classification trees for time series this paper proposes an extension of classification trees to time series input variables a new split criterion based on time series proximities is introduced first the criterion relies on an adaptive i e parameterized time series metric to cover both behaviors and values proximities the metrics parameters may change from one internal node to another to achieve the best bisection of the set of time series second the criterion involves the automatic extraction of the most discriminating subsequences the proposed time series classification tree is applied to a wide range of datasets public and new real and synthetic univariate and multivariate data we show through the experiments performed in this study that the proposed tree outperforms temporal trees using standard time series distances and performs well compared to other competitive time series classifiers

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

possibilistic nonlinear dynamical analysis for pattern recognition a nonlinear dynamical system can be defined as a study of any system that implies motion change or evolution in time where a change in one variable is not proportional to a change in a related variable the mathematical operations underlying such a system are very useful for pattern recognition with time series data one of the most recent developments in nonlinear dynamical analysis is the so called approximate entropy family however its algorithms are deterministic and do not consider uncertainty where the modeling of possibility can be appropriate and advantageous in many practical situations thus possibilistic entropy algorithms are proposed in this paper as a new methodology for nonlinear dynamical analysis the proposed approach is based on the notions of the approximate entropy family geostatistics and the theory of fuzzy sets furthermore for the first time nonlinear dynamical analysis of mass spectrometry data is presented for computer based recognition of potential protein biomarkers and classification which can be utilized for early disease prediction experimental results using proteomic and genetic data have shown the potential application of the proposed possibilistic nonlinear dynamical analysis to the study of complex biosignals

on generalized harmonic number sums we evaluate generalized harmonic number sums with parameter in terms of values of polylogarithm functions and several examples are given instances of such sums occur in diverse areas including analytic number theory and in calculations of high energy nuclear and atomic physics

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

soliton solutions of a few nonlinear wave equations this paper carries out the integration of a few nonlinear wave equations to obtain topological as well as non topological soliton solutions the mathematical techniques used to obtain the soliton solutions are he s variational iteration method the tanh method and the ansatz method the nonlinear wave equations that are studied are coupled mkdv equations drinfeld sokolov equation and its generalized version finally some numerical simulations are given to support the analytical solutions

on harmonic numbers and nonlinear euler sums abstract in this paper we are interested in euler type sums with products of harmonic numbers stirling numbers and bell numbers we discuss the analytic representations of euler sums through values of polylogarithm function and riemann zeta function moreover we provide explicit evaluations for almost all euler sums with weight 5 which can be expressed in terms of zeta values and polylogarithms furthermore we give explicit formula for several classes of euler related sums in terms of zeta values and harmonic numbers and several examples are given the given representations are new

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

explicit exact solutions for the generalized zakharov equations with nonlinear terms of any order in this paper an auxiliary ordinary differential equation with nonlinear terms of any order is introduced and its exact solutions are obtained by means of the auxiliary equation and its solutions abundant explicit exact solutions to the generalized zakharov equations with nonlinear terms of any order are obtained in a concise manner that include new solitary wave solutions and triangular periodic wave solutions

the sine cosine and the tanh methods reliable tools for analytic treatment of nonlinear dispersive equations the sine cosine method and the tanh method are used for analytic treatment of nonlinear dispersive equations nonlinear variants of boussinesq equation are used as vehicles to show the strength of these methods solutions of distinct physical structures solitons compactons solitary patterns solutions and periodic solutions are formally derived the results show that the change in physical structures of the obtained solutions depends mainly on exponents and on the coefficients of the derivatives involved

lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables we study here the coloured multiple zeta values obtained by extending the usual notion of the multiple zeta values by adding roots of unity we state their main combinatorial properties and we give as a result a maple algorithm which generates the relations table between these values

integrals of polylogarithmic functions recurrence relations and associated euler sums we show that integrals of the form 0 1 x m li p x li q x dx m 2 p q 1 and 0 1 log r x li p x li q x x dx p q r 1 satisfy certain recurrence relations which allow us to write them in terms of euler sums from this we prove that in the first case for all m p q and in the second case when p q r is even these integrals are reducible to zeta values in the case of odd p q r we combine the known results for euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight p q r the proofs are constructive giving a method for the evaluation of these and other similar integrals and we present a selection of explicit evaluations in the last section

variational approach to solitary wave solution of the generalized zakharov equation he s semi inverse method is applied to search for the solitary wave solution of the generalized zakharov equation gze the solution process reveals that the method is easy and straightforward

discrimination and clustering for multivariate time series abstract minimum discrimination information provides a useful generalization of likelihood methodology for classification and clustering of multivariate time series discrimination between different classes of multivariate time series that can be characterized by differing covariance or spectral structures is of importance in applications occurring in the analysis of geophysical and medical time series data for discrimination between such multivariate series kullback leibler discrimination information and the chernoff information measure are developed for the multivariate non gaussian case asymptotic error rates and limiting distributions are given for a generalized spectral disparity measure that includes the foregoing criteria as special cases applications to problems of clustering and classifying earthquakes and mining explosions are given

