large scale simulations of microstructural evolution in elastically stressed solids abstract we present a set of numerical methods for simulations of microstructural evolution in elastically stressed solids we combine three powerful tools to achieve computational efficiency the boundary integral method to provide excellent resolution the fast multipole method to reduce computational cost and the small scale decomposition technique with a two level time stepping scheme to remove the stiffness from the time advance although we apply these methods to study the topic of our interest the details of how they are implemented can be useful in many other situations we extend the fast multipole method to calculate the anisotropic stress field in periodic two dimensional domains and in doing so address issues associated with the conditional convergence of the summations in addition we introduce a new formula for the potential in periodically arranged two dimensional cells in the absence of an applied field through a summation in physical space without using the ewald sum furthermore we implement a time stepping scheme that enables us to speed up the calculation by an additional factor of 100 over a straightforward implementation of the small scale decomposition technique the computational complexity scales as the number of mesh points n and thus we are able to employ n 500 000 in a typical calculation

ramanujan s ternary quadratic form do not seem to obey any simple law following i kaplansky we call a non negative integer n eligible for a ternary form f x y z if there are no congruence conditions prohibiting f from representing n by the classical theory of quadratic forms it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer consequently if a genus consists of a single class with representative f x y z then f represents every eligible integer in the case of ramanujan s form this only implies that an eligible integer one not of the form 4 16 6 is represented by 1 or 2

integer points on spheres and their orthogonal lattices linnik proved in the late 1950 s the equidistribution of integer points on large spheres under a congruence condition the congruence condition was lifted in 1988 by duke building on a break through by iwaniec using completely different techniques we conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement we use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition

instrumental variables estimation in errors in variables models when instruments are correlated with errors abstract consistency of an instrumental variable iv estimator is based on the orthogonality assumption of the iv s and the regression equation errors if this assumption is in doubt there is not much hope left for the iv estimator the assumption can be tested but such a test usually requires another set of iv s for which the assumption holds in this paper it is shown that in an errors in variables model tighter bounds can be found in the presence of iv s even if they are correlated with the regression equation errors and that these bounds can be made tighter still if one has prior information restricting the extent of this correlation moreover the result is used to establish a sufficient condition for the ols estimator to be inconsistent when orthogonality is suspect so that the wu hausman test is not applicable this condition is useful even when the above type of prior information is not directly available

identification of complete information games this paper establishes sufficient conditions for point identification of the utility functions in generalized complete information game models these models allow generalized interaction structures and generalized behavioral assumptions the generalized interaction structures allow that the dependence of an agent s utility function on the other agents actions can itself depend on characteristics of the agents including an endogenous network of connections among the agents the generalized behavioral assumptions relax the solution concept from nash equilibrium play to weaker solution concepts like rationalizability the results allow a non parametric specification of the unobservables

laplace s equation and the dirichlet neumann map a new mode for mikhlin s method mikhlin s method for solving laplace s equation in domains exterior to a number of closed contours is discussed with particular emphasis on the dirichlet neumann map in the literature there already exist two computational modes for mikhlin s method here a new mode is presented the new mode is at least as stable as the previous modes furthermore its computational complexity in the number of closed contours is better as a result highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation

traces of cm values of modular functions zagier proved that the traces of singular moduli i e the sums of the values of the classical j invariant over quadratic irrationalities are the fourier coefficients of a modular form of weight 3 2 with poles at the cusps using the theta correspondence we generalize this result to traces of cm values of weakly holomorphic modular functions on modular curves of arbitrary genus we also study the theta lift for the weight 0 eisenstein series for sl2 and realize a certain generating series of arithmetic intersection numbers as the derivative of zagier s eisenstein series of weight 3 2 this recovers a result of kudla rapoport and yang

bounding parameter estimates with nonclassical measurement error abstract the bias introduced by errors in the measurement of independent variables has increasingly been a topic of interest among researchers estimating economic parameters however studies typically use the assumption of classical measurement error that is the variable of interest and its measurement error are uncorrelated and the expected value of the mismeasured variable is equal to the expected value of the true measure these assumptions often arise from convenience rather than conviction when a variable is bounded it is likely that the measurement error and the true value of the variable are negatively correlated we consider the case of a noisily measured variable with a negative covariance between the measurement error and the true value of the variable we show that asymptotically the parameter in a univariate regression is bounded between the ordinary least squares ols estimator and an instrumental variables iv estimator further we demonstrate that the ols bound can be improved in

bounding the effects of proxy variables on regression coefficients we consider a regression in which one of the observed variables is a proxy for some unobserved true variable given a lower bound for the correlation between the proxy and the unobserved true variable for which it substitutes we derive intervals in which the coefficients of the unobserved true regression must lie regardless of any other correlations involving unobserved variables or disturbances we present a simple solution for the important special case in which only the signs of the coefficients are of concern and one seeks the smallest correlation between the proxy and the true variable that guarantees the correctness of the signs of the coefficients in the proxy regression we also present an algorithm for extending these results to the multiple proxy problem attemprs to confront economic theories with data are often hampered by the problem that some of the relevant variables are unavailable or even unobservable in the case of regression models it is customary to substitute proxies for the unavailable variables though the coefficients in the proxy regression will not coincide with those in the regression of the dependent variable on the true explanatory variables it is clear by continuity that the differences will be small if each proxy is sufficiently highly correlated with the unobserved true variable for which it substitutes even if its error is correlated with other errors or variables moreover in many instances the theory predicts not the magnitudes but only the signs of the regression coefficients correct inferences will then be drawn from the model provided that the pairwise correlations between the proxies and the true variables are high enough to guarantee that the signs of the regression coefficients would be unchanged if the proxies were replaced by the true variables in applications econometricians can generally assess those pairwise correlations subjectively but how high do they need to be in this paper we present a complete answer to this question when only one variable is a proxy and describe an algorithm for determining the required correlations when there are several proxies the effects of proxy variables have been extensively studied in the special case of errors in variables eiv in which the differences between the proxies and the corresponding true variables the measurement errors are independent of the true variables of each other and of the disturbances in the true regression we require no such independence in the eiv case it is sometimes possible to guarantee correctness of the signs of the coefficients even without prior information about the variances of the measurement errors for example if the coefficient vector from the direct regression and the coefficient vectors from the reverse regressions as one varies the left side variable all lie in the same orthant then the coefficient vector in the true regression must also lie in that orthant see

second kind integral equations for the classical potential theory on open surfaces ii a second kind integral equation formulation is presented for the dirichlet problem for the laplace equation in two dimensions with the boundary conditions specified on a collection of open curves the performance of the obtained apparatus is illustrated with several numerical examples the formulation is a simplification of the equation previously constructed by the authors

gauss class number problem for imaginary quadratic fields a result first proved by heilbronn h in 1934 the disquisitiones also contains tables of binary quadratic forms with small class numbers actually tables of imaginary quadratic fields of small class number with even discriminant which is a much easier problem to deal with and gauss conjectured that his tables were complete in modern parlance we can rewrite gauss tables we are including both even and odd discriminants in the following form

preconditioned adaptive multipole accelerated iterative methods for three dimensional first kind integral equations of potential theory this paper presents a preconditioned krylov subspace iterative algorithm where a modified multipole algorithm with a novel adaptation scheme is used to compute the iterates for solving dense matrix problems generated by galerkin or collocation schemes applied to three dimensional first kind integral equations that arise in potential theory a proof is given that this adaptive algorithm reduces both matrix vector product computation time and storage to order n and experimental evidence is given to demonstrate that the combined preconditioned adaptive multipole accelerated pama method is nearly order n in practice examples from engineering applications are given to demonstrate that the accelerated method is substantially faster than standard algorithms on practical problems

consistent sets of estimates for regressions with correlated or uncorrelated measurement errors in arbitrary subsets of all variables over the last decade the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians in the fifties and sixties the problem was considered to be more or less hopeless due to its inherent underidentification e g theil 1971 apart from instrumental variables the most frequently cited textbook solution was wald s method of grouping wald 1940 recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases pakes 1982 since about 1970 new approaches to the problem have been explored basically along three lines viz embedding the error ridden equation into a set of multiple equations e g zellner 1970 goldberger 1972 into a set of simultaneous equations e g hsiao 1976 geraci 1976 and using the dynamics of the equation if present e g maravall and aigner 1977 in view of the underidentification of the basic model it is clear that all these methods invoke additional information of some kind if this information takes the form of exact or stochastic knowledge of certain parameters in the model the construction of consistent estimators is fairly straightforward e g fuller 1980 kapteyn and wansbeek 1984 for an overview of the state of the art see aigner et al 1984 an approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is examples are blomqvist 1972 hodges and moore 1972 and davies and hutton 1975 leamer 1983 starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference that is to say when measurement error is large enough to make it impossible to put bounds on regression parameters in an empirical example he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference at least without additional prior information the most systematic analysis of the information loss caused by measurement error is due to klepper and leamer 1984 they start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p in the special case where the measurement errors are assumed uncorrelated and the k 1 estimates of 3 obtained by regressing each of the k 1 variables involved i e the one dependent variable and the k independent variables on the remaining k variables are all in the same orthant one can bound the ml estimates of p in that case the convex hull of the k 1 regressions contains all possible ml estimates and any point in the hull is a possible ml estimate if the k 1 regressions are not all in the same orthant then the set of ml estimates is unbounded in that case klepper and leamer 1984 introduce extra prior information which allows them to bound the set of maximum likelihood estimates the prior information comes in two forms firstly a researcher is supposed to be able to specify a maximum value of r2 if all exogenous variables were measured accurately it is shown that if this maximum is low enough one can again bound the set of ml estimates by a convex hull secondly if

consistent estimation of linear panel data models with measurement error abstract measurement error causes a bias towards zero when estimating a panel data linear regression model the panel data context offers various opportunities to derive instrumental variables allowing for consistent estimation we consider three sources of moment conditions i restrictions on the covariance matrix of the errors in the equations ii nonzero third moments of the regressors and iii heteroskedasticity and nonlinearity in the relation between the error ridden regressor and another error free regressor in simulations these approaches appear to work well

problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak abstract we draw attention to two problems associated with the use of instrumental variables iv the importance of which for empirical work has not been fully appreciated first the use of instruments that explain little of the variation in the endogenous explanatory variables can lead to large inconsistencies in the iv estimates even if only a weak relationship exists between the instruments and the error in the structural equation second in finite samples iv estimates are biased in the same direction as ordinary least squares ols estimates the magnitude of the bias of iv estimates approaches that of ols estimates as the r 2 between the instruments and the endogenous explanatory variable approaches 0 to illustrate these problems we reexamine the results of a recent paper by angrist and krueger who used large samples from the u s census to estimate wage equations in which quarter of birth is used as an instrument for educational attainment we find evidence that despite huge sample sizes th

maass forms and the mock theta function f q let f q 1 sum n 1 infty alpha n q n be the well known third order mock theta of ramanujan in 1964 george andrews proved an asymptotic formula of the form begin aligned alpha n sum c le sqrt n psi n o epsilon left n epsilon right end aligned where psi n is an expression involving generalized kloosterman sums and the i bessel function andrews conjectured that the series converges to alpha n when extended to infinity and that it does not converge absolutely bringmann and ono proved the first of these conjectures here we obtain a power savings bound for the error in andrews formula and we also prove the second of these conjectures our methods depend on the spectral theory of maass forms of half integral weight and in particular on an average estimate for the fourier coefficients of such forms which gives a power savings in the spectral parameter as a further application of this result we derive a formula which expresses alpha n with small error as a sum of exponential terms over imaginary quadratic points this is similar in spirit to a recent result of masri we also obtain a bound for the size of the error term incurred by truncating rademacher s analytic formula for the ordinary partition function which improves a result of the first author and andersen when 24n 23 is squarefree

parameter estimation in regression models with errors in the variables and autocorrelated disturbances abstract for regression models with first order autocorrelated disturbances the traditional prescription of econometricians is to correct for serial correlation by using appropriate estimation techniques such as the cochrane orcutt hildreth lu or prais winsten procedures this suggestion however does not take into account the fact that in most economic applications the independent variables contain measurement errors when there are errors in the variables ordinary least squares without any attempt to correct for serial correlation may in several cases yield less unreliable and misleading results in terms of biases and mean squared errors of the estimators this paradoxical finding is demonstrated analytically for the asymptotic case and illustrated for finite samples by monte carlo experiments for the regression model with a single stationary explanatory variable the monte carlo experiments also suggest that the same conclusion holds if the independent variable follows a random walk at least for medium size samples

structure of the scalar field around unidirectional circular cylinders we study transport properties of equal infinite unidirectional circular cylinders that are arbitrarily distributed in a uniform host the problem is reduced to a conjugation problem for the two dimensional laplace equation the flux is written exactly in the form of a power series in bergman s contrast parameter asymptotic formulae for the flux are deduced for densely packed inclusions these formulae involve two small parameters the ratio of the distance between the close discs to their radius and a dimensionless parameter that characterizes the degree of perfect conductor isolator validity of the structural approximation is discussed

a local target specific quadrature by expansion method for evaluation of layer potentials in 3d accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations quadrature by expansion qbx is a recently introduced m

regularized liml for many instruments the use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators however in finite samples the inclusion of an excessive number of moments increases the bias to solve this problem we propose regularized versions of the limited information maximum likelihood liml based on three different regularizations tikhonov landweber fridman and principal components our estimators are consistent and asymptotically normal under heteroskedastic error moreover they reach the semiparametric efficiency bound assuming homoskedastic error we show that the regularized liml estimators possess finite moments when the sample size is large enough the higher order expansion of the mean square error mse shows the dominance of regularized liml over regularized two staged least squares estimators we devise a data driven selection of the regularization parameter based on the approximate mse a monte carlo study and two empirical applications illustrate the relevance of our estimators

the first nontrivial genus of positive definite ternary forms the first nontrivial genus of positive ternary forms has discriminant 7 the paper presents all known results concerning this genus including some computations

matrix algebra for higher order moments abstract a large part of statistics is devoted to the estimation of models from the sample covariance matrix the development of the statistical theory and estimators has been greatly facilitated by the introduction of special matrices such as the commutation matrix and the duplication matrix and the corresponding matrix algebra some more extensive models require however estimation based on higher order moments typically third and fourth order moments an example is the popular kenny judd model that includes interactions between latent variables this paper introduces some special matrices that can be used to simplify the model expressions for third fourth and higher order moments gives some relationships between these matrices and related matrices and gives some formulas for kronecker products of three and four matrices the theory is applied to derive convenient expressions for third and fourth order moments of some structural equation models

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

an adaptive fast multipole accelerated poisson solver for complex geometries abstract we present a fast direct and adaptive poisson solver for complex two dimensional geometries based on potential theory and fast multipole acceleration more precisely the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition the volume integral is computed by applying the fmm on a square box that encloses the domain of interest for the sake of efficiency and convergence acceleration we first extend the source distribution the right hand side in the poisson equation to the enclosing box as a c 0 function using a fast boundary integral based method we demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and as a result to an extremely efficient black box fast solver

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

nonparametric identification and semiparametric estimation of classical measurement error models without side information virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information e g validation data known error distributions repeated measurements or instruments in contrast we establish that the fully nonparametric classical errors in variables model is identifiable from data on the regressor and the dependent variable alone unless the model takes a very specific parametric form this parametric family includes but is not limited to the linear specification with normally distributed variables as a well known special case this result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions assumptions our approach can handle both monotone and nonmonotone specifications provided the latter oscillate a finite number of times given that the very specific unidentified parametric functional form is arguably the exception rather than the rule this identification r

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

regressor diagnostics for the classical errors in variables model abstract a series of diagnostics are developed which can be used to probe the sensitivity of regression coefficient estimates to measurement error in each regressor when all the regressors may be measured with error the diagnostics are based on the large sample properties of the classical errors in variables model the diagnostics provide information about the bounds on the measurement error variances of the regressors needed to bound any coefficient to a desired interval algorithms to compute the diagnostics are provided and the use of the diagnostics is illustrated in an example

stress calculations on multiply connected domains the outstanding problem of finding a simple muskhelishvili type integral equation for stress problems on multiply connected domains is solved complex potentials are represented in a way which allows for the incorporation of cracks and inclusions several numerical examples demonstrate the generality and extreme stability of the approach the stress field is resolved with a relative error of less than 10 10 on a large yet simply reproducible setup with a loaded square plate containing 4096 holes and cracks comparison with previous results in the literature indicates that general purpose finite element software may perform better than many special purpose codes

errors in variables in linear systems this paper extends the simple errors in variable bound to the setting of systems of equations both diagonal and nondiagonal measurement error covariance matrices are considered in the nondiagonal case the analogue of the simple errors in variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two stage least squares estimate for the diagonal case the set of estimates under some conditions must lie within the convex hull of 2k points

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

on almost universal mixed sums of squares and triangular numbers in 1997 k ono and k soundararajan invent math 130 1997 proved that under the generalized riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous ramanujan form x 2 y 2 1oz 2 equivalently the form 2x 2 5y 2 4t z represents all integers greater than 1359 where t z denotes the triangular number z z 1 2 given positive integers a b c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 by 2 ct z represents sufficiently large integers and to establish similar results for the forms ax 2 bt y ct z and at x bt y ct z here are some consequences of our main theorems i all sufficiently large odd numbers have the form 2 x 2 y 2 z 2 if and only if all prime divisors of a are congruent to 1 modulo 4 ii the form x 2 y 2 t z is almost universal i e it represents sufficiently large integers if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8 iii x 2 t y t z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4 iv when 2 3 the form t x t y tz is almost universal if and only if all odd prime divisors of are congruent to 1 modulo 4 and 2 5 7 where 2 a is the 2 adic order of

higher moment estimators for linear regression models with errors in the variables abstract this paper proposes consistent instrumental variable estimators for linear regression models with errors in the variables that require no extraneous information these estimators are based on sample moments of order higher than two while similar estimators proposed previously in the literature seem to be quite erratic our experimental f findings suggest that our estimators perform better than ordinary least squares estimators in terms of root mean squared errors and also in terms of size of type i errors of standard tests in many typical situations of economic analyses tests for the presence of errors in the variables are also described

simulated minimum distance estimation of dynamic models with errors in variables abstract empirical analysis often involves using inexact measures of the predictors suggested by economic theory the bias created by the correlation between the mismeasured regressors and the error term motivates the need for instrumental variable estimation this paper considers a class of estimators that can be used in dynamic models with measurement errors when external instruments may not be available or are weak the idea is to exploit the relation between the parameters of the model and the least squares biases in cases when the latter are not analytically tractable a special algorithm is designed to simulate the model without completely specifying the processes that generate the latent predictors the proposed estimators perform well in simulations of the autoregressive distributed lag model the methodology is used to estimate the long run risks model

the eligible numbers of positive definite ternary forms several nontrivial genera of positive ternary forms with small discriminants have been studied in this paper especially we prove that there are only finitely many square free eligible numbers in the genus of f 1 x 2 y 2 7z 2 and in the genus of f 2 x 2 7y 2 7z 2 which cannot be represented by f 1 and f 2 respectively our method is to use modular forms of weight 3 2

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

modeling discrete choice with response error food stamp participation abstract validation of food stamp program participation by marquis and moore revealed net bias of 13 in mean estimates of participation in the 1984 survey of income and program participation we extend the analysis incorporating demographic and economic covariates in models for underreporting and overreporting we use the resulting models of response error to refine estimates of participation models using pseudo maximum likelihood estimation methods we find that the probability of underreporting rises with increased family income estimates of the participation model indicate the presence of substantial bias in income and asset related parameters compared to values obtained without incorporating response error

quadratic forms representing all odd positive integers we consider the problem of classifying all positive definite integer valued quadratic forms that represent all positive odd integers kaplansky considered this problem for ternary forms giving a list of 23 candidates and proving that 19 of those represent all positive odds jagy later dealt with a 20th candidate assuming that the remaining three forms represent all positive odds we prove that an arbitrary positive definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451 this result is analogous to bhargava and hanke s celebrated 290 theorem in addition we prove that these three remaining ternaries represent all positive odd integers assuming the generalized riemann hypothesis this result is made possible by a new analytic method for bounding the cusp constants of integer valued quaternary quadratic forms q with fundamental discriminant this method is based on the analytic properties of rankin selberg l functions and we use it to prove that if q is a quaternary form with fundamental discriminant the largest locally represented integer n for which q vec x n has no integer solutions is o d 2 epsilon

partial identification of probability distributions with misclassified data abstract this paper addresses the problem of data errors in discrete variables when data errors occur the observed variable is a misclassified version of the variable of interest whose distribution is not identified inferential problems caused by data errors have been conceptualized through convolution and mixture models this paper introduces the direct misclassification approach the approach is based on the observation that in the presence of classification errors the relation between the distribution of the true but unobservable variable and its misclassified representation is given by a linear system of simultaneous equations in which the coefficient matrix is the matrix of misclassification probabilities formalizing the problem in these terms allows one to incorporate any prior information into the analysis through sets of restrictions on the matrix of misclassification probabilities such information can have strong identifying power the direct misclassification approach fully exploits it to derive identification regions for any real functional of the distribution of interest a method for estimating the identification regions and construct their confidence sets is given and illustrated with an empirical analysis of the distribution of pension plan types using data from the health and retirement study

bounding the effects of measurement error in regressions involving dichotomous variables abstract in a multiple regression involving dichotomous independent variables the effects of measurement error in the dichotomous variables is analyzed unlike the case of classical i e white noise measurement error the measurement error in a dichotomous variable must be correlated with the true variable and any other variables correlated with the true variable nonetheless it is shown that the results for the classical errors in variables model can be used to analyze the implications of measurement error in dichotomous variables it is shown how the bounding procedure developed by klepper and leamer 1984 for the classical errors in variables model can be adapted to bound the coefficients when the regression contains mismeasured dichotomous variables the classical results are also used to analyze the inconsistency in least squares regression coefficient estimators computed from data on the mismeasured variables

errors in survey reports of earnings hours worked and hourly wages abstract using administrative records on earnings hours worked and hourly wages for hourly employees of a single manufacturing firm we found standard assumptions about errors in survey measures to be violated to varying extents errors are correlated with true scores and errors in reports of earnings and hours in different periods are generally positively correlated with one another particularly for reports given in a single interview although a high proportion of reporting errors are from an approximately normal distribution a small proportion comes from a distribution with a much greater variance and these cases often have considerable influence on estimates of relationships between variables

a regularization approach to the many instruments problem abstract this paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite in order to improve the small sample properties of standard instrumental variable iv estimators we propose three modified iv estimators based on three different ways of inverting the covariance matrix of the instruments these inverses involve a regularization or smoothing parameter it should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation we show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound higher order analysis of the mse reveals that the bias of the modified estimators does not depend on the number of instruments finally we suggest a data driven method for selecting the regularization parameter interestingly our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument

lattice points in rational ellipsoids we combine exponential sums character sums and fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids

proper posteriors from improper priors for an unidentified errors in variables model the problem considered is inference in a simple errors in variables model where consistent estimation is impossible without introducing additional exact prior information the probabilistic prior information required for bayesian analysis is found to be surprisingly light despirte the model s lack of identification a proper posterior is guaranteed for any bounded prior density including those representing improper priors this result is illustrated with the improper uniform prior which implies marginal posterior densities obtainable by integrating the likelihood function

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

boundary integral equations with the generalized neumann kernel for laplace s equation in multiply connected regions this paper presents a new boundary integral method for the solution of laplace s equation on both bounded and unbounded multiply connected regions with either the dirichlet boundary condition or the neumann boundary condition the method is based on two uniquely solvable fredholm integral equations of the second kind with the generalized neumann kernel numerical results are presented to illustrate the efficiency of the proposed method

groups of units of zero ternary quadratic forms the groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to fuchsian groups and which for zero forms are commensurable with the classical modular group this is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms the signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms this last result should be compared with the corresponding result by mennicke on non zero forms

a generalized cubic moment and the petersson formula for newforms using a cubic moment we prove a weyl type subconvexity bound for the quadratic twists of a holomorphic newform of square free level trivial nebentypus and arbitrary even weight this generalizes work of conrey and iwaniec in that the newform that is being twisted may have arbitrary square free level and also that the quadratic character may have even conductor one of the new tools developed in this paper is a more general petersson formula for newforms of square free level

almost strong approximations for definite quadratic spaces a fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group spin of a non degenerate inde nite quadratic form in three or more variables in practice one invokes the corresponding version for the kernel o0 of the spinor norm function this result is due to the works of eichler and kneser ei kn in the 1950s and has ever since played a central role in the proofs of a number of important theorems for both inde nite as well as de nite quadratic forms on the other hand it is well known that such a strong approximation cannot possibly hold for i the group so of a inde nite quadratic form as it is not simply connected and ii for the group spin of a de nite quadratic form as spin1 is compact the purpose of this paper is to show that there exists a version of an almost strong approximation theorem for de nite quadratic forms one forom and one foro m m 3 and show that they too have serious applications in the arithmetic theory e g in the asymptotic distributions of classes in a genus and in the representation theory of de nite forms it is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article we give a brief outline of the contents of this paper terminology and notations are generally those from om for simplicity as well as convenience we shall consider here only positive de nite integral z lattices a key step in the proof of the strong approximation theorem for o0 is the following result on inde nite representations with approximation property invent math 129 471 487 1997

some further results on the exact small sample properties of the instrumental variable estimator we present new results on the exact small sample distribution of the instrumental variable estimator in particular we compare the small sample distribution to the asymptotic distribution

the explicit sato tate conjecture and densities pertaining to lehmer type questions let f z n 1 af n q n s k 0 n be a newform with squarefree level n that does not have complex multiplication for a prime p define p 0 to be the angle for which af p 2p k 1 2 cos p let i 0 be a closed subinterval and let d st 2 sin 2 d be the sato tate measure of i assuming that the symmetric power l functions of f satisfy certain analytic properties all of which follow from langlands functoriality and the generalized riemann hypothesis we prove that if x is sufficiently large then p x p i st i x 2 dt log t x log nkx log x with an implied constant of 3 33 by letting i be a short interval centered at 2 and counting the primes using a smooth cutoff we compute a lower bound for the density of positive integers n for which af n 6 0 in particular if is the ramanujan tau function then under the aforementioned hypotheses we prove that lim x n x n 6 0 x 1 1 54 10 13 we also discuss the connection between the density of positive integers n for which af n 6 0 and the number of representations of n by certain positive definite integer valued quadratic forms

bounding the effects of proxy variables on instrumental variables coefficients abstract we consider instrumental variables models in which one of the observed variables is a proxy for some unobserved true variable we seek the smallest correlation between the proxy and the unobserved true variable sufficient to guarantee that regardless of any other correlations involving unobserved variables or disturbances a coefficient s sign has not been reversed by substituting the proxy we derive exact solutions when the proxy is the dependent variable the endogenous explanatory variable whose coefficient is of concern some other endogenous explanatory variable or a solely instrumental variable for an exogenous explanatory proxy we derive either exact solutions or useful lower bounds

faster convergence and higher accuracy for the dirichlet neumann map new techniques allow for more efficient boundary integral algorithms to compute the dirichlet neumann map for laplace s equation in two dimensional exterior domains novelties include a new post processor which reduces the need for discretization points with 50 a new integral equation which reduces the error for resolved geometries with a factor equal to the system size systematic use of regularization which reduces the error even further and adaptive mesh generation based on kernel resolution

on acoustic and electric faraday cages two dimensional problems involving many identical small circles are considered the circles are the cross sections of parallel wires modelling a cage or a grating both electrostatic and acoustic fields are considered the main emphasis is on periodic configurations of n circles distributed evenly around a large circle a ring foldy s theory is used for acoustic problems and then adapted for electrostatic problems in both cases circulant matrices are encountered the fields can be calculated explicitly then the limit n is studied a connection between the n circle problem and the limiting problem fields exterior to the ring is established using known results on the convergence of a defective form of the trapezoidal rule defective in that endpoint contributions are ignored because the integrand has logarithmic singularities at those points this shows that the solution of the limiting problem is approached very slowly as n 1 log n as n

minimum distance estimation of the errors in variables model using linear cumulant equations abstract we consider a multiple mismeasured regressor errors in variables model we develop closed form minimum distance estimators from any number of estimating equations which are linear in the third and higher cumulants of the observable variables using the cumulant estimators alters qualitative inference relative to ordinary least squares in two applications related to investment and leverage regressions the estimators perform well in monte carlos calibrated to resemble the data from our applications although the cumulant estimators are asymptotically equivalent to the moment estimators from erickson and whited 2002 the finite sample performance of the cumulant estimators exceeds that of the moment estimators

estimating a class of triangular simultaneous equations models without exclusion restrictions this paper provides a control function estimator to adjust for endogeneity in the triangular simultaneous equations model where there are no available exclusion restrictions to generate suitable instruments our approach is to exploit the dependence of the errors on exogenous variables e g heteroscedasticity to adjust the conventional control function estimator the form of the error dependence on the exogenous variables is subject to restrictions but is not parametrically specified in addition to providing the estimator and deriving its large sample properties we present simulation evidence which indicates the estimator works well

bounding mean regressions when a binary regressor is mismeasured in this paper i examine identification and estimation of mean regression models when a binary regressor is mismeasured i prove that bounds for the model parameters are identified and provide simple estimators which are consistent and asymptotically normal when stronger prior information about the probability of misclassification is available the bounds can be made tighter again a simple estimator for these cases is provided all results apply to parametric and nonparametric models the paper concludes with a short empirical example

consistent noisy independent component analysis we study linear factor models under the assumptions that factors are mutually independent and independent of errors and errors can be correlated to some extent under the factor non gaussianity second to fourth order moments are shown to yield full identification of the matrix of factor loadings we develop a simple algorithm to estimate the matrix of factor loadings from these moments we run monte carlo simulations and apply our methodology to data on cognitive test scores and financial data on stock returns

identification and estimation of nonlinear models with misclassification error using instrumental variables a general solution abstract this paper provides a general solution to the problem of identification and estimation of nonlinear models with misclassification error in a general discrete explanatory variable using instrumental variables the misclassification error is allowed to be correlated with all the explanatory variables in the model it is not enough to identify the model by simply generalizing the identification in the binary case with a claim that the number of restrictions is no less than that of the unknowns such a claim requires solving a complicated nonlinear system of equations this paper introduces a matrix diagonalization technique which allows one to easily find the unique solution of the system the solution shows that the latent model can be expressed as an explicit function of directly observed distribution functions therefore the latent model is nonparametrically identifiable and directly estimable using instrumental variables the results show that certain monotonicity restrictions on the latent model may lead to its identification with virtually no restrictions on the misclassification probabilities an alternative identification condition suggests that the nonparametric identification may rely on the belief that people always have a higher probability of telling the truth than of misreporting the nonparametric identification in this paper directly leads to a n consistent semiparametric estimator the monte carlo simulation and empirical illustration show that the estimator performs well with a finite sample and real data

second king integral equations for the classical potential theory on open surfaces i analytical apparatus a stable second kind integral equation formulation has been developed for the dirichlet problem for the laplace equation in two dimensions with the boundary conditions specified on a collection of open curves the performance of the obtained apparatus is illustrated with several numerical examples

measurement of higher education in the census and current population survey we examine measurement error in the reporting of higher education in the 1990 decennial census and the post 1991 current population survey cps we document that measurement error in the reporting of higher education is prevalent in census data further these errors violate models of classical measurement error in important ways the level of education is consistently reported as higher than it is errors are not mean 0 errors in the reporting of education are correlated with covariates that appear in earnings regressions and errors in the reporting of education appear correlated with the error term in a model of earnings determination thus neither well known results on classical measurement error nor recent models of nonclassical measurement error are likely valid when using census and cps data we find some evidence that the measurement error is lower in the cps than in the census presumably because first interviews are generally conducted in person

bounding parameters in a linear regression model with a mismeasured regressor using additional information this paper discusses a linear regression model with a mismeasured regressor in which the measurement error is correlated with both the latent variable and the regression error we use a linear structure to capture the correlation between the measurement error and the latent variable this paper shows that the variance of the latent variable is very useful for revealing information on the parameters which otherwise cannot be obtained with such a nonclassical measurement error the main result is that the finite bounds on the parameters can be found using the variance of the latent variable regardless of how severely the measurement error and the regression error are correlated if the mismeasured regressor contains enough information on the latent one this paper also discusses the special but interesting case of the latent variable being dichotomous in this case the mean of the latent variable may even reveal information on the correlation between the measurement error and the regression error all the bounds developed in the paper are tight