increasing the sample size during clinical trials with t distributed test statistics without inflating the type i error rate in clinical trials with t distributed test statistics the required sample size depends on the unknown variance taking estimates from previous studies often leads to a misspecification of the true value of the variance hence re estimation of the variance based on the collected data and re calculation of the required sample size is attractive we present a flexible method for extensions of fixed sample or group sequential trials with t distributed test statistics the method can be applied at any time during the course of the trial and does not require the necessity to pre specify a sample size re calculation rule all available information can be used to determine the new sample size the advantage of our method when compared with other adaptive methods is maintenance of the efficient t test design when no extensions are actually made we show that the type i error rate is preserved

integrable nonlocal wave interaction models the equations we investigate describe the nonlinear and nonlocal interaction of waves in 1 1 dimensions with potential applicability although these equations are integrable the solution of the initial and boundary value problem requires a judicious use of spectral methods in particular we show that spectral data evolve linearly only for special boundary values

a group sequential t test with updating of sample size if the maximum sample size for a group sequential t test is calculated on the basis of a prior estimate of the variance power at a fixed distance from the null hypothesis is not robust to misspecification of the variance we propose a new group sequential t test based on stein s 1945 two stage test in which we use internal estimates of the variance to update the maximum sample size and allow early stopping to reject the null hypothesis through a lan demets 1983 error spending function we evaluate our procedure s performance and show that it closely controls the type i and ii error rates

coupled error spending functions for parallel bivariate sequential tests sequential procedures are developed to facilitate marginal monitoring of bivariate response vectors in clinical trials the general approach is based on an extension of the error spending function methodology of lan and demets 1983 biometrika 70 659 663 and is sufficiently flexible that one may elect to fix the experimental type i error rate cook 1994 controlled clinical trials 15 3 187 200 or the marginal type i error rates sample size calculations are described to ensure power requirements are satisfied for marginal tests of significance reformulating the procedures in terms of repeated confidence intervals jennison and turnbull 1989 journal of the royal statistical society series b 51 305 361 lends added flexibility to the monitoring process the developments are discussed in the context of responses with a bivariate normal distribution data from an asthma intervention trial are used for illustrative purposes

moving breathers and breather to soliton conversions for the hirota equation we find that the hirota equation admits breather to soliton conversion at special values of the solution eigenvalues this occurs for the first order as well as higher orders of breather solutions an analytic expression for the condition of the transformation is given and several examples of transformations are presented the values of these special eigenvalues depend on two free parameters that are present in the hirota equation we also find that higher order breathers generally have complicated quasi periodic oscillations along the direction of propagation various breather solutions are considered including the particular case of second order breathers of the nonlinear schrodinger equation

on solutions of constrained kp equations we derive solutions of general wronskian form for the vector constrained kp hierarchies as one explicit example we discuss rational solutions in order to introduce our method we give a direct elementary proof of the existence of wronskian solutions for the l modified kp hierarchies l 0 1

rogue waves rational solutions the patterns of their zeros and integral relations the focusing nonlinear schrodinger equation which describes generic nonlinear phenomena including waves in the deep ocean and light pulses in optical fibres supports a whole hierarchy of recently discovered rational solutions we present recurrence relations for the hierarchy the pattern of zeros for each solution and a set of integral relations which characterizes them

nonautonomous discrete rogue wave solutions and interactions in an inhomogeneous lattice with varying coefficients abstract in this paper we analytically investigate nonautonomous discrete rogue wave solutions and their interactions in the generalized ablowitz ladik hirota lattice with variable coefficients which possess complicated wave propagation in time and differ from the usual discrete rogue waves when the amplitude of the tunnel coupling coefficient between sites decreases these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values moreover we find that the interactions between nonautonomous discrete rogue wave solutions are elastic

the wronskian structure for solution of the k constrained kp hierarchy in this paper we study a class of 1 1 dimensional integrable systems called the k constrained kp hierarchy we first show by using the wronskian technique that the first flow of the 2 constrained kp hierarchy possesses solutions in generalized double wronskian form then we give a conjecture on the wronskian structure for solutions of the general k constrained kp hierarchy

