reducts of the random partial order abstract we determine up to the equivalence of first order interdefinability all structures which are first order definable in the random partial order it turns out that there are precisely five such structures we achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order and thus expose all symmetries of this structure our result lines up with previous similar classifications for the random graph and the order of the rationals it also provides further evidence for a conjecture due to simon thomas which states that the number of structures definable in a homogeneous structure in a finite relational language is up to first order interdefinability always finite in the proof we use the new technique of canonical functions originally invented in the context of theoretical computer science which allows for a systematic ramsey theoretic analysis of functions acting on the random partial order the technique identifies patterns in arbitrary functions on the random partial order which makes them accessible to finite combinatorial arguments

nonparametric inference of quantile curves for nonstationary time series the paper considers nonparametric specification tests of quantile curves for a general class of nonstationary processes using bahadur representation and gaussian approximation results for nonstationary time series simultaneous confidence bands and integrated squared difference tests are proposed to test various parametric forms of the quantile curves with asymptotically correct type i error rates a wild bootstrap procedure is implemented to alleviate the problem of slow convergence of the asymptotic results in particular our results can be used to test the trends of extremes of climate variables an important problem in understanding climate change our methodology is applied to the analysis of the maximum speed of tropical cyclone winds it was found that an inhomogeneous upward trend for cyclone wind speeds is pronounced at high quantile values however there is no trend in the mean lifetime maximum wind speed this example shows the effectiveness of the quantile regression technique

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

testing that a dependent process is uncorrelated an analysis is presented of a new testing procedure for the null hypothesis that a stochastic process is uncorrelated when the process is possibly dependent unlike with existing procedures the user does not need to choose any arbitrary number to implement the proposed test the asymptotic null distribution of the proposed test statistic is not standard but it is tabulated by means of simulations the test is compared with two alternative test procedures that require selection of user chosen numbers on the basis of asymptotic local power and finite sample behavior although the asymptotic local power of the proposed test is lower than those corresponding to the alternative tests in a monte carlo study i show that in small samples the test typically better controls the type i error and that the loss of power is not substantial

lie families theory and applications we analyze the families of non autonomous systems of first order ordinary differential equations admitting a common time dependent superposition rule i e a time dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants we next study the relations of these families called lie families with the theory of lie and quasi lie systems and apply our theory to provide common time dependent superposition rules for certain lie families

lie hamilton systems on the plane applications and superposition rules a lie hamilton lh system is a nonautonomous system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of hamiltonian vector fields with respect to a poisson structure we provide new algebraic geometric techniques to easily determine the properties of such lie algebras on the plane e g their associated poisson bivectors we study new and known lh systems on with physical biological and mathematical applications new results cover cayley klein riccati equations the here defined planar diffusion riccati systems complex bernoulli differential equations and projective schrodinger equations constants of motion for planar lh systems are explicitly obtained which in turn allow us to derive superposition rules through a coalgebra approach

unsupervised self normalized change point testing for time series abstractwe propose a new self normalized method for testing change points in the time series setting self normalization has been celebrated for its ability to avoid direct estimation of the nuisance asymptotic variance and its flexibility of being generalized to handle quantities other than the mean however it was developed and mainly studied for constructing confidence intervals for quantities associated with a stationary time series and its adaptation to change point testing can be nontrivial as direct implementation can lead to tests with nonmonotonic power compared with existing results on using self normalization in this direction the current article proposes a new self normalized change point test that does not require prespecifying the number of total change points and is thus unsupervised in addition we propose a new contrast based approach in generalizing self normalized statistics to handle quantities other than the mean which is specifically tailored for change point testing monte car

discrete matrix riccati equations with superposition formulas an ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution nonlinear ode s with superposition formulas include matrix riccati equations here we shall describe discretizations of riccati equations that preserve the superposition formulas the approach is general enough to include q derivatives and standard discrete derivatives

testing parametric assumptions of trends of a nonstationary time series the paper considers testing whether the mean trend of a nonstationary time series is of certain parametric forms a central limit theorem for the integrated squared error is derived and a hypothesis testing procedure is proposed the method is illustrated in a simulation study and is applied to assess the mean pattern of lifetime maximum wind speeds of global tropical cyclones from 1981 to 2006 we also revisit the trend pattern in the central england temperature series copyright 2011 oxford university press

