enhancing convolution and interpolation methods for nonparametric regression summary following earlier work of jones davies park 1994 we develop estimators based on convolution and interpolation methods with the aim of reducing their variance and making them competitive with local linear smoothers we argue that the classic convolution and interpolation approaches due to gasser miller 1979 and clark 1977 exhibit inflated variance on account of the relatively high variability of weights used in their construction we show that by convolving or interpolating over only a small additional number of design points variance is reduced by a constant factor theoretical and numerical properties are summarised

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

the generalized basis reduction algorithm let fx be a convex function defined in rn which is symmetric about the origin and homogeneous of degree 1 and let l be the lattice of integers zn a definition of a reduced basis b1 bn of the lattice with respect to the distance function f is presented and we describe an algorithm which yields a reduced basis in polynomial time for fixed n in the special case in which the bodies x fx t are ellipsoids the definition of a reduced basis is identical with that given by lenstra lenstra and lovasz 1982 and the algorithm is the well known basis reduction algorithm we show that the basis vector b1 in a reduced basis is an approximation to a shortest nonzero lattice point with respect to f and relate the basis vectors bi to minkowski s successive minima the results lead to an algorithm for integer programming which executes in polynomial time for fixed n but which avoids the ellipsoidal approximations required by lenstra s algorithm we also discuss the properties of a korkine zolotarev basis for the lattice

high performance numerical algorithms and software for structured total least squares we present a software package for structured total least squares approximation problems the allowed structures in the data matrix are block toeplitz block hankel unstructured and exact combination of blocks with these structures can be specified the computational complexity of the algorithms is o m where m is the sample size we show simulation examples with different approximation problems application of the method for multivariable system identification is illustrated on examples from the database for identification of systems daisy

structured total least squares and l2 approximation problems abstract it is shown how structured and weighted total least squares and l 2 approximation problems lead to a nonlinear generalized singular value decomposition an inverse iteration scheme to find a local minimum is proposed the emphasis of the paper is not on the convergence analysis of the algorithm rather the purpose is to illustrate its use in numerous applications in systems and control including total least squares with relative errors and or fixed elements inverse singular value problems an errors in variables variant of the kalman filter impulse response realization from noisy data h 2 model reduction h 2 system identification and calculating the largest stability radius of uncertain linear systems several numerical examples are given

small solutions to inhomogeneous linear equations over number fields we consider a system of m independent inhomogeneous linear equations in n m variables having coefficients in an algebraic number field k we give a best possible lower bound on the inhomogeneous height of a solution vector in k n and determine when a solution exists in o s n where o s is the ring of s integers in k if such a system has a solution vector in o s n we show that it has a solution in o s n such that the inhomogeneous height of is relatively small we give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system our method uses geometry of numbers over adele spaces and local to global arguments

small zeros of quadratic forms over number fields let f be a nontrivial quadratic form in n variables with coefficients in a number field k and let a be a k x n matrix over k we show that if the simultaneous equations f x 0 and ax 0 hold on a subspace x of dimension l and l is maximal then such a subspace x can be found with the height of x relatively small in particular the height of x can be explicitly bounded by an expression depending on the height of f and the height of a we use methods from geometry of numbers over adele spaces and local to global techniques which generalize recent work of h p schlickewei

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

solving a system of linear diophantine equations with lower and upper bounds on the variables we develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables the algorithm is based on lattice basis reduction it first finds a short vector satisfying the system of diophantine equations and a set of vectors belonging to the nullspace of the constraint matrix due to basis reduction all these vectors are relatively short the next step is to branch on linear combinations of the null space vectors which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists the research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing our algorithm is tested with good results on real life data and on instances from the literature

lower bounds for norms of products of polynomials via bombieri inequality in this paper we give a different interpretation of bombieri s norm this new point of view allows us to work on a problem posed by beauzamy about the behavior of the sequence sn p supqn pqn 2 where p is a fixed m homogeneous polynomial and qn runs over the unit ball of the hilbert space of n homogeneous polynomials we also study the factor problem for homogeneous polynomials defined on c and we obtain sharp inequalities whenever the number of factors is no greater than n in particular we prove that for the product of homogeneous polynomials on infinite dimensional complex hilbert spaces our inequality is sharp finally we use these ideas to prove that any set zk k 1 of unit vectors in a complex hilbert space for which sup z 1 z z1 z zn is minimum must be an orthonormal system

variable projection for affinely structured low rank approximation in weighted 2 norms the structured low rank approximation problem for general affine structures weighted 2 norms and fixed elements is considered the variable projection principle is used to reduce the dimensionality of the optimization problem algorithms for evaluation of the cost function the gradient and an approximation of the hessian are developed for mxn mosaic hankel matrices the algorithms have complexity o m 2n

the multiplicity of binary recurrences of the recurrence relation 0 1 it follows from the skolem mahler lech theorem that if u then one at least of and is a root of unity consequently we call um nondegenerate if none of these quantities is a root of unity in the thirties m ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity u 5 5 this conjecture was proved by kubota 4 kubota also proved that any nondegenerate binary recurrence um with elements um in a number eld k of degree d has multiplicity u 5 c d cf 5 beukers and tijdeman 2 established for this case the bound

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

variable projection methods for approximate greatest common divisor computations we consider the problem of finding for a given n tuple of polynomials real or complex the closest n tuple that has a common divisor of degree at least d extended weighted euclidean seminorm of the coefficients is used as a measure of closeness two equivalent representations of the problem are considered i direct parameterization over the common divisors and quotients image representation and ii sylvester low rank approximation kernel representation we use the duality between least squares and least norm problems to show that i and ii are closely related to mosaic hankel low rank approximation this allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic hankel low rank approximation we develop optimization methods based on the variable projection principle both for image and kernel representation these methods have linear complexity in the degrees of the polynomials for small and large d we provide a software implementation of the developed methods which is based on a software package for structured low rank approximation

a gradient system approach for hankel structured low rank approximation abstract rank deficient hankel matrices are at the core of several applications however in practice the coefficients of these matrices are noisy due to e g measurements errors and computational errors so generically the involved matrices are full rank this motivates the problem of hankel structured low rank approximation structured low rank approximation problems in general do not have a global and efficient solution technique in this paper we propose a local optimization approach based on a two levels iteration experimental results show that the proposed algorithm usually achieves good accuracy and shows a higher robustness with respect to the initial approximation compared to alternative approaches

the geometry of weighted low rank approximations the low rank approximation problem is to approximate optimally with respect to some norm a matrix by one of the same dimension but smaller rank it is known that under the frobenius norm the best low rank approximation can be found by using the singular value decomposition svd although this is no longer true under weighted norms in general it is demonstrated here that the weighted low rank approximation problem can be solved by finding the subspace that minimizes a particular cost function a number of advantages of this parameterization over the traditional parameterization are elucidated finding the minimizing subspace is equivalent to minimizing a cost function on the grassmann manifold a general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework within this framework two novel algorithms a steepest descent algorithm and a newton like algorithm are derived for solving the weighted low rank approximation problem they are compared with other algorithms for low rank approximation as well as with other algorithms for minimizing a cost function on a grassmann manifold

software for weighted structured low rank approximation a software package is presented that computes locally optimal solutions to low rank approximation problems with the following features mosaic hankel structure constraint on the approximating matrix weighted 2 norm approximation criterion fixed elements in the approximating matrix missing elements in the data matrix and linear constraints on an approximating matrix s left kernel basis it implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem for an mxn data matrix with n m the computational complexity of the cost function and derivative evaluation is o m 2n the package is suitable for applications with n m in statistical estimation and data modeling the main application areas of the package n m corresponds to modeling of large amount of data by a low complexity model performance results on benchmark system identification problems from the database daisy and approximate common divisor problems are presented

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

feasibility of integer knapsacks given a matrix a in mathbb z m times n satisfying certain regularity assumptions we consider the set mathcal f a of all vectors boldsymbol b in mathbb z m such that the associated knapsack polytope p a boldsymbol b boldsymbol x in mathbb r n geq0 a boldsymbol x boldsymbol b contains an integer point when m 1 the set mathcal f a is known to contain all consecutive integers greater than the frobenius number associated with a in this paper we introduce the diagonal frobenius number mathrm g a which reflects in an analogous way feasibility properties of the problem and the structure of mathcal f a in the general case we give an optimal upper bound for mathrm g a and also estimate the asymptotic growth of the diagonal frobenius number on average

realization and identification of autonomous linear periodically time varying systems abstract the subsampling of a linear periodically time varying system results in a collection of linear time invariant systems with common poles this key fact known as lifting is used in a two step realization method the first step is the realization of the time invariant dynamics the lifted system computationally this step is a rank revealing factorization of a block hankel matrix the second step derives a state space representation of the periodic time varying system it is shown that no extra computations are required in the second step the computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system a modification of the realization method is proposed which makes the complexity independent of the parameter variation period replacing the rank revealing factorization in the realization algorithm by structured low rank approximation yields a maximum likelihood identification method existing methods for structured low rank approximation are used to identify efficiently a linear periodically time varying system these methods can deal with missing data

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

the most powerful unfalsified model for data with missing values abstract the notion of the most powerful unfalsified model plays a key role in system identification since its introduction in the mid 80s many methods have been developed for its numerical computation all currently existing methods however assume that the given data is a complete trajectory of the system motivated by the practical issues of data corruption due to failing sensors transmission lines or storage devices we study the problem of computing the most powerful unfalsified model from data with missing values we do not make assumptions about the nature or pattern of the missing values apart from the basic one that they are a part of a trajectory of a linear time invariant system the identification problem with missing data is equivalent to a hankel structured low rank matrix completion problem the method proposed selects rank deficient complete submatrices of the incomplete hankel matrix under specified conditions the kernels of the submatrices form a nonminimal kernel representation of the data generating system the final step of the algorithm is reduction of the nonminimal kernel representation to a minimal one apart from its practical relevance in identification missing data is a useful concept in systems and control classic problems such as simulation filtering and tracking control can be viewed as missing data estimation problems for a given system the corresponding identification problems with missing data are data driven equivalents of the classical simulation filtering and tracking control problems

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

on the structure of reduced kernel lattice bases lattice based reformulation techniques have been used successfully both theoretically and computationally one such reformulation is obtained from the kernel lattice associated with an input matrix some of the hard instances in the literature that have been successfully tackled by lattice based techniques have randomly generated input since the considered instances are very hard even in low dimension less experience is available for larger instances recently we have studied larger instances and observed that the lll reduced basis of the kernel lattice has a specific sparse structure in particular this translates into a map in which some of the original variables get a rich translation into a new variable space whereas some variables are only substituted in the new space if an original variable is important in the sense of branching or cutting planes this variable should be translated in a nontrivial way in this paper we partially explain through a probabilistic analysis the obtained struct

covariate matched one sided tests for the difference between functional means abstract when testing hypotheses about the effects of different treatments variation among covariates can become confounded with that between treatments unless the treatments are applied using paired covariates in the context of unpaired covariates we propose implicit covariate matching methods for testing the hypothesis that one treatment effect is greater than another the methods are founded on the assumption that the mean treatment effect conditional on the covariate is a smooth function of the covariate they are implemented using new interpolation techniques for nonparametric curve estimation bootstrap arguments are used to construct critical points we show that even when the covariate distributions are identical for both treatments covariate matching of the type that we propose produces tests of greater power than methods that do not attempt matching our techniques have application to two sided hypothesis testing

generalized inverses of hankel and toeplitz mosaic matrices abstract hankel and toeplitz mosaic matrices are block matrices with hankel or toeplitz blocks respectively it is shown that hankel and toeplitz mosaic matrices possess reflexive generalized inverses which are bezoutians furthermore the bezoutian structure of the moore penrose and group inverses is investigated

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

mesures d independance lineaire de logarithmes dans un groupe algebrique commutatif this work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group defined over the field of algebraic numbers overline mathbb q let g be such a group let w be a hyperplane of the tangent space at the origin of g defined over overline mathbb q and u be a complex point of this tangent space such that the image of u by the exponential map of the lie group g is an algebraic point then we obtain a lower bound for the distance between u and w which improves the results known before and which is in particular the best possible for the height of the hyperplane w the proof rests on baker s method and hirata s reduction as well as a new arithmetic argument chudnovsky s process of variable change which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

combining problem structure with basis reduction to solve a class of hard integer programs recently aardal et al 2000 have successfully solved some small difficult equality constrained integer programs by using basis reduction to reformulate the problems as inequality constrained integer programs in a different space here we adapt their method to solve integer programs that are larger but have special structure the practical problem motivating this work is a variant of the market share problem more formally the problem can be viewed as finding a matrix x zmn satisfying xa c bx d where a b c d are matrices of compatible dimensions and the approach requires us to find a reduced basis of the lattice l x zm n xa 0 bx 0 the main topic of this paper is a study of the lattice l it is shown that an integer basis of l can be obtained by taking the kronecker product of vectors from integer bases of two much smaller lattices furthermore the resulting basis is a reduced basis if the integer bases of the two small lattices are reduced bases and a suitable ordering is chosen finally some limited computational results are presented showing the benefits of making use of the problem structure

integer knapsacks average behavior of the frobenius numbers the largest integer that cannot be represented as a nonnegative integral combination of given set of positive integers is called the frobenius number of these integers we show that the asymptotic growth of the frobenius number on average is significantly slower than the growth of the maximum frobenius number

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

a gauss newton like optimization algorithm for weighted nonlinear least squares problems the gauss newton algorithm is often used to minimize a nonlinear least squares loss function instead of the original newton raphson algorithm the main reason is the fact that only first order derivatives are needed to construct the jacobian matrix some applications as for instance multivariable system identification give rise to weighted nonlinear least squares problems for which it can become quite hard to obtain an analytical expression of the jacobian matrix to overcome that struggle a pseudo jacobian matrix is introduced which leaves the stationary points untouched and can be calculated analytically moreover by slightly changing the pseudo jacobian matrix a better approximation of the hessian can be obtained resulting in faster convergence

a unifying approach to nonparametric regression estimation abstract there exists a variety of nonparametric regression estimators with cubic smoothing splines k nearest neighbor k nn estimators and various types of kernel estimators among the most popular a class of kernel estimators with local bandwidth depending on the density of the design points is introduced where the degree of design adaptation may be expressed by a single parameter in 0 1 the adaptation to the design is such that the bandwidth is made larger to a degree depending on where the design is thin special values of this parameter correspond approximately to the ordinary fixed width kernel estimator the smoothing spline and the k nn estimator hence this method offers a synthesis of some classical methods the same method allows estimation of derivatives by using appropriate kernels the influence of this degree of design adaptation on the integrated mean squared error is investigated there is no uniformly optimal solution the optimal solution depends in a complex way on de

complexity of makanin s algorithm the exponent of periodicity is an important factor in estimates of complexity of word unification algorithms we prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1 07d where d is the length of the equation we also give a lower bound 2 0 29d so our upper bound is almost optimal and exponentially better than the original bound 6d 22d4 2 consequently our result implies an exponential improvement of known upper bounds on complexity of word unification algorithms

an ode based method for computing the distance of coprime polynomials to common divisibility the problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor gcd appears in computer algebra signal processing and control theory it has been studied in the literature under the names approximate common divisor varepsilon gcd and distance to uncontrollability existing solution methods use different types of local optimization methods and require a user defined initial approximation in this paper we propose a new method that allows us to include constraints on the coefficients of the polynomials moreover the method proposed in the paper is more robust to the initial approximation than the newton type optimization methods available in the literature our approach consists of two steps 1 reformulate the problem as a problem of determining the structured distance to singularity of an associated sylvester matrix and 2 integrate a system of odes which describes the gradient associated to the functional to be minim

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

identification of linear time invariant systems from multiple experiments a standard assumption for consistent estimation in the errors in variables setting is persistency of excitation of the noise free input signal we relax this assumption by considering data from multiple experiments consistency is obtained asymptotically as the number of experiments tends to infinity the main theoretical and algorithmic difficulties are related to the growing number of to be estimated initial conditions the method proposed in the paper is based on analytic elimination of the initial conditions and optimization over the remaining parameters the resulting estimator is consistent however achieving asymptotically efficiency is an open problem

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

hard equality constrained integer knapsacks we consider the following integer feasibility problem given positive integer numbersa0 a1 a n with gcd a1 a n 1 anda a1 a n does there exist a vectorx z n 0satisfyinga x a0 we prove that if the coefficientsa1 a nhave a certain decomposable structure then the frobenius number associated witha1 a n i e the largest value ofa0 for whicha x a0 does not have a nonnegative integer solution is close to a known upper bound in the instances we consider we takea0 to be the frobenius number furthermore we show that the decomposable structure ofa1 a nmakes the solution of a lattice reformulation of our problem almost trivial since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small for branch and bound such instances are difficult to solve since they are infeasible and have large values ofa0 a i 1 i n we illustrate our results by some computational examples

versions of kernel type regression estimators abstract we explore the aims of and relationships between various kernel type regression estimators to do so we identify two general types of direct kernel estimators differing in their treatment of the nuisance density function associated with regressor variable design we look at the well known gasser muller nadaraya watson and priestley chao methods in this light in the random design case none of these methods is totally adequate and we mention a novel direct kernel method with appropriate properties disadvantages of even the latter idea are remedied by kernel weighted local linear fitting a well known technique that is currently enjoying renewed popularity we see how to fit this approach into our general framework and hence form a unified understanding of how these kernel type smoothers interrelate though the mission of this article is unificatory and even pedagogical the desire for better understanding of superficially different approaches is motivated by the need to improve prac

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

the asymptotic distribution of frobenius numbers the frobenius number f a of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a we show that if a is taken to be random in an expanding d dimensional domain then f a has a limit distribution which is given by the probability distribution for the covering radius of a certain simplex with respect to a d 1 dimensional random lattice this result extends recent studies for d 3 by arnold bourgain sinai and shur sinai ustinov the key features of our approach are a a novel interpretation of the frobenius number in terms of the dynamics of a certain group action on the space of d dimensional lattices and b an equidistribution theorem for a multidimensional farey sequence on closed horospheres

small zeros of quadratic forms we give upper and lower bounds for zeros of quadratic forms in the rational real and p adic fields for example given r 0 s 0 there are infinitely many forms a with integer coefficients in r s variables of the type r s i e equivalent over r to x2 xr2 3 x2 j such that every nontrivial integer zero x has lxl fr 28 where f is the maximum modulus of the coefficients of a

testing integer knapsacks for feasibility abstract we present a new approach for determining whether there exist nonnegative integers x1 x2 xn satisfying a1x1 a2x2 anxn b where a1

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

lattice based extended formulations for integer linear equality systems we study different extended formulations for the set x boldsymbol x in mathbb z n mid boldsymbol a boldsymbol x boldsymbol a boldsymbol x 0 with boldsymbol a in mathbb z m times n in order to tackle the feasibility problem for the set x cap mathbb z n pursuing the work of aardal lenstra et al using the reformulation x boldsymbol x in mathbb z n mid boldsymbol x boldsymbol x 0 boldsymbol q boldsymbol lambda boldsymbol lambda in mathbb z n m our aim is to derive reformulations of the form boldsymbol x in mathbb z n mid boldsymbol p boldsymbol x boldsymbol x 0 boldsymbol t boldsymbol mu boldsymbol mu in mathbb z s with 0 s n m where preferably all the coefficients of p are small compared to the coefficients of a and t in such cases the new variables appear to be good branching directions and in certain circumstances permit one to deduce rapidly that the instance is infeasible we give a polynomial time algorithm for identifying such p t if possible and for the case that a has one row a we analyze the reformulation when s 1 that is one variable is introduced in particular we determine the integer width of the extended formulations in the direction of the variable and derive a lower bound on the frobenius number of a we conclude with some preliminary tests to see if the reformulations are effective when the number s of additional constraints and variables is limited

optimization on a grassmann manifold with application to system identification abstract in this paper we consider the problem of optimization of a cost function on a grassmann manifold this problem appears in system identification in the behavioral setting which is a structured low rank approximation problem we develop an optimization approach based on switching coordinate charts this method reduces the optimization problem on the manifold to an optimization problem in a bounded domain of a euclidean space we compare the proposed approach with state of the art methods based on data driven local coordinates and riemannian geometry and show the connections between the methods compared to the methods based on the local coordinates the proposed approach allows to use arbitrary optimization methods for solving the corresponding subproblems in the euclidean space

small zeros of quadratic forms outside a union of varieties let f be a quadratic form in n 2 variables defined on a vector space vk n over a global field k and zk n be a finite union of varieties defined by families of homogeneous polynomials over k we show that if v z contains a nontrivial zero of f then there exists a linearly independent collection of small height zeros of f in v z where the height bound does not depend on the height of z only on the degrees of its defining polynomials as a corollary of this result we show that there exists a small height maximal totally isotropic subspace w of the quadratic space v f such that w is not contained in z our investigation extends previous results on small zeros of quadratic forms including cassels theorem and its various generalizations the paper also contains an appendix with two variations of siegel s lemma all bounds on height are explicit

on the distribution of points in projective space of bounded height in this paper we consider the uniform distribution of points in compact metric spaces we assume that there exists a probability measure on the borel subsets of the space which is invariant under a suitable group of isometries in this setting we prove the analogue of weyl s criterion and the erdos tur an inequality by using orthogonal polynomials associated with the space and the measure in particular we discuss the special case of pro jective space over completions of number elds in some detail an invariant measure in these projective spaces is introduced and the explicit formulas for the orthogonal polynomials in this case are given finally using the analo gous erdos tur an inequality we prove that the set of all projective points over the number eld with bounded arakelov height is uniformly distributed with respect to the invariant measure as the bound increases