group sequential methods for clinical trials allowing early acceptance of ho and incorporating costs summary when interim accumulated results from a clinical trial clearly indicate that the null hypothesis should be accepted or rejected ethical and cost considerations may dictate terminating the trial and making the appropriate decision pocock 1977 has described a group sequential procedure for clinical trials that permits early termination only for rejecting the null hypothesis this paper extends pocock s procedure to allow early termination with acceptance of the null hypothesis theoretical and simulation results demonstrate that designs allowing early acceptance of the null hypothesis can have lower expected costs than designs that do not even when the alternative hypothesis is true

other 2n 2 parameters solutions of the nls equation and 2n 1 highest amplitude of the modulus of the n th order ap breather in this paper we construct new deformations of the akhmediev peregrine ap breather of order n or apn breather with real parameters other families of quasirational solutions of the nonlinear schrodinger nls equation are obtained we evaluate the highest amplitude of the modulus of the ap breather of order n we give the proof that the highest amplitude of the apn breather is equal to we get new formulas for the solutions of the nls equation which are different from these already given in previous works new solutions for the order 8 and their deformations according to the parameters are explicitly given we simultaneously get triangular configurations and isolated rings moreover the appearance for certain values of the parameters and of new configurations of concentric rings are underscored

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

an approach to the conditional error rate principle with nuisance parameters summary in the presence of nuisance parameters the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters to use the conditional error rate principle with nuisance parameters we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters we show that the optimization problem that defines such a test can be solved efficiently by existing algorithms

multi rogue waves for a higher order nonlinear schrodinger equation in optical fibers in this paper the rational solutions with arbitrary constants of a hirota equation are presented employing the darboux transformations then the similarity transformation allows us to relate certain class of multi rogue wave solutions of the higher order nonlinear schrodinger equation nlse to the solutions of 1 1 dimensional integrable hirota equation the multi rogue wave solutions with arbitrary constants are considered the obtained rogue wave solutions can be used to describe the possible formation mechanisms for optical and bose einstein condensates

dressing method and quadratic bundles related to symmetric spaces vanishing boundary conditions we consider quadratic bundles related to hermitian symmetric spaces of the type su m n s u m u n the simplest representative of the corresponding integrable hierarchy is given by a multi component kaup newell derivative nonlinear schrodinger equation which serves as a motivational example for our general considerations we extensively discuss how one can apply zakharov shabat s dressing procedure to derive reflectionless potentials obeying zero boundary conditions those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy one can distinguish between generic soliton type solutions and rational solutions

degenerate determinant representation of solutions of the nonlinear schrodinger equation higher order peregrine breathers and multi rogue waves we present a new representation of solutions of the focusing nls equation as a quotient of two determinants this work is based on a recent paper in which we have constructed a multi parametric family of solutions of this equation in terms of wronskians this formulation was written in terms of a limit involving a parameter here we give a very compact formulation without presence of a limit this is a completely new result which gives a very efficient procedure to construct families of quasi rational solutions of the nls equation with this method we construct peregrine breathers of orders n 4 6 and multi rogue waves associated by deformation of parameters

statistics of vector manakov rogue waves we present a statistical analysis based on the height and return time probabilities of high amplitude wave events in both focusing and defocusing manakov systems we find that analytical rational or semirational solutions associated with extreme rogue wave rw structures are the leading high amplitude events in this system we define the thresholds for classifying an extreme wave event as a rw our results indicate that there is a strong relationship between the type of rw and the mechanism which is responsible for its creation initially high amplitude events originate from modulation instability upon subsequent evolution the interaction among these events prevails as the mechanism for rw creation we suggest a strategy for confirming the basic properties of different extreme events this involves the definition of proper statistical measures at each stage of the rw dynamics our results point to the need for redefining criteria for identifying rw events

vector quantization for multiple classes abstract vector quantization algorithms have long been used to find a finite set of exemplars which represent a data set to within an a priori error tolerance such a representation is essential in codebook based data compression and transmission the present study considers the situation where the data to be encoded consists of subclasses the codebook must provide information compression within the several subclasses however minimization of interclass errors is of equal importance we present modifications to a basic vector quanitization algorithm which adapts it to the multiclass vector quantizing setting we then explore the behavior of the modified algorithm on selected benchmark applications we show in particular that overlapping subclasses can be accommodated by the algorithm

darboux transformation and rogue wave solutions for the variable coefficients coupled hirota equations abstract we construct the lax pair and darboux transformation for the variable coefficients coupled hirota equations based on modulation instability and by taking the limit approach we derive two types of nth order rogue wave solutions with different dynamic structures in compact determinant representations the explicit first order rogue wave solution is presented prolific vector rogue wave patterns such as the dark bright composite three sister quadruple and sextuple rogue waves with multiple compression points are demonstrated in particular in contrast to the standard peregrine combs unusual vector rogue wave combs such as the dark bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes further some wave characteristics such as the difference between light intensity and continuous wave background and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail

multiple comparison procedures based on the maximal component of the cumulative chi squared statistic summary multiple comparison procedures based on the maximal component of the cumulative chi squared statistic are discussed the markov properties for the successive components of the cumulative chi squared statistic are proved and applied for obtaining the p value of the maximal component in each of the one way and the two way analysis of variance models and the two way contingency table the power of the multiple comparison procedure is compared with those for some well known procedures in the setting of williams 1971 for detecting the change point from the zero level

on the continuum limit for a semidiscrete hirota equation in this paper we propose a new semidiscrete hirota equation which yields the hirota equation in the continuum limit we focus on the topic of how the discrete space step affects the simulation for the soliton solution to the hirota equation the darboux transformation and explicit solution for the semidiscrete hirota equation are constructed we show that the continuum limit for the semidiscrete hirota equation including the lax pair the darboux transformation and the explicit solution yields the corresponding results for the hirota equation as formula see text

rogue waves of the hirota and the maxwell bloch equations in this paper we derive a darboux transformation of the hirota and the maxwell bloch h mb system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the n fold darboux transformation of this system this n fold darboux transformation implies the determinant representation of n th solutions of e n p n ensuremath eta n generated from the known solution of e p ensuremath eta the determinant representation of e n p n ensuremath eta n provides soliton solutions positon solutions and breather solutions both bright and dark breathers of the h mb system from the breather solutions we also construct a bright and dark rogue wave solution for the h mb system which is currently one of the hottest topics in mathematics and physics surprisingly the rogue wave solution for p and ensuremath eta has two peaks because of the order of the numerator and denominator of them meanwhile after fixing the time and spatial parameters and changing two other unknown parameters ensuremath alpha and ensuremath beta we generate a rogue wave shape

rational solutions of the kadomtsev petviashvili hierarchy and the dynamics of their poles i new form of a general rational solution a new approach to the construction of the rational solutions to the hierarchy of the kadomtsev petviashvili equation is presented the generalization of the superposition formula is found to permit the construction of a new general solution from some partial ones the features of the general polynomial factorization and the types of scattering in the many body calogero moser problem are investigated

solutions for the vector k constrained kp hierarchy a class of 1 1 dimensional integrable systems which are constrained from the kadomtsev petviashvili kp hierarchy is studied the bilinear equations of these integrable systems are shown first and then by using the free fermion operators their rational and soliton solutions which are closely related to the functions of the kp hierarchy are obtained the result implies that a large family of solutions of the 2 1 dimensional kp hierarchy can be obtained by solving these 1 1 dimensional integrable systems in particular the n soliton solutions of the kp hierarchy can be deduced from the soliton solutions of these 1 1 dimensional integrable systems

closed testing procedures for group sequential clinical trials with multiple endpoints summary a simple approach is given for conducting closed testing in clinical trials with multiple end points in which group sequential monitoring is planned the approach allows a flexible stopping time the earliest and latest stopping times are described the paradigm is applicable both to clinical trials with multiple endpoints and to the one sided multiple comparison problem of several treatments versus a control the approach leads to enhancements of previous methods and suggestions for new methods an example of a respiratory disease trial with four endpoints is given

multicomponent integrable wave equations ii soliton solutions the darboux dressing transformations developed in degasperis and lombardo 2007 j phys a math theor 40 961 77 are here applied to construct soliton solutions for a class of boomeronic type equations the vacuum i e vanishing solution and the generic plane wave solution are both dressed to yield one soliton solutions the formulae are specialized to the particularly interesting case of the resonant interaction of three waves a well known model which is of boomeronic type for this equation a novel solution which describes three locked dark pulses simulton is introduced

integrable models in nonlinear optics and soliton solutions nonlinear optics more than other subjects has surprisingly produced quite a number of approximate models which are integrable here we briefly mention these models with the aim of showing contacts between the mathematical and applicative sides of the theory of integrable dynamical systems in particular we illustrate the recent impact of boomeronic type wave equations on applicative issues in the special context of the resonant interaction of three waves

exact solutions of the 3 wave resonant interaction equation the darboux dressing transformations are applied to the lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1 1 dimensions we display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary x and which are not pure soliton solutions these solutions depend on an arbitrary function and allow us to deal with collisions of waves with various profiles

interactive learning of monotone boolean functions this paper presents some optimal interactive algorithms for some problems related to learning of monotone boolean functions these algorithms are based on the fundamental hansel theorem the advantage of the algorithms is that they are not heuristics as is often the case of many known algorithms for general boolean functions but they are optimal in the sense of the shannon function this paper also formulates a new problem for the joint restoration of two nested monotone boolean functions f1 and f2 this formulation allows one to further decrease the dialogue with an expert and restore nonmonotone functions of the form f2 f1 the effectiveness of the proposed approaches is demonstrated by some illustrative computational experiments

universality and integrability of the nonlinear evolution pde s describing n wave interactions the universality of the equations describing n wave interactions is demonstrated by deriving them from a very large class of nonlinear evolution equations essentially all those whose linear part is dispersive various forms of these equations are displayed the fact that these universal nonlinear evolution equations obtain by an appropriate asymptotic limit from such a large class of nonlinear evolution equations suggests that they should be integrable since for this it is sufficient that the large class from which they are obtainable contain just one integrable equation this expectation is validated in several cases by deriving the equations from known integrable equations in this manner an explanation may be provided of the already known integrable nature of certain equations and new integrable equations may be obtained both s integrable and c integrable equations are discussed namely both equations integrable via an appropriate spectral transform and solvable via an appropriate change of variables in this paper the treatment is limited to equations in 1 1 dimensions

integrability of nonlinear wave equations and solvability of their initial value problem we investigate one particular prototypical system of evolution equations in 1 1 dimensions which models the nonlocal interaction of two waves in quadratic nonlinear media these equations are integrable as they possess a lax pair we consider the spectral method of solving the associated initial value problem and as recently pointed out we show that generically the associated initial value problem cannot be linearized we further elaborate on the integrability issue for this model by displaying the corresponding conservation laws and the motion of a particular soliton

a simple generalization of the o brien and fleming group sequential test procedure to more than two treatment groups as a modification of a fixed single sample trial o brien and fleming 1979 proposed a group sequential test procedure for clinical trials for comparing only two treatments the purpose of this paper is to provide a direct generalization of this procedure to accomodate more than two treatment groups on the basis of monte carlo simulations a table that summarizes the critical values for levels equal to 05 and 10 in the situation where the number of treatment groups ranges from 2 to 10 and the number of sequential tests ranges from 1 to 10 has been derived as well

general high order rogue waves and their dynamics in the nonlinear schrodinger equation general high order rogue waves in the nonlinear schrodinger equation are derived by the bilinear method these rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions it is shown that the general n th order rogue waves contain n 1 free irreducible complex parameters in addition the specific rogue waves obtained by akhmediev et al akhmediev et al 2009 phys rev e 80 026601 doi 10 1103 physreve 80 026601 correspond to special choices of these free parameters and they have the highest peak amplitudes among all rogue waves of the same order if other values of these free parameters are taken however these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns

two parameters wronskian representation of solutions of nonlinear schrodinger equation eighth peregrine breather and multi rogue waves in this paper we present a representation of solutions of the one dimensional focusing nonlinear schrodinger equation as a quotient of two wronskians depending on two parameters here we give the complete proof of this representation we have already constructed peregrine breathers and their two parameter deformations until order 7 with this method the construction of the explicit analytical expressions of peregrine breather of order 8 was made for the first time when parameters a or b are equal to 0 we recover the peregrine breather of order 8 we obtain multi rogue waves by deformation of parameters a and b these expressions enable us to understand the evolution of the solutions in the case of order 8 it is shown for high values of parameters a or b the appearance of peregrine breather of order 6

an o brien fleming sequential trial for comparing three treatments we consider a sequential procedure for comparing three treatments with the goal of ultimately selecting the best treatment this procedure starts with a sequential test to detect an overall treatment difference and eliminates the apparently inferior treatment if this test rejects the equality of the treatments it then proceeds with a sequential test of the remaining two treatments we base these sequential tests on the stopping boundaries popularized by o brien and fleming our procedure is similar in structure to that used by siegmund in conjunction with modified repeated significance tests we compare the performances of the two procedures via a simulation experiment we derive analytic approximations for an error probability the power and the expected sample size of our procedure which we compare to simulated values furthermore we propose a modification of the procedure for the comparison of a standard treatment with experimental treatments

six parameters deformations of fourth order peregrine breather solutions of the nonlinear schrodinger equation we construct solutions of the focusing nonlinear schrodinger nls equation as a quotient of two determinants this formulation gives in the case of the order 4 new deformations of the peregrine breather with 6 real parameters we construct families of quasi rational solutions of the nls equation and describe the apparition of multi rogue waves with this method we construct the analytical expressions of deformations of the peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves

on general rogue waves in the parity time symmetric nonlinear schrodinger equation abstract this article addresses the question of general rogue wave solutions in the nonlocal parity time symmetric nonlinear schrodinger equation by generalizing the previous bilinear method large classes of rogue waves are derived as gram determinants with schur polynomial elements it is shown that these rogue waves contain previously reported ones as special cases more importantly they contain many new rogue wave families it is conjectured that the rogue waves derived in this article are all rogue wave solutions in the parity time symmetric nonlinear schrodinger equation

fuzzy hmc classifiers in this study we present a design strategy for hierarchical modular classifiers applicable to fuzzy and or overlapping categories the design includes techniques for decomposing a large scale problem into a hierarchy of subproblems a neural classification module is then designed for each of the subproblems the collection of classification modules and a rule book governing their use comprise the resultant design as an ancillary but necessary issue we demonstrate the uses of a specific neural network for generating membership functions we also explore the use of maximum membership as a classification criterion

new integrable pdes of boomeronic type novel integrable systems of coupled first order autonomous pdes in 1 1 dimensions space x and time t are presented integrable covariant 2 vector and 3 vector pdes as well as higher order integrable pdes for a single or a couple of scalar dependent variables including an extension of the sine gordon equation and a remarkably neat highly nonlinear third order pde are also obtained by appropriate reductions of the basic matrix equations the lax pairs that characterize the integrable character of the basic matrix pdes are also exhibited as well as their single soliton solutions these solitons generally exhibit the boomeronic and less generically the trapponic phenomenology namely they do not move uniformly but rather in an appropriate reference system come in from one end in the remote past and boomerang back to that same end in the remote future boomerons or are trapped to oscillate around a value fixed by the initial data trappons

inverse scattering transform for the vector nonlinear schrodinger equation with nonvanishing boundary conditions the inverse scattering transform for the vector defocusing nonlinear schrodinger nls equation with nonvanishing boundary values at infinity is constructed the direct scattering problem is formulated on a two sheeted covering of the complex plane two out of the six jost eigenfunctions however do not admit an analytic extension on either sheet of the riemann surface therefore a suitable modification of both the direct and the inverse problem formulations is necessary on the direct side this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions the discrete spectrum bound states and symmetries of the direct problem are then discussed in the most general situation a discrete eigenvalue corresponds to a quartet of zeros poles of certain scattering data the inverse scattering problem is formulated in terms of a generalized riemann hilbert rh problem in the upper lower half planes of a suitable uniformization variable special soliton solutions are constructed from the poles in the rh problem and include dark dark soliton solutions which have dark solitonic behavior in both components as well as dark bright soliton solutions which have one dark and one bright component the linear limit is obtained from the rh problem and is shown to correspond to the fourier transform solution obtained from the linearized vector nls system

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

kp symmetry reductions and a generalized constraint we study the link between symmetry reductions and constraints of the kadomtsev petviashvili equations in terms of the tau function we propose a generalization adapted to non zero boundary conditions of the standard constraints and show a particular class of solutions solitons

akns and nls hierarchies mrw solutions pn breathers and beyond we describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the ablowitz kaup newell segur akns hierarchy and their mixed and deformed versions the definition of the akns hierarchy and its deformed versions is given in the sec ii we also consider the continuous symmetries of the related equations and the related spectral curves this work continues and summarises some of our previous studies dedicated to the rogue wave like solutions associated with akns nonlinear schrodinger and kp hierarchies the general scheme is illustrated by the examples of small rank n n 7 rational or quasi rational solutions in particular we consider rank 2 and rank 3 quasi rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics hydrodynamics and many other branches of science we describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the ablowitz kaup newell segur akns hierarchy and their mixed and deformed versions the definition of the akns hierarchy and its deformed versions is given in the sec ii we also consider the continuous symmetries of the related equations and the related spectral curves this work continues and summarises some of our previous studies dedicated to the rogue wave like solutions associated with akns nonlinear schrodinger and kp hierarchies the general scheme is illustrated by the examples of small rank n n 7 rational or quasi rational solutions in particular we consider rank 2 and rank 3 quasi rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics hydrodynamics and many other branches of science

rational solutions of the kp hierarchy and the dynamics of their poles ii construction of the degenerate polynomial solutions a general approach to constructing the polynomial solutions satisfying various reductions of the kadomtsev petviashvili kp hierarchy is described within this approach the reductions of the kp hierarchy are equivalent to certain differential equations imposed on the function of the hierarchy in particular the l reduction and the k constraint as well as their generalized counterparts are considered a general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the kdv and gardner equations the boussinesq and classical boussinesq systems the nls and yajima oikawa equations it is shown that the degenerate rational solutions of the kp hierarchy are related to stationary manifolds of the calogero moser cm hierarchy of dynamical systems the scattering dynamics of interacting particles in the cm systems may become complicated due to an anomalously slow fractional power rate of the particle motion along the stationary manifolds

rational solitons of wave resonant interaction models integrable models of resonant interaction of two or more waves in 1 1 dimensions are known to be of applicative interest in several areas here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear schrodinger equations and the equations describing the resonant interaction of three waves the darboux dressing construction of soliton solutions is applied under the condition that the solutions have rational or mixed rational exponential dependence on coordinates our algebraic construction relies on the use of nilpotent matrices and their jordan form we systematically search for all bounded rational mixed rational exponential solutions and find a broad family of such solutions of the three wave resonant interaction equations

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

group sequential large sample t2 like chi2 tests for multivariate observations in many studies a k degree of freedom large sample 2 test is used to assess the effect of treatment on a multivariate response such as an omnibus t2 like test of a difference between two treatment groups in any of k repeated measures alternately a k df 2 test may be used to test the equality of k 1 groups in a single outcome measure jennison and turnbull biometrika 1991 78 133 141 describe group sequential 2 and f tests for normal errors linear models and proschan follmann and geller statist med 1994 13 1441 1452 describe group sequential tests for k 1 group comparisons these methods apply to sequences of statistics that can be characterized as having an independent increments variance covariance structure thus simplifying the computation of the sequential variance covariance matrix and the resulting sequential test boundaries however many commonly used statistics do not share this structure including a liang zeger biometrika 1986 73 13 22 gee longitudinal analysis with an independence working correlation structure and a wei lachin j amer statist assoc 1984 79 653 661 multivariate wilcoxon rank test among others for such analyses this paper describes the computation of group sequential boundaries for the interim analysis of emerging results using k df tests that are expressed as quadratic forms in a statistics vector that is distributed as multivariate normal at least asymptotically we derive the elements of the covariance matrix of multiple successive k df 2 statistics based on established theorems on the distribution of quadratic forms this covariance matrix is estimated by augmenting the data from the successive interim analyses into a single analysis from which the component sequential tests and their variance covariance matrix can then be extracted boundary values for the sequential statistics can then be computed using the method of slud and wei j amer statist assoc 1982 77 862 868 or using the spending function of lan and demets biometrika 1983 70 659 663 with a surrogate measure of information an example is presented using the analysis of repeated cholesterol measurements in a clinical trial copyright 2003 john wiley sons ltd

rogue waves and rational solutions of the hirota equation the hirota equation is a modified nonlinear schrodinger equation nlse that takes into account higher order dispersion and time delay corrections to the cubic nonlinearity in describing wave propagation in the ocean and optical fibers it can be viewed as an approximation which is more accurate than the nlse we have modified the darboux transformation technique to show how to construct the hierarchy of rational solutions of the hirota equation we present explicit forms for the two lower order solutions each one is a regular nonsingular rational solution with a single maximum that can describe a rogue wave in this model numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution

semiparametric efficiency and its implication on the design and analysis of group sequential studies abstract authors have shown that the time sequential joint distributions of many statistics used to analyze data arising from group sequential time to event and longitudinal studies are multivariate normal with an independent increments covariance structure in theorem 1 of this article we demonstrate that this limiting distribution arises naturally when one uses an efficient test statistic to test a single parameter in a semiparametric or parametric model because we are able to think of many of the statistics in the literature in this fashion the limiting distribution under investigation is just a special case of theorem 1 using this general structure we then develop an information based design and monitoring procedure that can be applied to any type of model for any type of group sequential study provided that there is a unique parameter of interest that can be efficiently tested

steering high order moment calculations from lower dimensional spaces abstract in this study we consider some well established algorithms used to determine separating hyperplanes for sets in r n we implement these algorithms in a high dimensional tensor space that is affiliated with representing the higher order moment information for the sets in question we identify an inner product preserving morphism from tensor space to r n using the morphism the calculations of the algorithm are brought down to a space of much lower dimension our results make it clear that higher order moment computations can often avoid the burden of computation in a high dimensional vector space

exact calculations for sequential t x2 and f tests summary sequential and group sequential procedures are proposed for monitoring repeated t x2or f statistics these can be used to construct sequential hypothesis tests or repeated confidence intervals when the parameter of interest is a normal mean with unknown variance or a multivariate normal mean with variance matrix known or known up to a scale factor exact methods for calculating error probabilities and sample size distributions are described and tables of critical values needed to implement the procedures are provided

generalized korteweg de vries equation induced from position dependent effective mass quantum models and mass deformed soliton solution through inverse scattering transform we consider one dimensional stationary position dependent effective mass quantum model and derive a generalized korteweg de vries kdv equation in 1 1 dimension through lax pair formulation one being the effective mass schrodinger operator and the other being the time evolution of wave functions we obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized kdv equation is an integrable system inverse scattering transform method is applied to obtain general solution of the nonlinear equation and then n soliton solution is derived for reflectionless potentials finally a special choice has been made for the variable mass function to get mass deformed soliton solution the influence of position and time dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail the remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional kdv soliton associated with constant mass quantum model

darboux transformations for the nonlinear schrodinger equations darboux transformations for the akns zs system are constructed in terms of grammian type determinants of vector solutions of the associated lax pairs with an operator spectral parameter a study of the reduction of the darboux transformation for the nonlinear schrodinger equations with standard and anomalous dispersion is presented two different families of new solutions for a given seed solution of the nonlinear schrodinger equation are given being one family related to a new vector lax pair for it in the first family and associated to diagonal matrices we present topological solutions with different asymptotic argument for the amplitude and nonzero background for the anomalous dispersion case they represent continuous deformations of the bright n soliton solution which is recovered for zero background in particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear schrodinger equation associated with jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions the second family contains not only the solutions mentioned above but also broader classes of solutions for example in the standard dispersion case we are able to obtain the dark soliton solutions

nonparametric sequential evaluation of diagnostic biomarkers we consider evaluation and comparison of the diagnostic accuracy of biomarkers with continuous test outcomes possibly correlated due to repeated measurements we develop nonparametric group sequential testing procedures to evaluate and compare the area of biomarkers under their receiver operating characteristic curves with either independent or paired test outcomes these procedures rely on the construction of a two dimensional statistic of whitehead statist med 1999 18 2271 2286 so that design methods based on brownian motion can be applied

vector rogue waves in the manakov system diversity and compossibility we employ a nonrecursive darboux transformation formalism for obtaining a hierarchy of rogue wave solutions to the focusing vector nonlinear schrodinger equations manakov system the exact explicit rogue wave solutions up to the second order are presented typical rogue wave patterns such as peregrine type triple quadruple and sextuple vector rogue waves either bright dark or bright bright in their respective components are put forward despite the diversity there exists a universal compossibility that different rogue wave states could coexist for the same background parameters it is also shown that the higher order rogue wave hierarchy can indeed be thought of as a nonlinear superposition of a fixed well prescribed number of fundamental rogue waves these results may help understand the protean rogue wave manifestations in areas ranging from hydrodynamics to nonlinear optics

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

multicomponent integrable wave equations i darboux dressing transformation the darboux dressing transformations are applied to the lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bright and dark soliton solutions the general formalism is set up and the relevant equations are explicitly solved several instances of multicomponent wave equations of applicative interest such as vector nonlinear schrodinger type equations and three resonant wave equations are considered

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

modulation instability the beginning abstract we discuss the early history of an important field of sturm and drang in modern theory of nonlinear waves it is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability a phenomenon resulting in a variety of nonlinear processes such as envelope solitons envelope shocks freak waves etc examples from water wave hydrodynamics electrodynamics nonlinear optics and convection theory are given

variants of self organizing maps self organizing maps have a bearing on traditional vector quantization a characteristic that makes them more closely resemble certain biological brain maps however is the spatial order of their responses which is formed in the learning process a discussion is presented of the basic algorithms and two innovations dynamic weighting of the input signals at each input of each cell which improves the ordering when very different input signals are used and definition of neighborhoods in the learning algorithm by the minimal spanning tree which provides a far better and faster approximation of prominently structured density functions it is cautioned that if the maps are used for pattern recognition and decision process it is necessary to fine tune the reference vectors so that they directly define the decision borders

dynamic behaviors of interaction solutions of 3 1 dimensional shallow water wave equation abstract rational solutions and interaction solutions of the 3 1 dimensional shallow water wave equation are obtained depending on n soliton solution by the hirota bilinear method and long wave limit method and breather lump solutions and line rogue waves are obtained from two soliton solution as well as their interaction solutions are derived from four soliton solution after the collision breathers and lumps remain unchanged but the line rogue wave has a difference furthermore three soliton solution is discussed the interactions between some hybrid solutions such as hybrid solution between the stripe soliton and breather hybrid solution between stripe soliton and lump solution and so on are demonstrated in detail more interestingly stripe soliton remain unchanged but the amplitudes of the periodic line waves lump solutions and rogue waves have a great change after the collision