a proposal for a residual autocorrelation test in linear models summary this note proposes a test of goodness of fit for time series models based on the sum of the squared residual partial autocorrelations the test statistic is asymptotically x2 its small sample performance is studied through a monte carlo experiment it appears sensitive to erroneous specifications especially when the fitted model understates the order of the moving average component residual analysis is a fundamental step in building empirical time series models when checking the adequacy of the model one usually tests for the absence of residual autocorrelation there is a number of tests designed for this purpose both in the time and frequency domains see for instance quenouille 1947 1949 bartlett 1954 box pierce 1970 ljung box 1978 ansley newbold 1979 godfrey 1979 in this paper a test based on residual partial autocorrelations is proposed which is particularly powerful in cases when the fitted model underestimates the order of the moving average component let x be a zero mean process generated by a arma model o b x 0 b at where b is the backshift operator b is a polynomial of order p 0 b is a polynomial of order q and a is a white noise let a atn be the residuals obtained after estimating the model a very popular goodness of fit test proposed by box pierce 1970 and improved by ljung box 1978 is based on the statistic m

a note on the sum of the sample autocorrelation function abstract it is shown that the sum of the sample autocorrelation function at lag h 1 is always 1 2 for any stationary time series with arbitrary length t 2 hassani 2009 1 in this paper the distribution of a set of the sample autocorrelation function using the properties of this quantity is considered it is found that the distribution of a set of the sample autocorrelation estimates is not independent and identically distributed this finding implies that the result of diagnostic check and model building using the traditional assumption of iid can be quite misleading

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

the equilibrium of material forces in a 1d phase transition problem this paper aims at the use of material forces in estimating the interface location in an elastic one dimensional two phase problem a variational formulation based on variations of both the dependent and the independent variables results in the weak form of the momentum and the canonical momentum equations thus providing an appropriate framework for the use of the finite element method using this one can compute apart from the finite element solution to the equilibrium equation for the physical forces the corresponding one to the equilibrium equation for the material forces which provides the location of the interface in this way the total energy is minimized with respect to the deformations as well as to the interface location our theoretical expectations are confirmed by a numerical example concerning a one dimensional deformation induced phase transition problem with known analytical solution

automorphisms of borcherds superalgebras and fixed point subalgebras abstract let g be a borcherds superalgebra and let be a finite order automorphism of g satisfying certain conditions we prove that the fixed point subalgebra g is equal to a borcherds superalgebra we also compute the trace of on g from the twisted denominator identity of g and derive closed form formula for root multiplicities of g

dimension formula for graded lie algebras and its applications abstract in this paper we investigate the structure of infinite dimensional lie algebras l l l graded by a countable abelian semigroup satisfying a certain finiteness condition the euler poincare principle yields the denominator identities for the graded lie algebras from which we derive a dimension formula for the homogeneous subspaces l our dimension formula enables us to study the structure of the graded lie algebras in a unified way we will discuss some interesting applications of our dimension formula to the various classes of graded lie algebras such as free lie algebras kac moody algebras and generalized kac moody algebras we will also discuss the relation of graded lie algebras and the product identities for formal power series

modular invariant representations of infinite dimensional lie algebras and superalgebras abstract in this paper we launch a program to describe and classify modular invariant representations of infinite dimensional lie algebras and superalgebras we prove a character formula for a large class of highest weight representations l of a kac moody algebra unk with a symmetrizable cartan matrix generalizing the weyl kac character formula kac v g 1974 funct anal appl 8 68 70 in the case of an affine unk this class includes modular invariant representations of arbitrary rational level m t u where t unk z and u unk n are relatively prime and m g g u g is the dual coxeter number we write the characters of these representations in terms of theta functions and calculate their asymptotics generalizing the results of kac and peterson kac v g peterson d h 1984 adv math 53 125 264 and of kac and wakimoto kac v g wakimoto m 1988 adv math 70 156 234 for the u 1 integrable case we work out in detail the case unk a1 1 in particular classifying all its modular invariant representations furthermore we show that the modular invariant representations of the virasoro algebra vir are precisely the minimal series of belavin et al belavin a a polyakov a m zamolodchikov a b 1984 nucl phys b 241 333 380 using the character formulas of feigin and fuchs feigin b l fuchs d b 1984 lect notes math 1060 230 245 we show that tensoring the basic representation and modular invariant representations of a1 1 produces all modular invariant representations of vir generalizing the results of goddard et al goddard p kent a olive d 1986 commun math phys 103 105 119 and of kac and wakimoto kac v g wakimoto m 1986 lect notes phys 261 345 371 in the unitary case we study the general branching functions as well all these results are generalized to the kac moody superalgebras introduced by kac kac v g 1978 adv math 30 85 136 and to n 1 super virasoro algebras we work out in detail the case of the superalgebra b 0 1 1 showing in particular that restricting to its even part produces again all modular invariant representations of vir these results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations

domain evolution in ferroelectric materials a continuum phase field model and finite element implementation a numerical tool for simulating the polarization distribution in ferroelectric materials is presented using the concept of a phase field a continuum physics model is established which is descretized with finite elements the main feature of the numerical implementation is an implicit time integration of the non linear evolution equation representative examples and comparison to experimental measurements from the literature show the main features of the model

inference for autocorrelations under weak assumptions abstract in this article we consider the large sample behavior of estimates of autocorrelations and autoregressive moving average arma coefficients as well as their distributions under weak conditions specifically the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration such is the case when the time series follows an arma process with uncorrelated innovations that may not be assumed to be independent and identically distributed as a specific case it is well known that if the process is independent and identically distributed then the sample autocorrelation estimates scaled by the square root of the sample size are asymptotically standard normal this result is used extensively as a diagnostic check on the residuals of a fitted model or as an initial test on the observed time series to determine whether further model fitting is warranted in this article we show that this result can be quite

shape topology optimization of stokes flow via variational level set method in this paper the conventional level set method is extended as an efficient approach for shape optimization the motion of the dynamic interface is governed by stokes equations although this method is not specifically designed for topology optimization it can handle topology changes easily furthermore a relatively smooth evolution can be maintained without re initialization the cost of this method is moderate the accuracy and efficiency of the proposed algorithm are illustrated by several numerical examples

a powerful portmanteau test of lack of fit for time series a new portmanteau test for time series more powerful than the tests of ljung and box and monti is proposed the test is based on the mth root of the determinant of the mth autocorrelation matrix it is shown that the proposed statistic is a function of all of the squared multiple correlation coefficients of the regressions of the residuals on their lags when the number of lags goes from 1 to m it can also be written as a function of the first m partial autocorrelation coefficients the asymptotic distribution of the test statistic is a linear combination of chi squared distributions and can be approximated by a gamma distribution it is shown depending on the model and sample size that this test can be up to 50 more powerful than the ljung and box and monti tests the test is applied to the detection of several types of nonlinearity by using the autocorrelation matrix of the squared residuals and it is shown that in general the new test is more powerful than the test of mcleod and li an example

a level set approach to domain wall kinetics and domain patterning in elastic ferroelectrics the level set method is a powerful tool for tracking a moving interface in ferroelectric materials the boundaries between domains of uniform polarization referred to as domain walls are viewed as interfaces in this work a variational level set method is used to formulate a model for the behavior of elastic ferroelectrics and more particularly to describe the domain formation and the domain wall kinetics a regularized version of the sharp interface theory is introduced by substitution of the singular surfaces with thin transition layers inserting so inhomogeneities into the energy density function it is proved that the forces driving domain wall motion are in fact inhomogeneity forces with this description the level set method becomes an appropriate framework to insert and exploit material forces the proposed model may interpret typical microstructures in 2 d elastic ferroelectrics with tetragonal crystal symmetry comparisons with phase field models and experiments show good agreement

diagnostic testing of univariate time series models summary the portmanteau statistic for testing the adequacy of an autoregressive moving average model is based on the first m autocorrelations of the residuals from the fitted model this paper examines the properties of this test for various choices of m a modification which allows the use of small values of m is shown to result in a more powerful test the lagrange multiplier statistic godfrey 1979 and a test statistic examined by newbold 1980 are also discussed

new weighted portmanteau statistics for time series goodness of fit testing we exploit ideas from high dimensional data analysis to derive new portmanteau tests that are based on the trace of the square of the mth order autocorrelation matrix the resulting statistics are weighted sums of the squares of the sample autocorrelation coefficients that unlike many other tests appearing in the literature are numerically stable even when the number of lags considered is relatively close to the sample size the statistics behave asymptotically as a linear combination of chi squared random variables and their asymptotic distribution can be approximated by a gamma distribution the proposed tests are modified to check for nonlinearity and to check the adequacy of a fitted nonlinear model simulation evidence indicates that the proposed goodness of fit tests tend to have higher power than other tests appearing in the literature particularly in detecting long memory nonlinear models the efficacy of the proposed methods is demonstrated by investigating nonlinear effects in apple inc an

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

free lie algebras generalized witt formula and the denominator identity abstract let be a countable abelian semigroup satisfying a suitable finiteness condition and let l l be the free lie algebra generated by a graded vector space v over c in this paper from the denominator identity we derive a dimension formula for the homogeneous subspaces of the free lie algebra l l and discuss numerous applications of our dimension formula to various interesting cases our dimension formula will be expressed in terms of the witt partition functions various expressions of the witt partition functions will give rise to a number of interesting combinatorial identities as a special case we obtain a recursive relation for the coefficients of the elliptic modular function j

graded lie superalgebras and the superdimension formula in this paper we investigate the structure of graded lie superalgebras l a a l a where is a countable abelian semigroup and a is a countable abelian group with a coloring map satisfying a certain finiteness condition given a denominator identity for the graded lie superalgebra l we derive a superdimension formula for the homogeneous subspaces l a a a which enables us to study the structure of graded lie superalgebras in a unified way we discuss the applications of our superdimension formula to free lie superalgebras generalized kac moody superalgebras and monstrous lie superalgebras in particular the product identities for normalized formal power series are interpreted as the denominator identities for free lie superalgebras we also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of monstrous lie superalgebras

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

a generalization of borcherds algebra and denominator formula abstract we study generalized lie superalgebras an extension of kac s generalized lie superalgebras and show that they are transformed forms ofl graded lie superalgebras for some abelian groupl we then introduce a generalized lie superalgebra version of the generalized kac moody algebra borcherds algebra since it is a transformed borcherds superalgebra it has several properties similar to those of borcherds superalgebra for example it is defined by similar relations and has a similar denominator formula and character formulas

a character formula for generalized kac moody superalgebras abstract in 1 borcherds defined generalized kac moody denoted by gkm for short algebras and gave a character formula for lowest weight modules of these algebras in this paper we give a character formula for lowest weight modules of gkm superalgebras for any standard result about contragredient superalgebras that have an immediate generalization to gkm superalgebras see 3 and 4 all vector spaces are assumed to be over the real field

robustness of maximum likelihood estimates for multi step predictions the exponential smoothing case summary we extend the argument initiated by cox 1961 that the exponential smoothing formula can be made more robust for multi step forecasts if the smoothing parameter is adjusted as a function of the forecast horizon 1 the consistency property of the estimator which minimizes the sum of squares of the sample 1 step ahead forecast errors makes the robustness result useful in practice we also generalize the consistency result to some other parsimonious nonstationary models which have been popular in use the asymptotic distribution of the estimated smoothing parameter adjusted for forecast horizon l leads to the development of diagnostic tools which are based on i step forecasts

on covariance function tests used in system identification abstract tests on the autocovariance function of the residuals of an estimated model or on the cross covariance function between the model residuals and the past values of the input signal are frequently used in system identification for model validation purposes usually the threshold of such a test is derived under the assumption that the model and the identified system coincide the present paper shows that this simplifying assumption leads to a risk of overfitting less than expected and accordingly to a greater risk of underfitting the correct test threshold is problem dependent and cannot be determined easily in the general case however it can be shown to lie in a certain interval and to be equal to one of the bounds defining this interval under certain sufficient conditions the implication of these results for model validation based on covariance tests is discussed and illustrated by means of some numerical examples

borcherds superalgebras and a monstrous lie superalgebra this research was supported in part by nondirected research fund korea research foundation and garc kosef at seoul national university

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

graded lie algebras of kac moody type the first author is supported in pat by nsf under grant dms 88 00691 the 3rd author is supported in part by nsa msp grant mda 90 h 4039

skill luck and streaky play on the pga tour in this study we implement a random effects model that estimates cubic spline based time dependent skill functions for 253 active pga tour golfers over the 1998 2001 period our model controls for the first order autocorrelation of residual 18 hole scores and adjusts for round course and player course random effects using the model we are able to estimate time varying measures of skill and luck for each player in the sample estimates of spline based player specific skill functions provide strong evidence that the skill levels of pga tour players change through time using the model we are able to rank golfers at the end of 2001 on the basis of their estimated mean skill levels and identify the players who experienced the most improvement and deterioration in their skill levels over the 1998 2001 period we find that some luck is required to win pga tour events even for the most highly skilled players player course interactions contribute very little to variability in golfer performance but the par

on the finite sample distribution of modified portmanteau tests for randomness of a gaussian time series this paper proposes three modified portmanteau tests based on the applications of the fisher 1921 and hotelling 1953 transformations to sample autocorrelations the test statistics are asymptotically chi square our simulation results indicate that the empirical significance levels of the proposed tests are more reliable than the portmanteau tests of box pierce 1970 ljung box 1978 and dufour roy 1986 when the sample size is small 15 n 40 and the number of sample autocorrelations is large m 7

sum of squared acf and the ljung box statistics abstract the ljung box test is one of the most important tests for time series diagnostics and model selection the hassani s 1 2 theorem however indicates that the sum of sample autocorrelation function is always 1 2 for any stationary time series with arbitrary length in this paper hassani s 1 2 theorem is used to assess the sensitivity of the ljung box test results to the number of lags bitcoin price return data is also used to examine the applicability of the theoretical results obtained here the results confirm sensitivity of the ljung box test to the number of lags involved in the test and therefore it should be used with extra caution

spatial analysis of field experiments an extension to two dimensions the one dimensional spatial analysis procedure proposed by gleeson and cullis 1987 biometrics 43 277 288 is extended to two dimensions using the subclass of separable lattice processes to model the errors residual maximum likelihood estimation of the models is described and diagnostics for testing model adequacy are derived results from the analysis of 24 sets of uniformity data indicate the frequent need for a two dimensional analysis even when the plot shape is highly rectangular these results also indicate the potential gain from using a two dimensional spatial analysis rather than a row column analysis an example is presented of the analysis of a field experiment on tobacco

on the comparison of two approaches to compute material forces for inelastic materials application to single slip crystal plasticity abstract the main goal of this contribution is the application of the material force method to dissipative materials whereby special emphasis is placed on the application to crystal plasticity balance equations on the material manifold as e g the material motion balance of momentum generally necessitate the computation of the spatial gradient of additional internal degrees of freedom or rather internal variables we therefore focus in this work on the comparison of two different numerical approaches to compute these gradient fields within a finite element setting namely a node point based and an integration point based formulation several numerical examples underline that the latter framework which is the convenient numerically comparatively cheap and well established approach in computational inelasticity turns out to be sufficiently accurate for the problem at hand

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

the sample autocorrelation function and the detection of long memory processes the detection of long range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long memory or long range dependent process is based on the autocorrelation function it is not possible for long memory to be identified using the sum of the sample autocorrelations as usually defined the reason for this is that the sample sum is a predetermined constant for any stationary time series a result that is independent of the sample size diagnostic or estimation procedures such as those in the frequency domain that embed this sum are equally open to this criticism we develop this result in the context of long memory extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process the results are further extended to higher order sample autocorrelations and the bispectral density the corresponding result is that the sum of the third order sample auto bicorrelations at lags h k 1 is also a predetermined constant different from that in the second order case for any stationary time series of arbitrary length

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

generalized kac moody algebras and the modular function j the generalized kac moody algebras were introdcued by borcherds in his study of the vertex algebras and monstrous moonshine b1 b3 the structure and the representation theory of generalized kac moody algebras are very similar to those of kac moody algebras and a lot of facts about kac moody algebras can be generalized to generalized kac moody algebras the main difference is that the generalized kac moody algebras may have imaginm 3 simple roots with norms 1 for unitarizable irreducible highest weight modules over symmetrizable generalized kac moody algebras borcherds proved a character formula b2 called the weyl kac borcherds formula which generalizes the weylkac formula for integrable highest weight modules over symmetrizable kac moody algebras k as in the case of kac moody algebras the weyl kac borcherds formula when applied to the l dimensional trivial representation yields the denominator identio in this paper using the weyl kac borcherds formula and the denominator identity we obtain a closed form root multiplicity formula for all symmetrizable generalized kac moody algebras theorem 2 1 our formula enables us to study the structure of a symmetrizable generalized kac moody algebra 9 as a representation of a kac moody algebra 0o contained in the choice of 0 gives rise to various expressions of the root multiplicities of 13 which would yield some interesting combinatorial identities in sect 3 we apply this principle to a rank 2 generalized kac moody algebra to obtain a combinatorial identity for the pascal numbers in b3 borcherds proved the moonshine conjecture cni by constructing the monster lie algebra using the monster vertex algebra flm and the vertex algebra associated with the 2 dimensional even lorentzian lattice iil b 1 the monster lie

use and misuse of some hurst parameter estimators applied to stationary and non stationary financial time series abstract the detection of long range dependence lrd is an important task in time series analysis lrd is often summarized by the well known hurst parameter or exponent h 0 1 which can be estimated by a number of methods some of these techniques are designed to be applied to signals behaving as a stationary fractional gaussian noise fgn whereas others imply that the analyzed time series behave as a non stationary fractional brownian motion fbm moreover some estimators do not yield the hurst parameter but indexes related to h and ranging outside the unit interval therefore the fgn or fbm nature of the studied time series has to be preliminarily analyzed before applying any estimation method and the relationships between h and the indexes resulting from the analyses have to be taken into account to obtain coherent results since fgn like series represent the increments of fbm like processes and both the signals are characterized by the same h value by definition estimators designed for fgn like series can be applied to fbm like sequences after preventive differentiation and conversely estimators designed for fbm like processes can be applied to fgn like series after preventive integration the signal characterization is particularly important when h is estimated on financial time series because the returns represent the first difference of price time series which are often assumed to behave like self affine sequences the analysis of simulated fgn and fbm time series shows that all the considered methods yield comparable h values when properly applied the reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators supported by a preventive signal classification yields homogeneous h values allowing for a useful cross validation of results reported in different works moreover some conclusions reported in the literature about the anti persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods

selecting optimal lag order in ljung box test abstract theoretical results and empirical evidences indicate that the ljung box test is sensitive to the number of lags h involved in the test in time series literature different values are suggested for h this paper is concerned with the selecting optimal number of lags h in ljung box test such that the actual size of the test does not exceed the test s level while the power of the test does not fall under specific value a simulation study is employed to investigate the effect of selecting an improper values of h on the actual size and power of the ljung box test the results confirm that an optimal value of h depends on the time series length as well as the test s level the comparison results with currently used approaches in the literature confirm that the commonly used techniques are not suggesting a proper value for h