the dependency of timbre on fundamental frequency the dependency of the timbre of musical sounds on their fundamental frequency f0 was examined in three experiments in experiment i subjects compared the timbres of stimuli produced by a set of 12 musical instruments with equal f0 duration and loudness there were three sessions each at a different f0 in experiment ii the same stimuli were rearranged in pairs each with the same difference in f0 and subjects had to ignore the constant difference in pitch in experiment iii instruments were paired both with and without an f0 difference within the same session and subjects had to ignore the variable differences in pitch experiment i yielded dissimilarity matrices that were similar at different f0 s suggesting that instruments kept their relative positions within timbre space experiment ii found that subjects were able to ignore the salient pitch difference while rating timbre dissimilarity dissimilarity matrices were symmetrical suggesting further that the absolute displacement of the set of i

a time aggregation approach to markov decision processes we propose a time aggregation approach for the solution of infinite horizon average cost markov decision processes via policy iteration in this approach policy update is only carried out when the process visits a subset of the state space as in state aggregation this approach leads to a reduced state space which may lead to a substantial reduction in computational and storage requirements especially for problems with certain structural properties however in contrast to state aggregation which generally results in an approximate model due to the loss of markov property time aggregation suffers no loss of accuracy because the markov property is preserved single sample path based estimation algorithms are developed that allow the time aggregation approach to be implemented on line for practical systems some numerical and simulation examples are presented to illustrate the ideas and potential computational savings

aggregation of the policy iteration method for nearly completely decomposable markov chains a steady state optimal control problem is considered for nearly completely decomposable markov chains in order to apply the policy iteration method of r a howard dynamic programming and markov processes cambridge ma mit press 1960 a high dimensional ill conditioned system of algebraic equations must be solved in the value determination step although algorithms exist for aggregation of the steady state probability distribution problem they only provide methods for computing the cost not the dual variables using a singular perturbation approach an aggregation method for the value determination equation is developed the aggregation method is developed in three steps first a class of similarity transformations that transform the system into a singularly perturbed form is developed second an aggregation method to compute the steady state probability distribution is derived third this aggregation method is applied to the value determination step of howard s method 1960

iterative algorithms for large stochastic matrices markov chains have always constituted an efficient tool to model discrete systems many performance criteria for discrete systems can be derived from the steady state probability vector of the associated markov chain however the large size of the state space of the markov chain often allows this vector to be determined by iterative methods only various iterative methods exist but none can be proved a priori to be the best in this paper we propose a practical measure which allows the convergence rate of the various existing methods to be compared this measure is an approximation of the modulus of the second largest eigenvalue of the iteration matrix and can be determined a priori the model of a queueing network is used as an example to compare the convergence of several iterative methods and to show the accuracy of the measure

a local convergence proof for the iterative aggregation method the iterative aggregation method for the solution of a system of linear algebraic equations x ax b where a 0 b 0 s 0 and s a s is proved to be locally convergent it is shown that the method can be considered a consistent nonstationary iterative method where the iteration matrix depends on the current iterate and that some norm of the iteration matrix is less than one in the vicinity of the solution

perceptual components of pitch spatial representation using a multidimensional scaling technique paired comparison experiments were carried out on the pitch of 18 computer generated complex tones synthesized by slight modification of the frequency structure of shepard s endless scale sounds the results were analyzed by a multidimensional scaling technique and simple helixes were obtained in which two components of pitch tone height and tone chroma were represented some individual differences in perception of pitch were observed according to the weight that the subject gives to each component

response distributions in intensity resolution and speech discrimination in this paper the assumption of an equal gaussian distribution of the response to each stimulus in an experiment an assumption which has to be met if d is to be estimated by calculating the difference between z h and z fa is tested for two different sets of stimuli 1000 hz tones differing in level only and a continuum of stop consonants obtained by full spectral interpolation between p t and k response distributions were measured directly by means of a form of non numerical magnitude estimation in which subjects had to indicate the position of each stimulus on a quasi continuous rating scale it could be shown that in general all distributions were sufficiently unimodal but that their variances differed the consequences for the calculation of d are unlikely to be serious