multi modal and multi scale temporal metric learning for a robust time series nearest neighbors classification abstract the definition of a metric between time series is inherent to several data analysis and mining tasks including clustering classification or forecasting time series data present naturally several modalities covering their amplitude behavior or frequential spectrum that may be expressed with varying delays and at multiple temporal scales exhibited globally or locally combining several modalities at multiple temporal scales to learn a holistic metric is a key challenge for many real temporal data applications this paper proposes a multi modal and multi scale temporal metric learning m 2 tml approach for a robust time series nearest neighbors classification the solution lies in embedding time series into a dissimilarity space where a pairwise svm is used to learn both linear and non linear combined metric a sparse and interpretable variant of the solution shows the ability of the learned temporal metric to localize accurately discriminative modalities as well as their temporal scales a wide range of 30 public and challenging datasets encompassing images traces and ecg data are used to show the efficiency and the potential of m 2 tml for an effective time series nearest neighbors classification

1 soliton solution of the d m n equation with generalized evolution abstract this paper obtains the 1 soliton solution of the nonlinear dispersive drinfel d sokolov equation with power law nonlinearity in the first case the soliton solution is without the generalized evolution the solitary wave ansatz method is used to carry out the integration subsequently the he s semi inverse variational principle is used to integrate the equation with power law nonlinearity parametric conditions for the existence of envelope solitons are given

1 soliton solution of the generalized zakharov equation in plasmas by he s variational principle this paper obtains the 1 soliton solution to zakharov equation with power law and dual power law nonlinearities the he s variational principle is used to carry out the integration of this equation

cats clustering after transformation and smoothing cats clustering after transformation and smoothing is a technique for nonparametrically estimating and clustering a large number of curves our motivating example is a genetic microarray experiment but the method is very general the method includes transformation and smoothing multiple curves multiple nonparametric testing for screening out flat curves clustering curves with similar shape and nonparametrically inferring the clustering estimation error rate

functional echo state network for time series classification abstract echo state networks esns are a new approach to recurrent neural networks rnns that have been successfully applied in many domains nevertheless an esn is a predictive model rather than a classifier and methods to employ esns in time series classification tsc tasks have not yet been fully explored in this paper we propose a novel esn approach named functional echo state network fesn for time series classification the basic idea behind fesn is to replace the numeric variable output weights of an esn with time varying output weight functions and introduce a temporal aggregation operator to the output layer that can project temporal signals into discrete class labels thereby transforming the esn from a predictive model into a true classifier subsequently to learn the output weight functions a spatio temporal aggregation learning algorithm is proposed based on orthogonal function basis expansion by leveraging the nonlinear mapping capacity of a reservoir and the accumulation of temporal information in the time domain fesn can not only enhance the separability of different classes in a high dimensional functional space but can also consider the relative importance of temporal data at different time steps according to dynamic output weight functions theoretical analyses and experiments on an extensive set of ucr data were conducted on fesn the results show that fesn yields better performance than single algorithm methods has comparable accuracy with ensemble based methods and exhibits acceptable computational complexity interestingly for some time series datasets we visualized some interpretable features extracted by fesn via specific patterns within the output weight functions

fuzzy rule based decision trees this paper presents a new architecture of a fuzzy decision tree based on fuzzy rules fuzzy rule based decision tree frdt and provides a learning algorithm in contrast with traditional axis parallel decision trees in which only a single feature variable is taken into account at each node the node of the proposed decision trees involves a fuzzy rule which involves multiple features fuzzy rules are employed to produce leaves of high purity using multiple features for a node helps us minimize the size of the trees the growth of the frdt is realized by expanding an additional node composed of a mixture of data coming from different classes which is the only non leaf node of each layer this gives rise to a new geometric structure endowed with linguistic terms which are quite different from the traditional oblique decision trees endowed with hyperplanes as decision functions a series of numeric studies are reported using data coming from uci machine learning data sets the comparison is carried out with regard to traditional decision trees such as c4 5 ladtree bftree simplecart and nbtree the results of statistical tests have shown that the proposed frdt exhibits the best performance in terms of both accuracy and the size of the produced trees highlightsthis paper presents a new architecture of a fuzzy decision tree based on fuzzy rules the growth of the proposed tree is realized by expanding an additional node the proposed tree can capture the oblique geometric structure of class regions the main idea is creating a node for each class to be predicted at every level of the tree

dynamical survey of a generalized zakharov equation and its exact travelling wave solutions we investigate the dynamical behavior of a generalized zakharov equation for the complex envelope of the high frequency wave and the real low frequency field by analyzing its phase portraits following a dynamical system approach in different parameter regions we depict phase portraits of a travelling wave system through discussing the bifurcation of phase portraits we unwrap explicit miscellaneous travelling waves including localized and periodic ones