identification of arx models subject to missing data parameter estimation when the measurement information may be incomplete is discussed an arx model is used as a basic system representation the presentation covers both missing output and missing input first reconstruction of the missing values is discussed the reconstruction is based on a state space formulation of the system and is performed using kalman filtering or fixed interval smoothing formulas several approaches to the identification problem are presented including a new method based on the em expectation maximization algorithm the different approaches are tested and compared using monte carlo simulations the choice of method is always a tradeoff between estimation accuracy and computational complexity according to the simulations the gain in accuracy using the em method can be considerable if many data are missing

analytic derivatives for estimation of linear dynamic models abstract this paper develops two algorithms algorithm 1 computes the exact gaussian log likelihood function its exact gradient vector and an asymptotic approximation of its hessian matrix for discrete time linear dynamic models in state space form algorithm 2 derived from algorithm 1 computes the exact sample information matrix of this likelihood function the computed quantities are analytic not numerical approximations and should therefore be useful for reliably quickly and accurately i checking local identifiability of parameters by checking the rank of the information matrix ii using the gradient vector and hessian matrix to compute maximum likelihood estimates of parameters with newton methods and iii computing asymptotic covariances cramer rao bounds of the parameter estimates with the hessian or the information matrix the principal contribution of the paper is algorithm 2 which extends to multivariate models the univariate results of porat and friedlander 1 by relying on the kalman filter instead of the levinson durbin filter used by porat and friedlander algorithms 1 and 2 can automatically handle any pattern of missing or linearly aggregated data although algorithm 1 is well known it is treated in detail in order to make the paper self contained

krein adler transformations for shape invariant potentials and pseudo virtual states for 11 examples of one dimensional quantum mechanics with shape invariant potentials the darboux crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to krein adler transformations deleting multiple eigenstates with shifted parameters these are based upon infinitely many polynomial wronskian identities of classical orthogonal polynomials i e the hermite laguerre and jacobi polynomials which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape invariant potentials

rational extensions of the trigonometric darboux poschl teller potential based on para jacobi polynomials the possibility for the jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by calogero and yi who termed them para jacobi polynomials such polynomials are used here to build seed functions of a darboux backlund transformation for the trigonometric darboux poschl teller potential as a result one step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter are constructed for each n value the eigenstates of these extended potentials are associated with a novel family of dependent polynomials which are orthogonal on 1 1

on outlier detection in time series the estimation and detection of outliers in a time series generated by a gaussian autoregressive moving average process is considered it is shown that the estimation of additive outliers is directly related to the estimation of missing or deleted observations a recursive procedure for computing the estimates is given likelihood ratio and score criteria for detecting additive outliers are examined and are shown to be closely related to the leave k out diagnostics studied by bruce and martin the procedures are contrasted with those appropriate for innovational outliers

estimation prediction and interpolation for nonstationary series with the kalman filter abstract we show how our definition of the likelihood of an autoregressive integrated moving average arima model with missing observations alternative to that of kohn and ansley and based on the usual assumptions made in estimation of and forecasting with arima models permits a direct and standard state space representation of the nonstationary original data so that the ordinary kalman filter and fixed point smoother can be efficiently used for estimation forecasting and interpolation in this way the problem of estimating missing values in nonstationary series is considerably simplified the results are extended to regression models with arima errors and a computer program is available from the authors

conjugacy classes of elements in the borovik group the borovik groupbis a lie primitive finite subgroup ofe8 c which containsalt5 sym6with index 2 and is not a direct product we determine its elements as members ofe8 c conjugacy classes and discuss itsalt5subgroups

virasoro frames and their stabilizers for the e8 lattice type vertex operator algebra the concept of a framed vertex operator algebra fvoa is new cf dgh this article contributes to this theory with a full analysis of all virasoro frame stabilizers in v the important example of the e8 level 1 affine kac moody voa which is isomorphic to the lattice voa for the root lattice of e8 c we analyze the frame stabilizers both as abstract groups and as subgroups of aut v e8 c each frame stabilizer is a finite group contained in the normalizer of a 2b pure elementary abelian 2 group in aut v but is not usually a maximal finite subgroup of this normalizer in particular we prove that there are exactly five orbits for the action of aut v on the set of virasoro frames thus settling an open question about v in section 5 of dgh the results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories so this article contributes also to this theory there are two main viewpoints in our analysis the first is the theory of codes lattices markings and the resulting groups of automorphisms the second is the theory of finite subgroups of lie groups we expect our methods to be applicable to the study of other fvoas and their frame stabilizers appendices present aspects of the theory of automorphism groups of voas in particular there is a general result of independent interest on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness

a rank 3 simple group g of order 21433537 13 29 ii characters of g and ĝ abstract proper coverings of finite especially simple groups are discussed in general with emphasis on determining the conjugacy classes of elements of the covering group this is used to determine the conjugacy classes of elements of the 2 fold proper covering group g of the group g of the title during this determination it is proved that g must have a pair of complex conjugate characters of degree 28 with values in z i the ring of gaussian integers a brief general discussion is given of schur modules the irreducible components of the tensor powers of the natural module for the general linear group and their characters this is then applied to the above pair of 28 dimensional characters of g to obtain many irreducible characters of g and g the remaining irreducible characters of g and g are then obtained from certain induced characters

equivalences between fusion systems of finite groups of lie type when g is a finite group and p is a prime the fusion system fp g is the category whose objects are the p subgroups of g and whose morphisms are the homomorphisms between subgroups induced by conjugation in g if g is another finite group then fp g and fp g are isotypically equivalent if there is an equivalence of categories between them which commutes up to natural isomorphism of functors with the forgetful functors from fp to the category of groups alternatively fp g and fp g are isotypically equivalent if there is an isomorphism between sylow p subgroups of g and of g which is fusion preserving in the sense of definition 1 2 below the goal of this paper is to use methods from homotopy theory to prove that certain pairs of fusion systems of finite groups of lie type are isotypically equivalent our main result is the following theorem

filtering and smoothing algorithms for state space models abstract the paper reviews and generalizes recent filtering and smoothing algorithms for observations generated by a state model in particular the paper discusses the modified kalman filter derived by ansley and kohn 1985 and kohn and ansley 1986 to deal with state space models having partially diffuse initial conditions and shows how to compute the limiting normalized likelihood of the observations for such models the paper also discusses and generalizes the new smoothing algorithm presented by kohn and ansley 1987c 1989 and extends it to state space models with partially diffuse initial conditions

a rank 3 simple group of order 21433537 13 29 i abstract after a preliminary discussion of rank 3 permutation groups attention is focused on the case where the stabilizer of a point is the ree group f 2 f 4 2 and the subdegrees are 1755 and 2304 if there is a rank 3 permutation group with f as point stabilizer and the above subdegrees then it is shown to be a simple group g of order 2 14 3 3 5 3 7 13 29 and the conjugacy classes of elements of g are determined it is shown that such a group g has a proper covering group g with center of order two and also that g has no outer automorphisms

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

on linear equations with general polynomial solutions abstract we provide necessary and sufficient conditions for which an nth order linear differential equation has a general polynomial solution we also give necessary conditions that can directly be ascertained from the coefficient functions of the equation

missing observations in arima models skipping approach versus additive outlier approach optimal estimation of missing values in arma models is typically performed by using the kalman filter for likelihood evaluation skipping in the computations the missing observations obtaining the maximum likelihood ml estimators of the model parameters and using some smoothing algorithm the same type of procedure has been extended to nonstationary arima models in go mez and maravall 1994 an alternative procedure suggests filling in the holes in the series with arbitrary values and then performing ml estimation of the arima model with additive outliers ao when the model parameters are not known the two methods di er since the ao likelihood is a ected by the arbitrary values we develop the proper likelihood for the ao approach in the general non stationary case and show the equivalence of this and the skipping method finally the two methods are compared through simulation and their relative advantages assessed the comparison also includes the ao method with the uncorrected likelihood 1999 elsevier science s a all rights reserved

the exact likelihood of an autoregressive moving average model with incomplete data summary a transformation is developed which allows computationally efficient evaluation of the exact likelihood of an autoregressive moving average model with incomplete data timings are presented which indicate that the new method is preferable to the kalman filter in many instances extensions to nonstationary and vector models are discussed

a finite subgroup of the exceptional lie group g2 with a view to further refining the use of the exceptional group g2 in atomic and nuclear spectroscopy it is confirmed that a simple finite subgroup l168 psl2 7 of order 168 of the symmetric group s8 is also a subgroup of g2 it is established by character theoretic and other methods that there are two distinct embeddings of l168 in g2 analogous to the two distinct embeddings of so 3 in g2 relevant branching rules tensor products and symmetrized tensor products are tabulated as a stimulus to further applications the branching rules are given for the restriction from l168 to the octahedral crystallographic point group o

a computationally efficient method for vector autoregression with mixed frequency data abstract a linear transformation method is proposed to handle the vector autoregression with mixed frequency time series data temporally aggregated observations impose linear constraints on the distribution of latent variables which are converted such that each observation replaces a latent variable full sample transformation yields a closed form simulation smoother while partial sample transformation leads to a computationally efficient sampler suitable for parallel computing

efficient generalized cross validation for state space models summary for a signal plus noise model with a state space representation an efficient procedure is given for obtaining the trace of the influence matrix where the influence matrix expresses the estimated signal vector as a linear combination of the observed data this allows an 0 n evaluation of the generalized cross validation criterion function our approach is very efficient requiring the addition of only one equation to the ordinary kalman filter and extends to models with linear regressors in the observation equation important applications are to spline smoothing in regression and to variance components models for seasonal time series

estimation of missing values in possibly partially nonstationary vector time series summary ljung s 1989 method for estimating missing values and evaluating the corresponding likelihood function in scalar time series is extended to the vector case the series is assumed to be generated by a possibly partially nonstationary and noninvertible vector autoregressive moving average process no particular pattern of missing data is assumed future and past values are special cases of missing data that can be estimated in the same way the method does not use kalman filter iterations and hence avoids initialisation problems it does not require the series to be differenced and thus avoids complications caused by over differencing the estimators of the missing data are provided by the normal equations of an appropriate regression problem these equations are adapted to cope with temporally aggregated data the procedure parallels a matrix treatment of contour conditions in the analysis of variance autoregressive processes are considered in some detail

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

on conjugacy classes of maximal subgroups of finite simple groups and a related zeta function we prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of lie type of bounded rank is bounded for exceptional groups this solves a longstanding open problem the proof uses among other tools some methods from geometric invariant theory using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of lie type groups of fixed rank drawing conclusions regarding the behaviour of the corresponding zeta function g s m maxg g m s which appears in many probabilistic applications more specifically we are able to show that for simple groups g and for any fixed real number s 1 g s 0 as g this confirms a conjecture made in 27 we also apply these results to prove the conjecture made in 29 that the symmetric group sn has n o 1 conjugacy classes of primitive maximal subgroups 2000 mathematics classification numbers 20e28 20g15 20d06

a fast algorithm for signal extraction influence and cross validation in state space models summary a fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model the resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector the ideas of studentized residuals and leverage from regression analysis are generalized to state space models and the new algorithm is used to compute the various measures the results are also applied to obtain a new efficient algorithm for polynomial spline smoothing suppose observations are generated by a gaussian signal plus noise process with the signal described by a state space model such a model for the signal occurs often in practice for example when the signal is the output of a stochastic difference or differential equation this paper presents a new algorithm for signal extraction that is the computa tion of the conditional mean and variance of the signal given the observations the usual method for signal extraction is to use the kalman filter see for example anderson moore 1979 p 105 followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm anderson moore 1979 p 187 our approach also uses the kalman filter but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm as for the fixed interval smoothing algorithm recursions for estimating the signal and its variance are carried out separately so that considerable additional savings can be made if the signal estimate alone is required the development uses ideas introduced by ansley kohn 1987a a remarkable property of the new recursions is that they can be applied immediately

a solvable many body problem its equilibria and a second order ordinary differential equation whose general solution is polynomial some properties of a solvable n body problem featuring several free parameters coupling constants are investigated restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data isochrony the restrictions on its parameters which guarantee the existence of equilibria are also identified in this connection a remarkable second order ode generally not of hypergeometric type hence not reducible to those characterizing the classical polynomials is studied if its parameters satisfy a diophantine condition its general solution is a polynomial of degree n the n zeros of which identify the equilibria of the n body system

estimation prediction and interpolation for arima models with missing data abstract we show how to define and then compute efficiently the marginal likelihood of an arima model with missing observations the computation is carried out by using the univariate version of the modified kalman filter introduced by ansley and kohn 1985a which allows a partially diffuse initial state vector we also show how to predict and interpolate missing observations and obtain the mean squared error of the estimate

smoothing and interpolation with the state space model abstract a new result dealing with smoothing and interpolation in the state space model is developed and explored the result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages the connection to signal extraction and interpolation is explored and diffuse specifications are considered

recursive solution methods for dynamic linear rational expectations models abstract this paper develops recursive solution methods for linear rational expectations models the underlying structural model is transformed into a state space representation which can then be used to solve the model and to form the gaussian likelihood function the recursive solution method has several advantages over other approaches first the set of solutions to the model are summarized by a set of parameters that appear in the state space representation but are unspecified by the structural model next the likelihood function is formed as a byproduct of the solution to the model finally modifications in the likelihood function necessary to incorporate complications arising from temporal aggregation dynamic errors in variables etc are straightforward

signal extraction for finite nonstationary time series summary for a signal plus noise model where both signal and noise are generated by nonstation ary arima autoregressive integrated moving average models we use the transformation approach of ansley kohn 1985 to construct an estimate of the signal and obtain its mean squared error given a finite number of observations a method for efficient computa tion of the signal estimate and its mean squared error is also presented consider observations z t generated by the signal plus noise nmodel z t s t n t 1s1 where both signal s t and noise n t follow nonstationary arima processes we give a solution to the finite sample signal extraction problem by defining an estimate of the signal s t given observations z 1 z n and finding its mean squared error the major difficulty in handling the model with nonstationary components is that the distribution of the observations is not defined unless assumptions are made about the initial values of the series moreover it is usually very difficult to formulate reasonable assumptions about the initial values because they affect the correlation structure of the whole series see for example the discussion of bell 1984 p 652 kohn ansley 1986 use a transformation approach to overcome the initial value problem in defining the likelihood and obtaining predittors and interpolators for an arima model which may have missing values this approach does not require any assumptions about the initial values and generalizes the usual differencing approach of box jenkins 1976 for the case where there are no missing observations in the present paper we apply the transformation approach of kohn ansley 1986 and ansley kohn 1985 to solve the signal extraction problem we define a linear estimator a t of s t so that s t sa t is invariant to the initial values and furthermore is unbiased in the sense that e s t s t 0 moreover it has minimum mean squared error amongst all such estimators our approach allows missing observations and extends to signal extraction for several additive signals when each is an arima process we also show how to compute the signal estimate and its mean squared error efficiently by writing the model in state space form placing a diffuse prior on the initial values of the process and then using the modified kalman filter and the modified smoothing algorithm of ansley kohn 1985 and kohn ansley 1986 gersch kitagawa 1983 kitagawa gersch 1984 and harvey todd 1983 use

diagnostic checking in arma models with uncorrelated errors we consider tests for lack of fit in arma models with nonindependent innovations in this framework the standard box pierce and ljung box portmanteau tests can perform poorly specifically the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration in this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of arma models under weak assumptions on the noise the asymptotic distribution of the portmanteau statistics follows a consistent estimator of and a modification of the portmanteau tests are proposed this allows us to construct valid asymptotic significance limits for the residual autocorrelations and asymptotically valid goodness of fit tests when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference a set of monte carlo experiments and an application to the standard poor 500 returns illustrate the p

prediction mean squared error for state space models with estimated parameters summary we obtain a conditional prediction mean squared error for a state space model with estimated parameters an important application of our results is the derivation of conditional forecast and interpolation mean squared errors for autoregressive moving average models with estimated parameters we also obtain the conditional mean squared error for filtered and smoothed estimates of the state vector of z t and s t i n 0 var z t z t i n 0 is the mean squared error both z t i n 0 and s t i n 0 can be computed efficiently using the kalman filter and the smoothing algorithms in chapters 3 and 7 respectively of anderson moore 1979 if 0 is unknown then we can estimate it by maximum likelihood to obtain 0 say and a a then use z t i n 0 and s t in 0 in place of z t i n 0 and s t i n 0 however the estimate s t i n 0 of mean squared error underestimates the true mean squared error because it does not take into account the extra variation due to estimating the parameters one important example of this problem is the adjustment of the forecast mean squared error of an arma process to allow for the variability due to parameter estimation with the exception of phillips 1979 and fuller hasza 1981 whose work is confined to the autoregressive model results in the literature on arma and armax models with estimated parameters have dealt with the unconditional forecast mean squared error see for example bloomfield 1972 yamamoto 1976 1981 and baillie 1979 1980 their unconditional forecast mean squared error is an average over all realizations of y whereas the actual prediction error depends on the particular realization y we observe this will be made clear for the first order autoregressive model in 2 therefore the

exact initial kalman filtering and smoothing for nonstationary time series models abstract this article presents a new exact solution for the initialization of the kalman filter for state space models with diffuse initial conditions for example the regression model with stochastic trend seasonal and other nonstationary autoregressive integrated moving average components requires a partially diffuse initial state vector the proposed analytical solution is easy to implement and computationally efficient the exact solution for smoothing is also given missing observations are handled in a straightforward manner all proofs rely on elementary results

a linear second order ode with only polynomial solutions abstract a linear second order ode featuring an arbitrary number of free parameters is identified all solutions of which are polynomials

estimation filtering and smoothing in state space models with incompletely specified initial conditions on definit la vraisemblance pour un modele d espace d etats avec des conditions initiales incompletement specifiees par transformation des donnees pour eliminer la dependance sur des conditions non specifiees on etend cette approche pour obtenir des estimations sur les vecteurs d etat les predicteurs et les interpolateurs pour des observations manquantes

rigid local systems on gm with finite monodromy abstract we construct certain rigid local systems on g m over finite fields using the theory of deligne lusztig curves and some recent results on finite simple subgroups of complex lie groups

can the general solution of the second order ode characterizing jacobi polynomials be polynomial the answer to the question posed in the title is affirmative indeed while the second order ode 1 x2 p 2 x p n n 1 p 0 for generic values of the two parameters and and n a non negative integer identifies uniquely up to an overall multiplicative constant characterizing its normalization the jacobi polynomial p n x as its polynomial solution if instead n m and n n with m and n being two non negative integers satisfying the restriction m n n then the general solution of this ode is a polynomial of degree n depending in addition of course to the three parameters n m and n on two arbitrary parameters i e on an additional one besides the overall multiplicative parameter characterizing its normalization

diophantine properties of the zeros of certain laguerre and para jacobi polynomials matrices and of arbitrary rank n given by simple expressions in terms of the n zeros of certain laguerre or para jacobi polynomials of degree n feature a diophantine property in the laguerre case this property states that the 2n zeros of the polynomial are all integers indeed we conjecture that the results in the para jacobi case are somewhat analogous they refer to the zeros of the general solution of the ode generally characterizing jacobi polynomials in the special case in which this solution is a para jacobi polynomial featuring an additional arbitrary parameter

bayesian estimation of an autoregressive model using markov chain monte carlo abstract we present a complete bayesian treatment of autoregressive model estimation incorporating choice of autoregressive order enforcement of stationarity treatment of outliers and allowance for missing values and multiplicative seasonality the paper makes three distinct contributions first we enforce the stationarity conditions using a very efficient metropolis within gibbs algorithm to generate the partial autocorrelations second we show how to carry out the gibbs sampler when the autoregressive order is unknown third we show how to combine the various aspects of fitting an autoregressive model giving a more comprehensive and efficient treatment than previous work we illustrate our methodology with a real example

on the bias in estimates of forecast mean squared error abstract we examine the forecasting problem for a finite series of observations from either a nonseasonal or a seasonal autoregressive moving average process four sources of bias in the usual estimator of forecast mean squared error are identified and analyzed for particular models using maximum likelihood and least squares parameter estimates the usual estimator is found to be biased downwards especially near the boundary of a stationary or invertible region and the bias is severe for least squares estimators an alternative estimator is proposed for the maximum likelihood case which is shown generally to have reduced bias

fast evaluation of the distribution of the durbin watson and other invariant test statistics in time series regression abstract a method is given for evaluating p values in o n operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals the best known of these is the durbin watson statistic although several others have been discussed in the literature the method is numerically exact in the sense that the user specifies the error tolerance at the outset as with existing exact methods the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals which is evaluated by numerically inverting its characteristic function existing methods however calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form each of which requires o n 3 operations for sample size n whereas the new method uses a modification of the kalman filter to do it in o n operations moreover the new method has minimal storage requiremen

efficient estimation and prediction in time series regression models summary for a time series regression model with correlated residuals we show how to compute efficiently the likelihood predict future values and interpolate missing values as an illustration we consider a regression model with autoregressive moving average residuals

estimating missing observations in economic time series abstract two related problems are considered the first concerns the maximum likelihood estimation of the parameters in an arima model when some of the observations are missing or subject to temporal aggregation the second concerns the estimation of the missing observations both problems can be solved by setting up the model in state space form and applying the kalman filter

modeling of time series arrays by multistep prediction or likelihood methods abstract an estimation theory is provided for the fitting of possibly incorrect invertible short memory models to short or long memory time series or time series arrays by multistep prediction error minimization or gaussian likelihood maximization by array we mean data y t t 1 t t that depend on the number of observations t such as regression or other estimated model residuals or the outputs of time varying filters for example seasonal adjustments our theory only requires the modeled array to have basic properties for a s i p convergence of parameter estimates the array s sample lagged second moments must converge a s i p and its end values y 1 j t and y t j t must be of order less than t 1 2 or an appropriately differenced version of the observed array must have these properties in findley et al ann statist 29 2001 815 broad classes of arrays were shown to have these properties even for the special case of autoregressive moving average models fit to stationary gaussian time series data our result on the convergence of parameter estimates minimizing p step ahead prediction error sums of squares is new

leave k out diagnostics for time series abstract the authors propose diagnostics for arima model fitting for time series formed by deleting observations from the data and measuring the change in the estimates of the parameters the use of leave one out diagnostics is a well established tool in regression analysis this document demonstrates the efficacy of observation deletion based diagnostics for arima models addressing issues special to the time diagnostics based on the innovations variance it is shown that the dependency aspect of time series data gives rise to a smearing effect which confounds the diagnostics for the coefficients it is also shown that the diagnostics based on the innovations variance are much clearer and more sensitive than those for the coefficients a leave k out diagnostics approach is proposed to deal with patches of outliers and problems caused by masking are handled by use of iterative deletion an overall strategy for arima model fitting is given and applied to two data sets

a moonshine path for 5a and associated lattices of ranks 8 and 16 abstract we continue the program begun in griess and lam in press 18 to make a moonshine path between a node of the extended e 8 diagram and the monster our goal is to provide a context for observations of mckay glauberman and norton by realizing their theories in a more concrete form in this article we treat the 5a node most work in this article is a study of certain lattices of ranks 8 16 and 24 whose determinants are powers of 5

measuring influence in dynamic regression models this article presents a methodology for building measures of influence in regression models with time series data we introduce statistics that measure the influence of each observation on the parameter estimates and on the forecasts these statistics take into account the autocorrelation of the sample the first statistic can be decomposed to measure the change in the univariate autoregressive integrated moving average parameters the transfer function parameters and the interaction between both for independent data they reduced to the d statistic considered by cook in the standard regression model these statistics can be easily computed using standard time series software their performance is analyzed in an example in which they are shown to be useful in identifying important events such as additive outliers and trend shifts in time series data

short proofs of embeddings into exceptional groups of lie type the availability of effective character theoretic arguments makes the study of embeddings of a particular finite group into classical lie groups much easier than the study of its embeddings into exceptional lie groups in this paper our goal is to give short reasonably uniform character theoretic and module theoretic arguments to prove the existence of particular finite subgroups of exceptional groups of lie type as examples we apply the methods that we develop to give short proofs of the embeddings ru e7 5 hs e7 5 f4 2 e6 5 u3 8 e7 and j1 g2 11 apart from the much studied example j1 g2 11 these embeddings have previously only been settled on a case by case basis with intricate computation and group theoretic arguments our main tool is the following simple module theoretic condition which is sufficient to exhibit a g invariant subalgebra in a known g invariant lie algebra

can the generalized hypergeometric equation feature several independent polynomial solutions the answer to the question posed in the title is positive it is even possible provided the parameters of the generalized hypergeometric equation are suitably restricted that the general solution of this ordinary differential equation be polynomial

temporal aggregation and the power of tests for a unit root abstract the asymptotic local power of unit root tests with the same data span is shown to be independent of sampling frequency a measure of the power trade off between sampling frequency and time span for distinct alternatives is derived using an approximate slopes approach only small span increases are generally required to maintain power when reducing sampling frequency monte carlo results support the asymptotic analysis for finite samples an application is made to a consumption function for the uk cointegration of consumption and wealth is rejected with quarterly data but convincingly accepted with a longer span of annual data

an exact multivariate model based structural decomposition we propose a simple and structured procedure for decomposing a vector of time series into trend cycle seasonal and irregular components contrary to common practice we do not assume these components to be orthogonal conditional on their past however the state space representation employed ensures that their estimates converge to values with null variances and covariances null variances are very important as they ensure that the components do not change when the sample increases this lack of revisions is the most important feature of our method in comparison with most alternative procedures on the other hand null covariances provide a solid statistical foundation for the decomposition as it ensures that a given component can be analyzed and interpreted independently of any other component s other convenient properties of our method derive from the use of a state space approach first defining the problem in state space avoids dependence on particular model specifications so the same proc

fast likelihood evaluation and prediction for nonstationary state space models summary a recursive procedure for initializing the kalman filter is displayed the recursion is for nonstationary state space models the procedure imposes small computational and programming burden over and above the kalman filter the procedure is superior to other suggested approaches in both computational speed and general applicability general properties of the method are investigated details of the initialization for the arima p d q and basic structural models are considered

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

an analysis of tests for regression coefficient stability abstract an exact small sample point optimal test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a first order autoregressive process in addition it is shown how watson and engle s 1985 asymptotic test can be modified so that its small sample level can be computed and its small sample power properties investigated simulation results show that the point optimal test developed in this paperoutperforms the small sample watson engle test as well as other tests given in the literature a point optimal test is also constructed for a stochastic coefficient generated by an arima 1 1 0 process

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of