quadratic pseudopotentials for gl n c principal sigma models abstract matrix pseudopotentials for the gl n c principal sigma model in two dimensional minkowski space are derived through the method of closed differential ideals the structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix riccati type a set of further constraints is deduced which leads to inner backlund transformations depending on two complex parameters through a geometrical interpretation of such matrix riccati equations and the associated constraints a linearization procedure is derived and the equations are reduced to those of zakharov mikhailov and shabat 7 10 an inductive procedure is applied to explicitly solve an iterated sequence of such backlund transformations through a purely algebraic transformation of solutions to the first step the resulting nonlinear superposition formula is used to demonstrate a permutability theorem and to analyse the appearance of singularities fr nous deduisons les pseudo potentiels matriciels pour le modele sigma principal a valeurs dans gl n c defini sur l espace de minkowski bi dimensionnel en utilisant la methode des ideaux differentiels fermes on restreint la structure des equations des pseudopotentiels de facon a ce que n apparaissent que des non linearites quadratiques du type riccati matriciel nous deduisons un ensemble de contraintes supplementaires menant a des transformations de backlund internes dependantes de deux parametres complexes gr ce a l interpretation geometrique de ces equations de riccati matricielles et des contraintes associees nous deduisons un procede de linearisation et les equations se reduisent a celles de zakharov mikhailov et shabat 7 10 un procede inductif est utilise pour resoudre explicitement une succession de transformations de backlund par une transformation purement algebrique des solutions de la premiere etape la formule de superposition non lineaire qui en resulte permet de demontrer un theoreme de permutabilite et d analyser l apparition de singularites

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

solutions generating technique for abel type nonlinear ordinary differential equations abstract a second order nonlinear differential equation describing the dynamics of a bulk viscous fluid filled general relativistic friedmann robertson walker universe is considered by means of appropriate transformations the evolution equation of the space time is transformed into a third order polynomial abel differential equation with the use of some particular solutions of this equation repeated classes of exact solutions are generated

superposition formulas for rectangular matrix riccati equations a system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every lie group subgroup pair g g0 we consider the case when g sl n k c and g0 p k is a maximal parabolic subgroup of g leaving a k dimensional vector space invariant 1 k n the nonlinear ordinary differential equations ode s in this case are rectangular matrix riccati equations for a matrix w t cn k the special case n rk n r k n is considered and a superposition formula is obtained expressing the general solution in terms of r 3 particular solutions for r 2 k 2 for r 1 square matrix riccati equations five solutions are needed for r n projective riccati equations the required number is n 2

periodic solutions of abel differential equations for a class of polynomial non autonomous differential equations of degree n we use phase plane analysis to show that each equation in this class has n periodic solutions the result implies that certain rigid two dimensional systems have at most one limit cycle which appears through multiple hopf bifurcation

narrowest over threshold detection of multiple change points and change point like features we propose a new generic and flexible methodology for non parametric function estimation in which we first estimate the number and locations of any features that may be present in the function and then estimate the function parametrically between each pair of neighbouring detected features examples of features handled by our methodology include change points in the piecewise constant signal model kinks in the piecewise linear signal model and other similar irregularities which we also refer to as generalized change points our methodology works with only minor modifications across a range of generalized change point scenarios and we achieve such a high degree of generality by proposing and using a new multiple generalized change point detection device termed narrowest over threshold not detection the key ingredient of the not method is its focus on the smallest local sections of the data on which the existence of a feature is suspected for selected scenarios we show the consistency and near optimality of the not algorithm in detecting the number and locations of generalized change points the not estimators are easy to implement and rapid to compute importantly the not approach is easy to extend by the user to tailor to their own needs our methodology is implemented in the r package not

nonlinear equations with superposition formulas and the exceptional group g2 i complex and real forms of g2 and their maximal subalgebras in order to study nonlinear ordinary differential equations with superposition principles related to the exceptional simple lie group g2 the complex and real forms of its lie algebra are examined and their maximal subalgebras are summarized in particular the parabolic subalgebras of the noncompact real form gnc2 r are determined explicit matrix realizations of the fundamental representation d 1 0 are used and studied in connection with invariant subspaces in a seven dimensional complex or real vector space the results are collected in three tables of specific interest for the study of nonlinear differential equations which will be developed in paper ii of this series

new aspects of integrability of force free duffing van der pol oscillator and related nonlinear systems in this paper we show that the force free duffing van der pol oscillator is completely integrable for a specific parametric choice we derive a general solution for this parametric choice further we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature the procedure is shown to have a wider applicability through additional examples we also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example

mixed superposition rules and the riccati hierarchy abstract mixed superposition rules i e functions describing the general solution of a system of first order differential equations in terms of a generic family of particular solutions of first order systems and some constants are studied the main achievement is a generalization of the celebrated lie scheffers theorem characterizing systems admitting a mixed superposition rule this somehow unexpected result says that such systems are exactly lie systems i e they admit a standard superposition rule this provides a new and powerful tool for finding lie systems which is applied here to studying the riccati hierarchy and to retrieving some known results in a more efficient and simpler way

structural break estimation for nonstationary time series models this article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive ar processes the number and locations of the piecewise ar segments as well as the orders of the respective ar processes are assumed unknown the minimum description length principle is applied to compare various segmented ar fits to the data the goal is to find the best combination of the number of segments the lengths of the segments and the orders of the piecewise ar processes such a best combination is implicitly defined as the optimizer of an objective function and a genetic algorithm is implemented to solve this difficult optimization problem numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties the segmentation of multivariate time series is also considered assuming that the true underlying model is a segmented autoregression this procedure is shown to be consistent for estimating the location of

quasi lie families schemes invariants and their applications to abel equations abstract this work analyses types of group actions on families of t dependent vector fields of a particular class the hereby called quasi lie families we devise methods to obtain the defined here quasi lie invariants namely a kind of functions constant along the orbits of the above mentioned actions our techniques lead to a deep geometrical understanding of quasi lie schemes and quasi lie systems giving rise to several new results our achievements are illustrated by studying abel and riccati equations we retrieve the liouville invariant and study other new quasi lie invariants of abel equations several abel equations with a superposition rule are described and we characterise abel equations via quasi lie schemes

estimating and testing linear models with multiple structural changes this paper develops the statistical theory for testing and estimating multiple change points in regression models the rate of convergence and limiting distribution for the estimated parameters are obtained several test statistics are proposed to determine the existence as well as the number of change points a partial structural change model is considered the authors study both fixed and shrinking magnitudes of shifts in addition the models allow for serially correlated disturbances mixingales an estimation strategy for which the location of the breaks need not be simultaneously determined is discussed instead the authors method successively estimates each break point

the dependent wild bootstrap we propose a new resampling procedure the dependent wild bootstrap for stationary time series as a natural extension of the traditional wild bootstrap to time series setting the dependent wild bootstrap offers a viable alternative to the existing block based bootstrap methods whose properties have been extensively studied over the last two decades unlike all of the block based bootstrap methods the dependent wild bootstrap can be easily extended to irregularly spaced time series with no implementational difficulty furthermore it preserves the favorable bias and mean squared error property of the tapered block bootstrap which is the state of the art block based method in terms of asymptotic accuracy of variance estimation and distribution approximation the consistency of the dependent wild bootstrap in distribution approximation is established under the framework of the smooth function model in addition we obtain the bias and variance expansions of the dependent wild bootstrap variance estimat

heteroscedasticity and autocorrelation robust structural change detection the assumption of weak stationarity is crucial for the validity of most of the conventional tests of structure change in time series under complicated nonstationary temporal dynamics we argue that traditional testing procedures result in mixed structural change signals of the first and second order and hence could lead to biased testing results the article proposes a simple and unified bootstrap testing procedure that provides consistent testing results under general forms of smooth and abrupt changes in the temporal dynamics of the time series monte carlo experiments are performed to compare our testing procedure with various traditional tests our robust bootstrap test is applied to testing changes in an environmental and a financial time series and our procedure is shown to provide more reliable results than the conventional tests

quasi lie schemes and emden fowler equations the recently developed theory of quasi lie schemes is studied and applied to investigate several equations of emden type and a scheme to deal with them and some of their generalizations is given as a first result we obtain t dependent constants of the motion for particular instances of emden equations by means of some of their particular solutions previously known results are recovered from this new perspective some t dependent constants of the motion for equations of emden type satisfying certain conditions are recovered finally new exact particular solutions are given for certain cases of emden equations

sources of nonmonotonic power when testing for a shift in mean of a dynamic time series sources of nonmonotonic power are uncovered for a wide variety of tests for a shift in the mean of a dynamic time series two main sources of nonmonotonic power are found the first source is the behavior of the estimate of error variance under the alternative hypothesis of a shift in mean in particular if the error variance is estimated under the null hypothesis nonmonotonic power can result the second source is the presence of a lagged dependent variable in the estimated regression

inference for linear models with dependent errors summary the paper is concerned with inference for linear models with fixed regressors and weakly dependent stationary time series errors theoretically we obtain asymptotic normality for the m estimator of the regression parameter under mild conditions and establish a uniform bahadur representation for recursive m estimators methodologically we extend the recently proposed self normalized approach of shao from stationary time series to the regression set up where the sequence of response variables is typically non stationary in mean since the limiting distribution of the self normalized statistic depends on the design matrix and its corresponding critical values are case dependent we develop a simulation based approach to approximate the critical values consistently through a simulation study we demonstrate favourable finite sample performance of our method in comparison with a block bootstrap based approach empirical illustrations using two real data sets are also provided

time dependent invariant associated to nonlinear schrodinger langevin equations via the quantum hydrodynamical method a time dependent invariant associated to the quantum dissipative time dependent harmonic oscillator tdho described by two classes of nonlinear schrodinger langevin equations with the following frictional nonlinear terms is constructed i w1 i ln ln which is the schuch chung hartman frictional nonlinear term and ii w2 x x cp 1 c p 1 2 i c which includes the sussmann c 1 the hasse c 1 2 and the albrecht kostin c 0 frictional nonlinear operators the associated invariant found is exact for the schuch chung hartman and hasse models and only approximate for the sussman and albrecht kostin models

schaefer s theorem for graphs schaefer s theorem is a complexity classification result for so called boolean constraint satisfaction problems it states that every boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time or is np complete we present an analog of this dichotomy result for the propositional logic of graphs instead of boolean logic in this generalization of schaefer s result the input consists of a set w of variables and a conjunction of statements constraints about these variables in the language of graphs where each statement is taken from a fixed finite set of allowed quantifier free first order formulas the question is whether is satisfiable in a graph we prove that either is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time or the problem is np complete this is achieved by a universal algebraic approach which in turn allows us to use structural ramsey theory to apply the universal algebraic approach we formulate the computational problems under consideration as constraint satisfaction problems csps whose templates are first order definable in the countably infinite random graph our method for classifying the computational complexity of those csps is based on a ramsey theoretic analysis of functions acting on the random graph and we develop general tools suitable for such an analysis which are of independent mathematical interest

a likelihood ratio approach to sequential change point detection for a general class of parameters in this article we propose a new approach for sequential monitoring of a general class of parameters of a d dimensional time series which can be estimated by approximately linear functionals of t

on sample reuse methods for dependent data we suggest a sample reuse method for dependent data based on a cross between the block bootstrap and richardson extrapolation instead of simulating a same size resample by resampling blocks and placing them end to end it analyses the blocks directly and employs a variant of richardson extrapolation to adjust for block size a simple empirical rule also based on richardson extrapolation is suggested for empirically selecting the block size performance in the contexts of distribution and bias estimation is discussed via theoretical analysis and numerical simulation

simple subgroups of simple lie groups and nonlinear differential equations with superposition principles nonlinear ordinary differential equations admitting a superposition principle based on the action of the group sl n c on the homogeneous spaces sl n c o n c and sl n c sp n c are derived the superposition formulas are presented explicitly in the o n c case the general solution is expressed in terms of three particular solutions for any n for the sp n c case three solutions are needed for n 2k 8 four solutions for n 6 and five solutions for n 4

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions

superposition formulas for pseudounitary matrix riccati equations the purpose of this article is to derive a superposition formula for the pseudounitary matrix riccati equation of dimension n 2 the superposition formula will be written in closed form in terms of five particular solutions satisfying certain well specified conditions defining a fundamental set examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations

a structural break test for extremal dependence in β mixing random vectors summarywe derive a structural break test for extremal dependence in beta mixing possibly high dimensional random vectors with either asymptotically dependent or asymptotically independent components existing tests require serially independent observations with asymptotically dependent components to avoid estimating a long run variance we use self normalization which obviates the need to estimate the coefficient of tail dependence when components are asymptotically independent simulations show favourable empirical size and power of the test which we apply to s p 500 and dax log returns we find evidence for one break in the coefficient of tail dependence for the upper and lower joint tail at the beginning of the 2007 08 financial crisis leading to more extremal co movement