mkdv and its birkhoff coordinates abstract we show that the modified korteweg de vries equation with periodic boundary conditions admits global birkhoff coordinates on any soblev space h n with n 1 the construction of these coordinates involves a new lax pair for mkdv

canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

algebraic signal processing theory foundation and 1 d time this paper introduces a general and axiomatic approach to linear signal processing sp that we refer to as the algebraic signal processing theory asp basic to asp is the linear signal model defined as a triple a m phi where familiar concepts like the filter space and the signal space are cast as an algebra a and a module m respectively the mapping phi generalizes the concept of a z transform to bijective linear mappings from a vector space of signal samples into the module m common concepts like filtering spectrum or fourier transform have their equivalent counterparts in asp once these concepts and their properties are defined and understood in the context of asp they remain true and apply to specific instantiations of the asp signal model for example to develop signal processing theories for infinite and finite discrete time signals for infinite or finite discrete space signals or for multidimensional signals we need only to instantiate the signal model to one that makes sense for that specific class of signals filtering spectrum fourier transform and other notions follow then from the corresponding asp concepts similarly common assumptions in sp translate into requirements on the asp signal model for example shift invariance is equivalent to a being commutative for finite duration signals shift invariance then restricts a to polynomial algebras we explain how to design signal models from the specification of a special filter the shift the paper illustrates the general asp theory with the standard time shift presenting a unique signal model for infinite time and several signal models for finite time the latter models illustrate the role played by boundary conditions and recover the discrete fourier transform dft and its variants as associated fourier transforms finally asp provides a systematic methodology to derive fast algorithms for linear transforms this topic and the application of asp to space dependent signals and to multidimensional signals are pursued in companion papers

optimal experimental design in the presence of nested factors abstracta common occurrence in practical design of experiments is that one factor called a nested factor can only be varied for some but not all the levels of a categorical factor called a branc

mean squared error of estimation or prediction under a general linear model abstract the problem considered is that of predicting a linear combination of the fixed and random effects of a mixed effects linear model more generally the problem considered is that of predicting an unobservable random variable from a set of observable random variables the best linear unbiased predictor depends on parameters which generally are unknown various exact or approximate expressions are given for the mean squared error mse of the predictor obtained by replacing the unknown parameters with estimates several estimators of the mse are investigated

on e functions of semisimple lie groups we develop and describe continuous and discrete transforms of class functions on a compact semisimple but not simple lie group g as their expansions into series of special functions that are invariant under the action of the even subgroup of the weyl group of g we distinguish two cases of even weyl groups one is the direct product of even weyl groups of simple components of g and the second is the full even weyl group of g the problem is rather simple in two dimensions it is much richer in dimensions greater than two we describe in detail e transforms of semisimple lie groups of rank 3

nonstandard drinfeld sokolov reduction subject to some conditions the input data for the drinfeld sokolov construction of kdv type hierarchies is a quadruplet where the are gradations of a loop algebra and is a semisimple element of the nonzero grade a new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented the proposal relies on splitting the grade zero part of into a vector space direct sum of two subalgebras this permits one to interpret certain gelfand dickey type systems associated with a nonstandard splitting of the algebra of pseudodifferential operators in the drinfeld sokolov framework

string theory and water waves we uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non perturbatively we are able to embed the type 0a and 0b a a minimal string theories into this single framework the string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy we observe that there are several other string like limits of the system and conjecture that some of them are type iia and iib a d minimal string backgrounds we explain how these and several string like special points arise and are connected in some cases the framework endows the theories with a non perturbative definition for the first time notably we discover that the painlev iv equation plays a key role in organizing the string theory physics joining its siblings painlev i and ii whose roles have previously been identified in this minimal string context

d optimal split plot designs with given numbers and sizes of whole plots the design of split plot experiments has received considerable attention during the last few years the goal of this article is to provide an efficient algorithm to compute d optimal split plot designs with given numbers of whole plots and given whole plot sizes the algorithm is evaluated and applied to a protein extraction experiment in addition it is shown that two level factorial and fractional factorial designs are d optimal for estimating first order response surface models for specific numbers and sizes of whole plots

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

on the geometry of darboux transformations for the kp hierarchy and its connection with the discrete kp hierarchy we tackle the problem of interpreting the darboux transformation for the kp hierarchy and its relations with the modified kp hierarchy from a geometric point of view this is achieved by introducing the concept of a darboux covering we construct a darboux covering of the kp equations and obtain a new hierarchy of equations which we call the darboux kp hierarchy dkp we employ the dkp equations to discuss the relationships among the kp equations the modified kp equations and the discrete kp equations our approach also handles the various reductions of the kp hierarchy we show that the kp hierarchy is a projection of the dkp the mkp hierarchy is a dkp restriction to a suitable invariant submanifold and that the discrete kp equations are obtained as iterations of the dkp ones

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

d optimal design of split split plot experiments in industrial experimentation there is growing interest in studies that span more than one processing step convenience often dictates restrictions in randomization in passing from one processing step to another when the study encompasses three processing steps this leads to split split plot designs we provide an algorithm for computing d optimal split split plot designs and several illustrative examples

a permutation test for effects in an unreplicated factorial design a permutation test is proposed for examining the significance of effects in unreplicated factorial experiments the procedure tests each effect with a separate sampling distribution using a test statistic that is equivalent to the optimal invariant decision rule of birnbaum the proposed test is more flexible than other methods proposed for the same situation because it requires no a priori assumptions regarding the underlying distribution of the data nor does it impose any practical restriction on the number of potentially significant effects present

a general strategy for analyzing data from split plot and multistratum experimental designs increasingly industrial experiments use multistratum designs such as split plot and strip plot designs often these experiments span more than one processing stage the challenge is to identify an appropriate multistratum design along with an appropriate statistical model in this article we introduce hasse diagrams in the response surface context as a tool to visualize the unit structure of the experimental design the randomization and sampling approaches used the stratum in which each experimental factor is applied and the degrees of freedom available in each stratum to estimate main effects interactions and variance components we illustrate their use on several responses measured in a large study of the adhesion properties of coatings to polypropylene we discuss quantitative binary and ordered categorical responses for designs ranging from a simple split plot to a strip plot that involves repeated measurements of the response the datasets discussed in this article are available online a

multistratum response surface designs response surface designs are usually described as if the treatments have been completely randomized to the experimental units however in practice there is often a structure to the units implying the need for blocking if in addition some factors are more difficult to vary between units than others a multistratum structure arises naturally we present a general strategy for constructing response surface designs in multistratum unit structures designs are constructed stratum by stratum starting in the highest stratum in each stratum a prespecified treatment set for the factors applied in that stratum is arranged to be nearly orthogonal to the units in the higher strata allowing for all the effects that have to be estimated three examples are given to show the applicability of the method and are also used to check the relationship of the final design to the choice of treatment set finally some practical considerations in randomization are discussed

newton s method on the complex exponential function we show that when newton s method is applied to the product of a polynomial and the exponential function in the complex plane the basins of attraction of roots have finite area

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

improved split plot and multistratum designs many industrial experiments involve some factors whose levels are harder to set than others the best way to deal with these is to plan the experiment carefully as a split plot or more generally a multistratum design several different approaches for constructing split plot type response surface designs have been proposed in the literature since 2001 which has allowed experimenters to make better use of their resources by using more efficient designs than the classical balanced ones one of these approaches the stratum by stratum strategy has been shown to produce designs that are less efficient than locally d optimal designs an improved stratum by stratum algorithm is given which though more computationally intensive than the old one makes better use of the advantages of this approach that is it can be used for any structure and does not depend on prior estimates of the variance components this is shown to be almost as good as the locally optimal designs in terms of their own criteria and more

the hartley transform in seismic imaging phase shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast fourier transform fft generally the hermitian symmetry of the complex valued fourier transform causes computational redundancies in terms of the number of operations and memory requirements in practice a combination of the fft with the well known real to complex fourier transform is often used to avoid such complications as an alternative means to the fourier transform we introduce the inherently real valued non symmetric hartley transform into phase shift migration techniques by this we automatically avoid the hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real to complex fft we derive the phase shift operator in the hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation split step migration and split step double square root prestack migration in terms of the

on discretization of tori of compact simple lie groups the discrete orthogonality of special function families called c and s functions which are derived from the characters of compact simple lie groups is described in hrivnak and patera 2009 j phys a math theor 42 385208 here the results of hrivnak and patera are extended to two additional recently discovered families of special functions called ss and sl functions the main result is an explicit description of their pairwise discrete orthogonality within each family when the functions are sampled on finite fragments fsm and flm of a lattice in any dimension n 2 and of any density controlled by m and of the symmetry of the weight lattice of any compact simple lie group with two different lengths of roots

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

split plot and multi stratum designs for statistical inference abstractit is increasingly recognized that many industrial and engineering experiments use split plot or other multi stratum structures much recent work has concentrated on finding optimum or near optimum designs for estimating the fixed effects parameters in multi stratum designs however often inference such as hypothesis testing or interval estimation will also be required and for inference to be unbiased in the presence of model uncertainty requires pure error estimates of the variance components most optimal designs provide few if any pure error degrees of freedom gilmour and trinca 2012 introduced design optimality criteria for inference in the context of completely randomized and block designs here these criteria are used stratum by stratum to obtain multi stratum designs it is shown that these designs have better properties for performing inference than standard optimum designs compound criteria which combine the inference criteria with traditional point estimation criteria are al

quick and easy analysis of unreplicated factorials box and meyer 1986 introduced a method for assessing the sizes of contrasts in unreplicated factorial and fractional factorial designs this is a useful technique and an associated graphical display popularly known as a bayes plot makes it even more effective this article presents a competing technique that is also effective and is computationally simple an advantage of the new method is that the results are given in terms of the original units of measurement this direct association with the data may make the analysis easier to explain

miura map between lattice kadomtsev petviashvili and its modification is canonical we consider the miura map between the lattice kadomtsev petviashvili hierarchy and the lattice modified kp hierarchy and prove that the map is canonical not only between the first hamiltonian structures but also between the second hamiltonian structures

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

graphene lattice field theory and symmetries borrowing ideas from tight binding model we propose a board class of lattice field models that are classified by non simply laced lie algebras in the case of an 1 su n series we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice lsun are governed by the complex fundamental representations n and n of su n and the second nearest neighbor interactions are described by its adjoint n n the lattice models associated with the leading su 2 su 3 and su 4 cases are explicitly studied and their fermionic field realizations are given it is also shown that the su 2 and su 3 models describe the electronic properties of the acetylene chain and the graphene respectively it is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the an roots through the typical dependence n 2 roots cosk with k the wave vector other features such as the so 2n extension and other application

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

the coordinate exchange algorithm for constructing exact optimal experimental designs we describe the cyclic coordinate exchange algorithm for constructing d optimal and linear optimal experimental designs the algorithm uses a variant of the gauss southwell cyclic coordinate descent algorithm within the k exchange algorithm to achieve substantive reductions in required computing among its advantages are the following candidate sets which grow exponentially in the number of factors need not be explicitly constructed or enumerated convex design spaces or mixed convex by discrete design spaces are handled directly without the need for sophisticated nonlinear programming routines or candidate set adjustment for design problems having 10 or more factors the reductions in execution time are typically two or more orders of magnitude when compared to standard candidateset based procedures such as k exchange yet the designs produced exhibit no loss of efficiency

generalized discrete orbit function transforms of affine weyl groups the affine weyl groups with their corresponding four types of orbit functions are considered two independent admissible shifts which preserve the symmetries of the weight and the dual weight lattices are classified finite subsets of the shifted weight and the shifted dual weight lattices which serve as a sampling grid and a set of labels of the orbit functions respectively are introduced the complete sets of discretely orthogonal orbit functions over the sampling grids are found and the corresponding discrete fourier transforms are formulated the eight standard one dimensional discrete cosine and sine transforms form special cases of the presented transforms

staggered level designs for experiments with more than one hard to change factor in many industrial experiments some of the factors are not independently set for each run this is due to time and or cost constraints and to the hard to change nature of the levels of these factors most of the literature restricts attention to split plot designs in which all the hard to change factors are independently reset at the same points in time this constraint is to some extent relaxed in split split plot designs because these allow the less hard to change factors to be reset more often than the most hard to change factors a key feature of the split split plot designs however is that the less hard to change factors are reset whenever the most hard to change factors are reset in this article we relax this constraint and present a new type of design which allows the hard to change factor levels to be reset at entirely different points in time we show that the new designs are cost efficient and that they outperform split plot and split split plot designs in terms of the d and a optimality

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

agricultural response surface experiments based on four level factorial designs treatment factors with four equally spaced quantitative levels can be expressed as direct products of pairs of pseudo factors each at two levels where the interaction between the two pseudo factors of a pair represents the quadratic trend component of a treatment factor assuming no cubic trend component in the response model the two pseudo factors of the pair each represent components of the pure linear trend term of the factor interactions between pseudo factors in different pairs represent linear by linear treatment interaction effects further the variances of the two independent estimates of a linear trend component derived from a pair of pseudo factor effects are in the ratio 1 4 so one pseudo factor of a pair is of major and the other of minor importance these relationships are exploited to construct quadratic response surface designs based on fourlevel factorial designs using the theory of aliasing and confounding applied to the equivalent twolevel pseudo factorial designs the advantages of these designs for agricultural experiments are discussed and a set of example designs is listed the design and analysis of a glasshouse experiment are outlined using this approach

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

on e discretization of tori of compact simple lie groups ii ten types of discrete fourier transforms of weyl orbit functions are developed generalizing one dimensional cosine sine and exponential each type of the weyl orbit function represents an exponential symmetrized with respect to a subgroup of the weyl group fundamental domains of even affine and dual even affine weyl groups governing the argument and label symmetries of the even orbit functions are determined the discrete orthogonality relations are formulated on finite sets of points from the refinements of the dual weight lattices explicit counting formulas for the number of points of the discrete transforms are deduced real valued hartley orbit functions are introduced and all ten types of the corresponding discrete hartley transforms are detailed

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

canonical miura maps between the modified kp and kp hierarchies we investigate the hamiltonian nature of two miura maps between the modified kp and kp hierarchies we show that they are canonical in the sense that the bi hamiltonian structure of the modified kp hierarchy is mapped to the bi hamiltonian structure of the kp hierarchy