periodic signals with minimal power fluctuations in this article pumplin s algorithm j pumplin j acoust soc am 78 100 104 1985 is used to find periodic waveforms with minimal power fluctuations starting with a particular power spectrum the algorithm can find a set of phases for the harmonics such that the variance in waveform power is a minimum or near a minimum such optimized waveforms are smooth and tend to have very small crest factors the particular power spectra chosen for study include narrow and wideband spectra with emphasis given to signals that are useful in research in psychoacoustics and in physiological acoustics

multirate and composite control of two time scale discrete time systems the design of stabilizing feedback control of singularly perturbed diserete time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control composite control strategies are developed for the case of single rate measurements all variables are measured at the same rate as well as for the case of multirate measurements slow variables are measured at a rate slower than that of fast variables

the first laurent series coefficients for singularly perturbed stochastic matrices there are a few procedures for computing the laurent series expansions for the mean passage time matrix and for the deviation matrix of a singularly perturbed markov chain we suggest here a method for computing the first terms in these expansions in a way which highlights the system dynamics in various time scales

signal densities and criterion variance in speech and nonspeech perception the actual shape of signal densities has become an important issue when studying speech perception within the framework of signal detection theory sdt using an sdt model that allowed unequal criterion variances lopez bascuas proc eurosp 3 2281 2283 1995 found that speech signals did not accommodate to the standard gaussian assumption however schouten and van hessen j acoust soc am 104 2980 2990 1998 measured response distributions directly and assuming an interval scale concluded that the gaussian assumption held for both continua nevertheless pastore and macmillan j acoust soc am 111 2432 2002 applied roc analysis to schouten and van hessen s data and their curves supported the gaussian assumption for the nonspeech signals only later lopez bascuas et al j acoust soc am 115 2465 2004 showed that non linear z transformed rocs are not sufficient evidence for postulating non gaussian signal densities in this paper we try to figure out whether unequal criterion variances could underlie non linear z transformed rocs by fitting a restricted thurstonian sdt like model to a nonspeech continuum composed by white noise and a square wave the results indicate that unequal criterion variances is the possible cause of the deviant rocs for nonspeech

convergence theory of some classes of iterative aggregation disaggregation methods for computing stationary probability vectors of stochastic matrices this contribution is a natural follow up of the paper of the same authors entitled convergence theory of an aggregation disaggregation methods for computing stationary probability vectors of stochastic matrices published in numer linear algebra appl 5 1998 253 in contrast to that paper in which the algorithm studied was based on the splitting whose iteration matrix was identical with the matrix whose stationary probability vectors are computed the present paper presents a convergence analysis of algorithms based on fully general splittings of nonnegative type together with this generalization another feature of the older paper and namely the independence of the convergence results on the size of the elements of the examined stochastic matrix is shown to remain valid for the new algorithms as well this concerns in particular the possibility of computing stationary probability vectors of markov chains containing rare events i e events whose stationary probabilities are substantially smaller than some of elements of the transition matrix of the chain

iterative aggregation disaggregation techniques for nearly uncoupled markov chains iterative aggregation disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled also known as nearly completely decomposable markov chains three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined specifically it is shown that the method of takahashi corresponds to a modified block gauss seidel step and aggregation whereas that of vantilborgh corresponds to a modified block jacobi step and aggregation the third method that of koury et al is equivalent to a standard block gauss seidel step and iteration for each of these methods a lemma is established which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix in addition conditions are established for the convergence of the first two of these methods convergence conditions for the third having already been established by stewart et al all three methods are shown to have the same asymptotic rate of convergence

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

pitch matching in bimodal cochlear implant patients effects of frequency spectral envelope and level this study systematically investigated the effects of frequency level and spectral envelope on pitch matching in twelve bimodal cochlear implant ci users the participants were asked to vary the frequency and level of a pure or complex tone adjustable sounds presented in the non implanted ear to match the pitch and loudness of different reference stimuli presented to the implanted ear three reference sounds were used single electrode pulse trains pure tones and piano notes the data showed a significant effect of the frequency and complexity of the reference sounds no significant effect of the level of the reference sounds was found the magnitude of effect of frequency was compressed in the implanted ear on average a difference of seven semitones in the non implanted ear induced the same pitch change as a difference of 19 to 24 semitones for a stimulus presented to the implanted ear the spectral envelope of the adjustable sound presented to the non implanted ear also had a significant effect the matched frequencies were higher by an average of six semitones for the pure tone compared to a complex tone overall the ci listeners might have matched the stimuli based on timbre characteristics such as brightness

frequency and frequency ratio resolution by possessors of absolute and relative pitch examples of categorical perception the methodology derived from the trace context theory of intensity resolution durlach and braida 1969 macmillan et al 1988 was applied to resolution over an octave range along two continua a sequential frequency ratio continuum for possessors of relative pitch rp and a pure tone frequency continuum for possessors of absolute pitch ap the performance of both rp and ap possessors was exceptional in that total identification sensitivity along both continua was much greater than identification sensitivity along unidimensional psychophysical continua characterized by the 7 2 rule in addition the performance of rp possessors was exceptional in that on average total sensitivity for identification resolution was greater than sensitivity for resolution in discrimination finally identification sensitivity between category prototypes chromatic semits along both continua was approximately the same as identification sensitivity between phonemic category prototypes along speech continua despite the fact that both the discrimination ranges and the total number of categories are much larger for the two pitch continua

reduced complexity estimation for large scale hidden markov models in this paper we address the problem of reduced complexity estimation of general large scale hidden markov models hmms with underlying nearly completely decomposable discrete time markov chains and finite state outputs an algorithm is presented that computes o spl epsi where spl epsi is the related weak coupling parameter approximations to the aggregate and full order filtered estimates with substantial computational savings these savings are shown to be quite large when the chains have blocks with small individual dimensions some simulation studies are presented to demonstrate the performance of the algorithm

a convergence analysis of the iterative aggregation method with one parameter a method for accelerating linear iterations in a banach space is studied as a linear iterative method in an augmented space and sufficient conditions for convergence are derived in the general case and in ordered banach spaces an acceleration of convergence takes place if an auxiliary functional is chosen sufficiently close to a dual eigenvector associated with a dominant simple eigenvalue of the iteration operator in this case the influence of this eigenvalue on the asymptotic rate of convergence is eliminated quantitative estimates and bounds on convergence are given

on a rayleigh ritz refinement technique for nearly uncoupled stochastic matrices abstract an analysis is presented for the convergence of an iterative technique for computing the dominant left eigenvector of a nearly uncoupled nearly completely decomposable stochastic matrix the technique combines the power method and rayleigh ritz refinement to force rapid convergence to the stationary probability vector of the associated markov chain

modeling phoneme perception i categorical perception on the basis of a number of vowel and stop consonant discrimination experiments ax and 2ifc fixed and roving with natural stimuli it is concluded that stop consonant perception is highly categorical there were few significant differences between the discrimination results and the phoneme identification results moreover the discrimination and identification response maxima differed significantly from the other data points vowel perception was much less categorical the maxima in the functions were much less significant and there were significant differences between the various paradigms consonant discrimination was much less if at all subject to range effects than vowel discrimination all these results point to different memory types for stop consonants and vowels and consequently to a combination of two different theories of speech sound discrimination dual process theory dpt for consonants and trace context theory tct for vowels

melodic interval perception by normal hearing listeners and cochlear implant users the perception of melodic intervals sequential pitch differences is essential to music perception this study tested melodic interval perception in normal hearing nh listeners and cochlear implant ci users melodic interval ranking was tested using an adaptive procedure ci users had slightly higher interval ranking thresholds than nh listeners both groups interval ranking thresholds although not affected by root note significantly increased with standard interval size and were higher for descending intervals than for ascending intervals the pitch direction effect may be due to a procedural artifact or a difference in central processing in another test familiar melodies were played with all the intervals scaled by a single factor subjects rated how in tune the melodies were and adjusted the scaling factor until the melodies sounded the most in tune ci users had lower final interval ratings and less change in interval rating as a function of scaling factor than nh listeners for ci users the root mean square error of the final scaling factors and the width of the interval rating function were significantly correlated with the average ranking threshold for ascending rather than descending intervals suggesting that ci users may have focused on ascending intervals when rating and adjusting the melodies

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

reduced complexity filtering for partially observed nearly completely decomposable markov chains this paper provides a systematic method of obtaining reduced complexity approximations to aggregate filters for a class of partially observed nearly completely decomposable markov chains it is also shown why an aggregate filter adapted from courtois 1977 aggregation scheme has the same order of approximation as achieved by the algorithm proposed in this paper this algorithm can also be used systematically to obtain reduced complexity approximations to the full order fitter as opposed to algorithms adapted from other aggregation schemes however the computational savings in computing the full order filters are substantial only when the large scale markov chain has a large number of weakly interacting blocks or superstates with small individual dimensions some simulations are carried out to compare the performance of our algorithm with algorithms adapted from various other aggregation schemes on the basis of an average approximation error criterion in aggregate slow filtering these studies indicate that the algorithms adapted from other aggregation schemes may become ad hoc under certain circumstances the algorithm proposed in this paper however always yields reduced complexity filters with a guaranteed order of approximation by appropriately exploiting the special structures of the system matrices

block iterative algorithms for stochastic matrices abstract the theory of regular splittings for singular m matrices is used to derive the necessary and sufficient conditions for the convergence of iterative decomposition and aggregation techniques in the computation of the perron frobenius eigenvector of a stochastic irreducible matrix these conditions turn out to be less restrictive than those which must be satisfied by the block jacobi or the block gauss seidel method

triangular and skew symmetric splitting method for numerical solutions of markov chains in this paper a theorem is presented to indicate that there exists a nonnegative constant e 0 such that the matrix a q email protected is a positive definite matrix where email protected r n x n is an identity matrix and q email protected r n x n is a matrix with positive diagonal and nonpositive off diagonal elements then a class of triangular and skew symmetric splitting tss iteration method is applied to solve the positive definite linear system ax b for obtaining the stationary probability vector of an irreducible markov chain theoretical analyses show that the tss iteration method converges unconditionally to the unique solution of the linear system with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved moreover the inexact triangular and skew symmetric splitting itss iteration method which employs certain krylov subspace methods as the inner iteration processes at each step of the outer tss iteration method is proposed to accelerate the convergence of the tss iteration method numerical experiments are used to illustrate the effectiveness of the tss and itss iteration methods

complexity reduction in fixed lag smoothing for hidden markov models we investigate approximate smoothing schemes for a class of hidden markov models hmms namely hmms with underlying markov chains that are nearly completely decomposable the objective is to obtain substantial computational savings our algorithm can not only be used to obtain aggregate smoothed estimates but can be used also to obtain systematically approximate full order smoothed estimates with computational savings and rigorous performance guarantees unlike many of the aggregation methods proposed earlier

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

on a two level multigrid solution method for finite markov chains abstract we study a two level algebraic multigrid scheme for computing the stationary distribution of a homogeneous markov chain with a finite state space we reveal its relationship to an iterative aggregation disaggregation method provide an error analysis and prove its local convergence moreover we discuss the stochastic meaning of a block sor procedure that can be used as iterative smoother of the multigrid scheme

aggregation disaggregation methods for computing the stationary distribution of a markov chain we implement and analyse aggregation disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite markov chain we define six of these methods and analyse them in detail in particular we show that some existing procedures lie in the aggregation disaggregation framework we set and hence can be considered as special cases also for all described methods we identify cases where they are promising numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well

multilevel adaptive aggregation for markov chains with application to web ranking a multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described the method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation disaggregation methods for markov chains in contrast to most existing approaches our aggregation process does not employ any explicit advance knowledge of the topology of the markov chain instead adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level the strength of connection is determined as in the algebraic multigrid method and the aggregation process is fully adaptive with optimized aggregates chosen in each step of the iteration and at all recursive levels the multilevel method is applied to a set of stochastic matrices that provide models for web page ranking numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods the tests also provide more insight into why google s pagerank model is a successful model for determining a ranking of web pages

travelling wave solutions for a class of generalized kdv equation in this work the k l p equation is investigated the sine cosine method the tanh method and the extended tanh method are efficiently used for analytic study of this equation new solitary patterns solutions and compactons solutions are formally derived the proposed schemes are reliable and manageable

a reliable treatment of the physical structure for the nonlinear equation k m n in this work the nonlinear equation k m n is studied for all possible values of m and n we show that this equation may exhibit compactons solitons or periodic solutions the analysis reveals the change of the physical structure of the solutions as a result of the change of m and n

optimal kullback leibler aggregation via information bottleneck in this paper we present a method for reducing a regular discrete time markov chain dtmc to another dtmc with a given typically much smaller number of states the cost of reduction is defined as the kullback leibler divergence rate between a projection of the original process through a partition function and a dtmc on the correspondingly partitioned state space finding the reduced model with minimal cost is computationally expensive as it requires an exhaustive search among all state space partitions and an exact evaluation of the reduction cost for each candidate partition our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost the proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition then we express the problem in the form of information bottleneck optimization and propose using the agglomerative information bottleneck algorithm for searching a suboptimal partition greedily rather than exhaustively the theory is illustrated with examples and one application scenario in the context of modeling bio molecular interactions

aggregation disaggregation for eigenvalue problems convergence of the power method for the algebraic eigenvalue problem is accelerated by an aggregation disaggregation process a lower dimensional aggregated eigenvalue problem is created from time to time during the course of the power iteration and the solution of this problem is used to produce acceleration analysis and numerical experiments show the effectiveness of the method the more so the more poorly conditioned is the original eigenvalue problem

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

iterative aggregation disaggregation method for computing stationary probability vectors of markov type operators a global convergence of a class of iterative aggregation disaggregation methods is presented and some applications as well as some conclusions of numerical experiments performed are shown

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

asynchronous two stage iterative methods parallel block two stage iterative methods for the solution of linear systems of algebraic equations are studied convergence is shown for monotone matrices and for h matrices two different asynchronous versions of these methods are considered and their convergence investigated

blockwise perturbation theory for nearly uncoupled markov chains and its application abstract let p be the transition matrix of a nearly uncoupled markov chain the states can be grouped into aggregates such that p has the block form p p ij i j 1 k where p ii is square and p ij is small for i j let t be the stationary distribution partitioned conformally as t 1 t k t in this paper we bound the relative error in each aggregate distribution i t caused by small relative perturbations in p ij the error bounds demonstrate that nearly uncoupled markov chains usually lead to well conditioned problems in the sense of blockwise relative error as an application we show that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve such structured backward errors and compute each aggregate distribution with high relative accuracy

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

the reduction of perturbed markov generators an algorithm exposing the role of transient states a new algorithm for the hierarchical aggregation of singularly perturbed finite state markov processes is derived the approach taken bridges the gap between conceptually simple results for a relatively restricted class of processes and the significantly more complex results for the general case the critical role played by almost transient states is exposed resulting in a straightforward algorithm for the construction of a sequence of aggregate generators associated with various time scales these generators together provide a uniform asymptotic approximation of the original probability transition function

numerical methods in markov chain modeling this paper describes and compares several methods for computing stationary probability distributions of markov chains the main linear algebra problem consists of computing an eigenvector of a sparse nonsymmetric matrix associated with a known eigenvalue it can also be cast as a problem of solving a homogeneous singular linear system we present several methods based on combinations of krylov subspace techniques single vector power iteration relaxation procedures and acceleration techniques we compare the performance of these methods on some realistic problems

a unified view of aggregation and coherency in networks and markov chains abstract this paper presents a unified treatment of aggregability lumpability coherency reversibility partial balance and similar properties appearing in different fields including power systems and queueing theory a coherency condition well known in power systems implies the existence of a finite state filter for markov chains while aggregability and coherency yield a new condition for decentralized computation of the invariant measure

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

bifurcations of travelling wave solutions in a class of generalized kdv equation by using the bifurcation theory of dynamical systems to a generalized kdv equation given by cooper et al the existence of solitary wave solutions solitary cusp wave solutions and uncountably infinite many smooth and non smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some exact explicit parametric representations of the above waves are determined

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

all traveling wave exact solutions of the variant boussinesq equations in this article we employ the complex method to obtain all meromorphic solutions of complex variant boussinesq equations 1 then find out related traveling wave exact solutions of system vb the idea introduced in this paper can be applied to other nonlinear evolution equations our results show that all rational and simply periodic solutions w r 1 k x t w r 2 k x t w s 1 k x t and w s 2 k x t of system vb are solitary wave solutions and there exist some rational solutions wr 2 z and simply periodic solutions ws 2 z which are not only new but also not degenerated successively by the elliptic function solutions we believe that this method should play an important role for finding exact solutions in the mathematical physics we also give some computer simulations to illustrate our main results

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations