solution vanishes or blows up in a finite time it is the same for the numerical solution

intrinsic dimension estimation via nearest constrained subspace classifier we consider the problems of classification and intrinsic dimension estimation on image data a new subspace based classifier is proposed for supervised classification or intrinsic dimension estimation the distribution of the data in each class is modeled by a union of a finite number of affine subspaces of the feature space the affine subspaces have a common dimension which is assumed to be much less than the dimension of the feature space the subspaces are found using regression based on the 0 norm the proposed method is a generalisation of classical nn nearest neighbor nfl nearest feature line classifiers and has a close relationship to ns nearest subspace classifier the proposed classifier with an accurately estimated dimension parameter generally outperforms its competitors in terms of classification accuracy we also propose a fast version of the classifier using a neighborhood representation to reduce its computational complexity experiments on publicly available datasets corroborate these claims highlightsa new classifier called ncsc and its fast version called ncsc ii are proposed classifiers of nn nfl and ns are the special cases of ncsc ncsc ncsc ii can be used as intrinsic dimension estimator using well tuned intrinsic parameter ncsc ncsc ii outperforms nn nfl and ns

uniform partitions and a dimensions spectrum for lacunar measures a multifractal analysis has been proposed to investigate complex systems which exhibit a nonuniform local structure multifractal measures are normalized mass distributions inhomogeneous at every scale to every point p is associated the exponent of the power law r describing the asymptotic behaviour as r 0 of the mass of a sphere of radius r and centre p the measure properties are specified by the dimension spectrum f of the set of points p with the same scaling exponent this spectrum is nontrivial even when the mass distribution has no holes measure with euclidean support and the box counting dimension is an integer in this case it can be evaluated by considering a sequence of uniform grids whose spacing decreases as 2 n known as uniform partitions we consider here a family of iterated function system fractal measures with support on the unit square with random defects specified by three parameters a scale a probability p and an attenuation factor given a partition with a diameter less than any cell is chosen with probability p and its mass is multiplied by the corresponding pre measure is obtained after normalization in all the examples examined the f spectrum appears to widen monotonically as decreases this result suggests the possibility of using the method to detect the presence of defects in sponge like structures such as cancellous bone sponge like by analysing their radiographs since the grey intensity is related to the mass density of the structure a widening of the f spectrum is indeed observed in osteoporotic bone radiographs compared to normal bone radiographs this may be attributed to the rarefaction of the cancellous bone

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

numerical solution of a nonlinear reaction diffusion equation in this paper the authors propose a numerical method to compute the solution of a cauchy problem with blow up of the solution the problem is split in two parts a hyperbolic problem which is solved by using hopf and lax formula and a parabolic problem solved by a backward linearized euler method in time and a finite element method in space it is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self similar solution remains bounded the convergence of the scheme is obtained

time delay embedding increases estimation precision of models of intraindividual variability this paper investigates the precision of parameters estimated from local samples of time dependent functions we find that time delay embedding i e structuring data prior to analysis by constructing a data matrix of overlapping samples increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data we show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function hence we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions we support these conclusions both by mathematical analysis and two simulations

frames a simple characterization of permutations realized by frequently used networks rearrangeable multistage networks such as the benes network realize any permutation yet their routing algorithms are not cost effective on the other hand there exist inexpensive routing algorithms for nonrearrangeable networks but no simple technique exists to characterize all the permutations realized on these networks this paper introduces the concept of frame and shows how it can be used to characterize all the permutations realized on various multistage interconnection networks they include subnetworks of baseline benes and cascaded baseline and shuffle exchange networks

kolmogorov sinai entropy from the ordinal viewpoint abstract in the case of ergodicity much of the structure of a one dimensional time discrete dynamical system is already determined by its ordinal structure we generally discuss this phenomenon by considering the distribution of ordinal patterns which describe the ups and downs in the orbits of a borel measurable map on a borel subset of the real line on this base we give a natural ordinal description of the kolmogorov sinai entropy of one dimensional dynamical systems and relate the kolmogorov sinai entropy to the permutation entropy recently introduced by bandt and pompe

the necessity for a time local dimension in systems with time varying attractors abstract we show that a simple non linear system for ordinary differential equations may possess a time varying attractor dimension this indicates that it is infeasible to characterize eeg and meg time series with a single time global dimension we suggest another measure for the description of non stationary attractors

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems