the degrees of freedom of partial least squares regression the derivation of statistical properties for partial least squares regression can be a challenging task the reason is that the construction of latent components from the predictor variables also depends on the response variable while this typically leads to good performance and interpretable models in practice it makes the statistical analysis more involved in this work we study the intrinsic complexity of partial least squares regression our contribution is an unbiased estimate of its degrees of freedom it is defined as the trace of the first derivative of the fitted values seen as a function of the response we establish two equivalent representations that rely on the close connection of partial least squares to matrix decompositions and krylov subspace techniques we show that the degrees of freedom depend on the collinearity of the predictor variables the lower the collinearity the higher the complexity in particular they are typically higher than the naive approach that defines the degree

global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights recently a dynamic neural system was presented and analyzed due to its good performance in optimization computation and low complexity for implementation the global asymptotic stability of such a dynamic neural system with symmetric connection weights was well studied in this note based on a new lyapunov function we investigate the global asymptotic stability of the dynamic neural system with asymmetric connection weights since the dynamic neural system with asymmetric weights is more general than that with symmetric ones the new results are significant in both theory and applications specially the new result can cover the asymptotic stability results of linear systems as special cases

tests of equal forecast accuracy and encompassing for nested models we examine the asymptotic and finite sample properties of tests for equal forecast accuracy and encompassing applied to 1 step ahead forecasts from nested linear models we first derive the asymptotic distributions of two standard tests and one new test of encompassing and provide tables of asymptotically valid critical values monte carlo methods are then used to evaluate the size and power of tests of equal forecast accuracy and encompassing the simulations indicate that post sample tests can be reasonably well sized of the post sample tests considered the encompassing test proposed in this paper is the most powerful we conclude with an empirical application regarding the predictive content of unemployment for inflation

fourier pseudospectral method with kepler mapping for travelling waves with discontinuous slope application to corner waves of the ostrovsky hunter equation and equatorial kelvin waves in the four mode approximation many species of travelling waves have a single branch of solutions which ends at finite amplitude with a singular wave whose slope is discontinuous a so called corner wave fourier pseudospectral methods converge exponentially fast with n the truncation of the series for the smooth waves but the error falls only as o 1 n for a wave with a slope discontinuity we show that the error rate can be accelerated to o 1 n3 by making the change of coordinate kepler map x z sin z unfortunately there is a subtlety the end in a singular solution bifurcation is possible only for infinite dimensional systems we show that the truncated fourier pseudospectral approximation has roots on both sides of the corner wave the bifurcation can be detected by i a step function like jump in the residual of the differential equation at the corner wave and ii by observing the slope of u x which becomes discontinuous only at the corner wave itself these concepts are illustrated for the equatorially trapped ocean kelvin wave in the so called four latitudinal mode approximation however corner waves arise in many species of waves and the concepts explained here are applicable to all

the generalized dynamic factor model consistency and rates abstract a factor model generalizing those proposed by geweke in d j aigner and a s goldberger latent variables in socio economic models north holland amsterdam 1977 sargent and sims new methods in business research federal reserve bank of minneapolis minneapolis 1977 engle and watson j amer statist assoc 76 1981 774 and stock and watson j business econom statist 20 2002 147 has been introduced in forni et al rev econ statist 80 2000 540 where consistent as the number n of series and the number t of observations both tend to infinity along appropriate paths n t n estimation methods for the common component are proposed rates of convergence associated with these methods are obtained here as functions of the paths n t n along which n and t go to infinity these results show that under suitable assumptions consistency requires t n to be at least of the same order as n whereas an optimal rate of n is reached for t n of the order of n2 if convergence to the space of common components is considered consistency holds irrespective of the path t n thus can be arbitrarily slow the optimal rate is still n but only requires t n to be of the order of n

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

analytical and numerical studies of weakly nonlocal solitary waves of the rotation modified korteweg de vries equation abstract a century ago the korteweg de vries kdv equation was derived as a model for weakly nonlinear long waves propagating down a channel when cross channel and depth variations are sufficiently weak in this article we study the steadily translating coherent structures of a generalization of this equation the rotation modified kdv equation which applies when coriolis forces are significant u t uu x u xxx x 2 u 0 where x is the down channel coordinate this is also called the ostrovosky equation the rmkdv solitary waves are weakly nonlocal due to the radiation of long waves where the radiation is proportional to the small parameter because of its simplicity the rmkdv coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter micropterons we extend the matched asymptotic expansions of hunter to third order we also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem new modes with multiple kdv like peaks flanked by small quasi sinusoidal oscillations are shown to be well described by matched asymptotics too

isospectral flows and abstract matrix factorizations a general framework for constructing isospectral flows in the space operatorname gl n of n by n matrices is proposed depending upon how operatorname gl n is split this framework gives rise to different types of abstract matrix factorizations when sampled at integer times these flows naturally define special iterative processes and each flow is associated with the sequence generated by the corresponding abstract factorization the proposed theory unifies as special cases the well known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

effects of exponentially small terms in the perturbation approach to localized buckling localized buckling solutions are known to exist in the heuristic model of an elastic strut resting on an elastic winkler foundation the primary localized solution emerging from the critical buckling state consists of an amplitude envelope of approximately the form of a hyperbolic secant function which modulates a fast varying sinusoidal oscillation in previous works such solutions have been tracked for the entire post buckling regime both numerically and through a rayleigh ritz approach a very rich structure is known to exist in the subcritical load range but has been proved to exist for only a certain family of reversible systems studies have concentrated on symmetric homoclinic solutions and the asymmetric solutions which bifurcate from these solution paths the primary solution is known to exist for the entire subcritical parameter range and all other symmetric and associated asymmetric solutions exist strictly for values less than critical here we uncover a new family of antisymmetric solutions and some asymptotic analysis suggests that the primary antisymmetric solution exists over the same range as does the primary symmetric solution a perturbation approach can be used so as to describe the bifurcation hierarchy for the novel antisymmetric forms we illustrate a unified approach which is able to predict the circumstances under which non divergent localized solutions are possible and the results of the analysis are compared with some numerical solutions

weakly non local solutions for capillary gravity waves fifth degree korteweg de vries equation hunter and scheurle have shown that capillary gravity water waves in the vicinity of bond number bo l 3 are consistently modelled by the korteweg de vries equation with the addition of a fifth derivative term this wave equation does not have strict soliton solutions for bo l 3 because the near solitons have oscillatory wings that extend indefinitely from the core of the wave however these solutions are arbitrarily small perturbations of solitary waves because the amplitude of the wings is exponentially small in the amplitude 6 of the core pomeau ramani and grammaticos have calculated the amplitude of the wings by applying matched asymptotics in the complex plane in the limit e 0 in this article we describe a mixed chebyshev radiation function pseudospectral method which is able to calculate the weakly non local solitons for all e we show that for fixed phase speed the solitons form a three parameter family because the linearized wave equation has three eigensolutions we also show that one may repeat the soliton with even spacing to create a three parameter of periodic solutions which we also compute because the amplitude of the wings is exponentially small these non local capillary gravity solitons are as interesting as the classical localized solitons that solve the korteweg de vries equation

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations ii periodic solutions in this second of two articles designated i and ii the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long wave rlw equation these solutions are expressed in terms of riemann theta functions and this approach leads to a new and compact expression for the important dispersion relation the periodic solution or cnoidal wave can be represented as an infinite sum of sech2 solitary waves this remarkable property may be interpreted in the context of a nonlinear superposition principle the rlw cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes similar results are presented in brief for the related rlw boussinesq rlwb equation

a differential equation approach to the singular value decomposition of bidiagonal matrices we consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one parameter family of differentiable matrix flows it is shown that this approach can be fully expressed as an autonomous homogeneous and cubic dynamical system asymptotic behavior is justified by the established theory of the toda lattice

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

on the periodic solution of the burgers equation a unified approach for the most part the various known expressions for the periodic solution to the burgers equation have been derived by a mixture of ad hoc argument and approximation methods these include the well known solutions due to r d fay and d f parker remarkably these solutions were obtained without recourse to the celebrated hopf cole transformation which linearizes the burgers equation to the heat conduction equation we present here an alternative approach to the periodic problem which uses the fact that the riemann theta functions solve the burgers equation via the hopf cole transformation and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place yet another expression for the periodic solution in the form of an amplitude modulated sine wave is given and would appear to be new the limiting forms of the periodic solution as t rightarrow 0 embryonic sawtooth profile and t rightarrow infty old age sinusoidal profile are examined in detail the representation due to d f parker which expresses the periodic wave as a superposition of spreading taylor shock profiles is discussed within the context of a nonlinear superposition principle the composite asymptotic expansion due to m b lesser and d g crighton is examined in the light of our results the curious behaviour of this expansion in the embryonic region t ll 1 whereby a first order approximation yields an exact periodic solution the fay solution is explained

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

ols or gls in the presence of specification error an expected loss approach omitted variables in regression analysis can lead to the erroneous conclusion that autocorrelation or heteroscedasticity is present the common response is to use the suggested gls procedure even if it is suspected that the error is a non zero disturbance mean the question addressed here is whether one is better off with the gls or with the ols estimator when the omitted portion of the regression cannot be incorporated into the regression using a loss function this paper relates the seriousness of ols and gls loss to identifiable parameters with consistent estimators of these parameters the researcher can choose between ols and gls

l2 index formulae for perturbed dirac operators the callias index theorem is generalized from the euclidean case to certain spin manifolds with warped ends making use of certain index preserving deformations

on the equivalence between sor type methods for linear systems and the discrete gradient methods for gradient systems abstract inspired by the iterative nature of many discretization methods for continuous dynamical systems connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s for stationary iterative methods solving linear systems chu 1988 2008 discussed a connection to continuous dynamical systems by using the explicit euler method however further understanding of stationary iterative methods might be limited due to the use of the explicit euler method this paper presents a new connection based on the so called discrete gradient methods between sor type methods and gradient systems there the key of the discussion is the equivalence between sor type methods and the discrete gradient methods applied to gradient systems the discussion leads to new interpretations for sor type methods for example a new derivation of sor type methods is found these methods monotonically decrease a certain quadratic function and a new interpretation of the relaxation parameter is obtained besides while studying the new connection a new discrete gradient is also obtained

peakons and coshoidal waves traveling wave solutions of the camassa holm equation abstract camassa and holm 1 have recently derived a new integrable wave equation for the special case 0 they showed it has solitary waves of the form c exp x ct which they named peakons in this work we derive a perturbation series for general which converges even at the peakon limit we also give three analytical representations for the spatially periodic generalization of the peakon the coshoidal wave the three representations are i a closed form analytical solution ii a fourier series with coefficients that are explicit rational functions and iii an imbricate fourier series which is the superposition of an infinite number of peakons each separated from its neighbors by distance p where p is the spatial period lastly we have numerically tested the soliton superposition principle although the camassa holm equation is integrable for general it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases 0 c 1 2 however the imbricate soliton series is a very good approximate solution for general even when the spatial period is small and the solution resembles a sine wave more than a solitary wave

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

the three pass regression filter a new approach to forecasting using many predictors we forecast a single time series using many predictor variables with a new estimator called the three pass regression lter 3prf it is calculated in closed form and conveniently represented as a set of ordinary least squares regressions 3prf forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large this requires only specifying the number of relevant factors driving the forecast target regardless of the total number of common including potentially irrelevant factors driving the cross section of predictors we derive inferential theory including limiting distributions for estimated relevant factors predictive coefcients and forecasts and provide consistent standard error estimators we explore two empirical applications forecasting macroeconomic aggregates with a large panel of economic indices and forecasting stock market aggregates with many individual assets price dividend ratios these combined with a range of monte carlo experiments demonstrate the 3prf s forecasting power

on a numerical treatment for the curve tracing of the homotopy method a newton like iteration scheme is proposed for the tracing of an implicitly defined smooth curve this scheme originates from the study of the continuous newton raphson method for underdetermined systems and hence inherits the characteristic property of orthogonality its domain of attraction is formed and makes it possible to trace this curve more efficiently

double cnoidal waves of the korteweg de vries equation a boundary value approach double cnoidal waves of the korteweg de vries equation are studied by direct solution of the nonlinear boundary value problem these double cnoidal waves which are the spatially periodic generalization of the well known double soliton are exact solutions with two independent phase speeds the equation is written in terms of two phase variables and expanded in two dimensional fourier series the small amplitude solution is obtained via the stokes perturbation expansion this solution is numerically extended to larger amplitude by employing a newton kantorovich continuation in amplitude galerkin algorithm the crests of the finite amplitude solution closely match the sech 2 solitary wave form and the three cases of solitary wave interaction described by lax are identified for the double cnoidal waves this simple approach reproduces specific features such as phase shift upon collision distinction between instantaneous and average phase speeds and a paradox of wavenumbers

numerical computations of a nearly singular nonlinear equation we numerically calculate bions which are bound states of two solitary waves which travel together as a single coherent structure with a fixed peak to peak separation for the fifth order korteweg devries equation r h j grimshaw and b a malomed j phys a26 1993 4087 4091 predicted such bions using perturbation theory we find that the nearly singular quasi translational eigenmode which is the heart of the theory is also numerically important in the sense that later iterations are approximately proportional to this eigenmode however the near singularity does not create any serious problems for our fourier pseudospectral newton kantorovich pseudoarclength continuation algorithms this type of theory for weakly overlapping solitary waves has been previously developed by gorshkov ostrovskii papko and others however grimshaw and malomed s work and our own are the first on bions which are weakly nonlocal that is decay for large x to small amplitude oscillations rather than to zero our numerical calculations confirm the main assertions of grimshaw and malomed however there are other features such as a complicated branch structure with multiple turning points and the existence of bions with narrow peak to peak separation which are not predicted by the theory

a homotopy approach for stabilizing single input systems with control structure constraints a linear state feedback gain vector used in the control of a single input dynamical system may be constrained because of the way feedback is realized some examples of feedback realizations which impose constraints on the gain vector are static output feedback constant gain feedback for several operating points of a system and two controller feedback we consider a general class of problems of stabilization of single input dynamical systems with such structural constraints and give a numerical method to solve them each of these problems is cast into a problem of solving a system of equalities and inequalities in this formulation the coefficients of the quadratic and linear factors of the closed loop characteristic polynomial are the variables to solve the system of equalities and inequalities a continuous realization of the gradient projection method and a barrier method are used under the homotopy framework our method is illustrated with an example for each class of control structure constraint

forecasting economic time series using targeted predictors abstract this paper studies two refinements to the method of factor forecasting first we consider the method of quadratic principal components that allows the link function between the predictors and the factors to be non linear second the factors used in the forecasting equation are estimated in a way to take into account that the goal is to forecast a specific series this is accomplished by applying the method of principal components to targeted predictors selected using hard and soft thresholding rules our three main findings can be summarized as follows first we find improvements at all forecast horizons over the current diffusion index forecasts by estimating the factors using fewer but informative predictors allowing for non linearity often leads to additional gains second forecasting the volatile one month ahead inflation warrants a high degree of targeting to screen out the noisy predictors a handful of variables notably relating to housing starts and interest rates are found to have systematic predictive power for inflation at all horizons third the targeted predictors selected by both soft and hard thresholding changes with the forecast horizon and the sample period holding the set of predictors fixed as is the current practice of factor forecasting is unnecessarily restrictive

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

forecasting with factor augmented regression a frequentist model averaging approach abstract this paper considers forecast combination with factor augmented regression in this framework a large number of forecasting models are available varying by the choice of factors and the number of lags we investigate forecast combination across models using weights that minimize the mallows and the leave h out cross validation criteria the unobserved factor regressors are estimated by principle components of a large panel with n predictors over t periods with these generated regressors we show that the mallows and leave h out cross validation criteria are asymptotically unbiased estimators of the one step ahead and multi step ahead mean squared forecast errors respectively provided that n t however the paper does not establish any optimality properties for the methods in contrast to well known results in the literature this result suggests that the generated regressor issue can be ignored for forecast combination without restrictions on the relation between n and t simulations show that the mallows model averaging and leave h out cross validation averaging methods yield lower mean squared forecast errors than alternative model selection and averaging methods such as aic bic cross validation and bayesian model averaging we apply the proposed methods to the us macroeconomic data set in stock and watson 2012 and find that they compare favorably to many popular shrinkage type forecasting methods

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

deleted residuals the qr factored newton iteration and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary cnoidal and shock waves abstract solitary waves cnoidal waves and shock waves can be computed by solving a nonlinear eigenvalue problem which in discretized form is a system of nonlinear algebraic equations unfortunately many such systems are singular because the solution is not unique until one or more additional constraints are imposed for example if the waves are translationally invariant and u x is a solution then so also is u x for arbitrary to obtain a unique solution one must impose an additional condition to reduce the one parameter family of solutions by constraining we describe five methods for coping with such singular systems i reformulation of the problem ii deleting residuals iii keller s bordered matrix scheme iv qr factored overdetermined newton iteration and v pseudoinverse newton iteration we illustrate these ideas using the cnoidal waves of the korteweg de vries equation the traveling shocks of the korteweg de vries burgers equation and the weakly nonlocal solitary waves of the nonlinear equatorial beta plane equations finite difference fourier and rational chebyshev pseudospectral methods and spectrally upgraded finite differences are applied reformulation and deleting residuals are the cheapest strategies but the qr factored newton iteration is needed for the shock waves which lack the symmetry of the other two wave species

tests of equal accuracy for nested models with estimated factors abstract in this paper we develop asymptotics for tests of equal predictive ability between nested models when factor augmented regressions are used to forecast we provide conditions under which the estimation of the factors does not affect the asymptotic distributions developed in clark and mccracken 2001 and mccracken 2007 this enables researchers to use the existing tabulated critical values when conducting inference despite the presence of estimated predictors as an intermediate result we derive the asymptotic properties of the principal components estimator over recursive windows we provide simulation evidence on the finite sample effects of factor estimation and apply the tests to the case of forecasting excess returns to the s p 500 composite index

confidence intervals for diffusion index forecasts and inference for factor augmented regressions we consider the situation when there is a large number of series n e ach witht observations and each series has some predictive ability for some variable of interest a methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors in this paper we show that the least squares estimates obtained from these factor augmented regressions are t consistent and asymptotically normal if t n 0 the conditional mean predicted by the estimated factors is min t n consistent and asymptotically normal except when t n goes to zero inference should take into account the effect of estimated regressors on the estimated conditional mean we present analytical formulas for prediction intervals that are valid regardless of the magnitude of n t and that can also be used when the factors are nonstationary

the arctan tan and kepler burgers mappings for periodic solutions with a shock front or internal boundary layer abstract many periodic solutions have internal regions of rapid change internal boundary layers shock waves and geophysical fronts are one class of examples a second class is composed of functions which decay rapidly away from a central peak or peaks spherical harmonics mathieu eigenfunctions prolate spheroidal wave functions and geophysical hough functions may all be locally approximated by hermite functions in the appropriate parameter range and decay exponentially fast outside a narrow subinterval similarly the large amplitude cnoidal waves of the korteweg devries equation are narrow isolated peaks which are well approximated by the sech 2 y form of the solitary wave in this article we show that a change of coordinate is a powerful tool for resolving such internal boundary layers in the first part we develop a general theory of mappings for the spherical harmonic cnoidal wave class of examples which decay rapidly away towards the edges of the spatial period the particular map y arc tan l tan x is a particularly effective choice four numerical examples show that this map and the fourier pseudospectral method are a good team in the second part we generalize the earlier theory to describe mappings which asymptote to a constant but non zero resolution at the ends of the periodicity interval we explain why the kepler burgers mapping is particularly suitable for shock and fronts

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

solitary wave interaction phenomena in a strut buckling model incorporating restabilisation the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure the primary solution can be detected by a double scale perturbation procedure and is reminiscent of a solitary wave essentially it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions particular interest is paid to the process of adapting the procedure to account for the post buckling behaviour of two packet or double humped solitary waves in this model we employ the methods of beyond all orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post buckling behaviour of the system including the interaction phenomenon of interest the analysis is reinforced by direct numerical computations which reveal the so called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions however additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

exact periodic solutions of the sixth order generalized boussinesq equation this paper examines a class of nonlinear sixth order generalized boussinesq like equations sgbe utt uxx 3 u2 xx uxxxx uxxxxxx r depending on the positive parameter hirota s bilinear transformation method is applied to the above class of non integrable equations and exact periodic solutions have been obtained the results confirmed the well known nonlinear superposition principle

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

a direct matrix method for computing analytical jacobians of discretized nonlinear integro differential equations in this article we present a simple direct matrix method for analytically computing the jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro differential equations the method is based on a formulation of the discretized equations in vector form using only matrix vector products and component wise operations by applying simple matrix based differentiation rules the matrix form of the analytical jacobian can be calculated with little more difficulty than required to compute derivatives in single variable calculus after describing the direct matrix method we present numerical experiments demonstrating the computational performance of the method discuss its connection to the newton kantorovich method and apply it to illustrative 1d and 2d example problems from electrochemical transport

boosting high dimensional predictive regressions with time varying parameters abstract high dimensional predictive regressions are useful in wide range of applications however the theory is mainly developed assuming that the model is stationary with time invariant parameters this is at odds with the prevalent evidence for parameter instability in economic time series but theories for parameter instability are mainly developed for models with a small number of covariates in this paper we present two l 2 boosting algorithms for estimating high dimensional models in which the coefficients are modeled as functions evolving smoothly over time and the predictors are locally stationary the first method uses componentwise local constant estimators as base learner while the second relies on componentwise local linear estimators we establish consistency of both methods and address the practical issues of choosing the bandwidth for the base learners and the number of boosting iterations in an extensive application to macroeconomic forecasting with many potential predictors we find that the benefits to modeling time variation are substantial and they increase with the forecast horizon furthermore the timing of the benefits suggests that the great moderation is associated with substantial instability in the conditional mean of various economic series

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

sufficient forecasting using factor models we consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect the dimensionality was first reduced via a high dimensional approximate factor model implemented by the principal component analysis using the extracted factors we develop a novel forecasting method called the sufficient forecasting which provides a set of sufficient predictive indices inferred from high dimensional predictors to deliver additional predictive power the projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi parametric approximate factor model is assumed our method is also applicable to cross sectional sufficient regression using extracted factors the connection between the sufficient forecasting and the deep learning architecture is explicitly stated the sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function the proposed method extends the sufficient dimension reduction to high dimensional regimes by condensing the cross sectional information through factor models we derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices we further show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace our method and theory allow the number of predictors to be larger than the number of observations we finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables

a legendre pseudospectral method for computing travelling waves with corners slope discontinuities in one space dimension with application to whitham s equation family if the dispersion in a nonlinear hyperbolic wave equation is weak in the sense that the frequency k of cos kx is bounded as k it is common that i travelling waves exist up to a limiting amplitude with wave breaking for higher amplitudes and ii the limiting wave has a corner that is a discontinuity in slope because corner waves are not smooth standard numerical methods converge poorly as the number of grid points is increased however the corner wave is important because at least in some systems it is the attractor for all large amplitude initial conditions here we devise a legendre pseudospectral method which is uncorrupted by the singularity the symmetric u x u x wave can be computed on an interval spanning only half the spatial period since u is smooth on this domain which does not include the corner except as an endpoint all numerical difficulties are avoided a key step is to derive an extra boundary condition which uniquely identifies the corner wave with both the grid point values of u x and phase speed c as unknowns the discretized equations imposing three boundary conditions on a second order differential equation are solved by a newton raphson iteration although our method is illustrated by the so called whitham s equation ut uux du dx where d is a very general linear operator the ideas are widely applicable

on a sequence of higher order wave equations abstract we undertake a symmetry analysis of a sequence of evolution equations of orders n 3 4 5 6 which includes at lower orders partial differential equations having important applications for n 3 4 and 5 the equations in this sequence are the korteweg de vries equation the dissipative kuramoto sivashinsky equation and the so called korteweg de vries equation of fifth order we give a detailed discussion of both classical and nonclassical symmetries for this sequence of equations it is in this latter case using an approach based on the compatibility of the members of this sequence of equations with a first order differential equation and in the case where the infinitesimal 0 that we make our main new insights further results are also given for two of the obtained reductions of this sequence of evolution equations to ordinary differential equations in addition a generalization of the approach to reductions based on compatibility is also considered and is found to provide much promise for future work

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation