vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

isentropic approximation of the compressible euler system in one space dimension using the stability results of bressan colombo bc for strictly hyperbolic 2 2 systems in one space dimension we prove that the solutions of isentropic and non isentropic euler equations in one space dimension with the respective initial data 0 u0 and 0 u0 theta 0 0 1 remain close as soon as the total variation of 0 u0 is sufficiently small

local minima in cross validation functions summary the method of least squares cross validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research through both theoretical analysis and simulation studies the method involves the minimization of a certain function of the bandwidth one of the less attractive features of this method which has been observed in simulation studies but has not previously been understood theoretically is that rather often the cross validation function has multiple local minima the theoretical results of this paper provide an explanation and quantification of this empirical observation through modelling the cross validation function as a gaussian stochastic process asymptotic analysis reveals that the degree of wiggliness of the cross validation function depends on the underlying density through a fairly simple functional but dependence on the kernel function is much more complicated a simulation study explores the extent to which the asymptotic analysis describes the actual situation our techniques may also be used to obtain other related results e g to show that spurious local minima of the cross validation function are more likely to occur at too small values of the bandwidth rather than at too large values

stochastic mathematical programs with hybrid equilibrium constraints this paper considers a stochastic mathematical program with hybrid equilibrium constraints smphec which includes either here and now or wait and see type complementarity constraints an example is given to describe the necessity to study smphec in order to solve the problem the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints limiting behaviors of the proposed approach are discussed preliminary numerical experiments show that the proposed approach is applicable

one sided cross validation abstract a new method of selecting the smoothing parameters of nonparametric regression estimators is introduced the method termed one sided cross validation oscv has the objectivity of cross validation and statistical properties comparable to those of a plug in rule the new method may be viewed as an application of the prequential model selection method of dawid as such our results identify a situation in which the prequential method is a more efficient model selector than cross validation an example simulations and theoretical results demonstrate the utility of oscv when used with local linear and kernel estimators

equivalence of the euler and lagrangian equations of gas dynamics for weak solutions abstract this paper demonstrates the equivalence of the euler and the lagrangian equations of gas dynamics in one space dimension for weak solutions which are bounded and measurable in eulerian coordinates the precise hypotheses include all known global solutions on r r in particular solutions containing vacuum states zero mass density are included furthermore there is a one to one correspondence between the convex extensions of the two systems and the corresponding admissibility criteria are equivalent in the presence of a vacuum the definition of weak solution for the lagrangian equations must be strengthened to admit test functions which are discontinuous at the vacuum as an application we translate a large data existence result of diperna for the euler equations for isentropic gas dynamics into a similar theorem for the lagrangian equations

adaptive methods for linear programming decoding detectability of failures of linear programming lp decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying lp problem in this paper we make a first step in studying this method and show that by starting from a simple lp problem and adaptively adding the necessary constraints the complexity of lp decoding can be significantly reduced in particular we observe that with adaptive lp decoding the sizes of the lp problems that need to be solved become practically independent of the density of the parity check matrix we further show that adaptively adding extra constraints such as constraints based on redundant parity checks can provide large gains in the performance

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

optimal control of cell mass and maturity in a model of follicular ovulation in this paper we study optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles changes in the age and maturity of follicular cells are described by a two dimensional conservation law where the control terms act on the velocities the control problem consists in optimizing the follicular cell resources so that the follicular maturity reaches a maximal value in fixed time formulating the optimal control problem within a hybrid framework we prove necessary optimality conditions in the form of the hybrid maximum principle then we derive the optimal strategy and show that there exists at least one optimal bang bang control with one single switching time

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

on the controllability of the non isentropic 1 d euler equation in this paper we examine the question of the boundary controllability of the one dimensional non isentropic euler equation for compressible polytropic gas in the context of weak entropy solutions we consider the system in eulerian coordinates and the one in lagrangian coordinates we obtain for both systems a result of controllability toward constant states with the limitation gamma frac 5 3 on the adiabatic constant for the lagrangian system the solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation

pseudocodewords of parity check codes over fields of prime cardinality this paper considers pseudocodewords of lowdensity parity check codes over alphabets with prime cardinality p for use over the p ary symmetric channel pseudocodewords are decoding algorithm outputs that may not be legitimate codewords here we consider pseudocodewords arising from graph cover decoding and linear programming decoding for codes over the binary alphabet such pseudocodewords correspond to rational points of the fundamental polytope they can be characterized via the fundamental cone which is the conic hull of the fundamental polytope the pseudocodewords are precisely those integer vectors within the fundamental cone that reduce modulo 2 to a codeword in this paper we determine a set of conditions that pseudocodewords of codes over f p the finite field of prime cardinality p must satisfy to do so we introduce a class of critical multisets and a mapping which associates a real number to each pseudocodeword over f p the real numbers associated with pseudocodewords are subject to lower bounds imposed by the critical multisets the inequalities are given in terms of the parity check matrix entries and critical multisets this gives a necessary and sufficient condition for pseudocodewords of codes over f 2 and f 3 and a necessary condition for those over larger alphabets in addition irreducible pseudocodewords of codes over f 3 are found as a hilbert basis for the lifted fundamental cone

unique continuation for the linearized benjamin bona mahony equation with space dependent potential in this paper we analyze a unique continuation problem for the linearized benjamin bona mahony equation with space dependent potential in a bounded interval with dirichlet boundary conditions the underlying cauchy problem is a characteristic one we prove two unique continuation results by means of spectral analysis and the generalized eigenvector expansion of the solution instead of the usual holmgren type method or carleman type estimates it is found that the unique continuation property depends very strongly on the nature of the potential and in particular on its zero set and not only on its boundedness or integrability properties

the admm penalized decoder for ldpc codes linear programming lp decoding for low density parity check codes was introduced by feldman et al and has been shown to have theoretical guarantees in several regimes furthermore it has been reported in the literature via simulation and via instanton analysis that lp decoding displays better error rate performance at high signal to noise ratios snr than does belief propagation bp decoding however at low snrs lp decoding is observed to have worse performance than bp in this paper we seek to improve lp decoding at low snrs while maintaining lp decoding s high snr performance our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers admm algorithm to a set of non convex optimization problems these non convex problems are constructed by adding a penalty term to the objective of lp decoding the goal of the penalty is to make pseudocodewords which are non integer vertices of the lp relaxation more costly we name this class of decoders admm penalized decoders for low and moderate snrs we simulate admm penalized decoding with ell 1 and ell 2 penalties we find that these decoders can outperform both bp and lp decoding for high snrs where it is difficult to obtain data via simulation we use an instanton analysis and find that asymptotically admm penalized decoding performs better than bp but not as well as lp unfortunately since admm penalized decoding is not a convex program we have not been successful in developing theoretical guarantees however the non convex program can be approximated using a sequence of linear programs an approach that yields a reweighted lp decoder we show that a two round reweighted lp decoder has an improved theoretical recovery threshold when compared with lp decoding in addition we find via simulation that reweighted lp decoding significantly attains lower error rates than lp decoding at low snrs

on the initial boundary value problem for temple systems this paper is devoted to the proof of the well posedness of the initial boundary value problem for a system of conservation laws of temple type no assumption on the monotonicity of the eigenvalues along the lax curves is assumed and the total variation of the initial data is not required to be small the solution is shown to be a lipschitz function of the initial and boundary data of time and of the boundary profile as well

analysis and control of a scalar conservation law modeling a highly re entrant manufacturing system abstract in this paper we study a scalar conservation law that models a highly re entrant manufacturing system as encountered in semi conductor production as a generalization of coron et al 2010 14 the velocity function possesses both the local and nonlocal character we prove the existence and uniqueness of the weak solution to the cauchy problem with initial and boundary data in l we also obtain the stability continuous dependence of both the solution and the out flux with respect to the initial and boundary data finally we prove the existence of an optimal control that minimizes in the l p sense with 1 p the difference between the actual out flux and a forecast demand over a fixed time period

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho

population shape regression from random design data regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent regressor variable and in particular it is applicable to the study of anatomical growth and shape change when the underlying process can be modeled by parameters in a euclidean space classical regression techniques hardle applied nonparametric regression 1990 wand and jones kernel smoothing 1995 are applicable and have been studied extensively however recent work suggests that attempts to describe anatomical shapes using flat euclidean spaces undermines our ability to represent natural biological variability fletcher et al ieee trans med imaging 23 8 995 1005 2004 grenander and miller q appl math 56 4 617 694 1998 in this paper we develop a method for regression analysis of general manifold valued data specifically we extend nadaraya watson kernel regression by recasting the regression problem in terms of frechet expectation although this method is quite general our driving problem is the study anatomical shape change as a function of age from random design image data we demonstrate our method by analyzing shape change in the brain from a random design dataset of mr images of 97 healthy adults ranging in age from 20 to 79 years to study the small scale changes in anatomy we use the infinite dimensional manifold of diffeomorphic transformations with an associated metric we regress a representative anatomical shape as a function of age from this population

on the boundary controllability of first order hyperbolic systems abstract we discuss some results concerning the boundary controllability and stabilizability of a hyperbolic system of conservation laws u t f u x 0 t 0 x 0 1 where we regard the boundary data or a partial number of their components as boundary input controls in particular we consider the problem of the global exact boundary controllability of a first order linear hyperbolic system with constant coefficients relative to the linear boundary conditions under generic orthogonality assumptions on the boundary and control matrices and assuming a non resonance condition of the characteristic speeds we show that one can steer in finite time the solution from any initial condition l 1 to any terminal state l 1 even in the case where only a partial control of the boundary values is available

a data dependent approach to density estimation a data dependent approach to density estimation is proposed here the proposed method requires boundedness and some weak integrability condition on the unknown density but not any assumption of smoothness applications to histogram kernel spline and orthogonal series methods are discussed

new blow up rates for fast controls of schrodinger and heat equations abstract we consider the null controllability problem for the schrodinger and heat equations with boundary control we concentrate on short time or fast controls we improve recent estimates see l miller geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 202 226 l miller how violent are fast controls for schrodinger and plate vibrations arch ration mech anal 172 2004 429 456 l miller controllability cost of conservative systems resolvent condition and transmutation j funct anal 218 2005 425 444 l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 on the norm of the operator associating to any initial state the minimal norm control driving the system to zero our main results concern the schrodinger and heat equations in one space dimension they yield new estimates concerning window problems for series of exponentials as described in t i seidman the coefficient map for certain exponential sums nederl akad wetensch indag math 48 1986 463 478 and in t i seidman s a avdonin s a ivanov the window problem for series of complex exponentials j fourier anal appl 6 2000 233 254 these results are used following l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 to deal with the case of several space dimensions

the sharp form of oleĭnik s entropy condition in several space variables we investigate the conditions under which the volpert kruzkov solution of a single conservation law in several space variables with flux f will satisfy the simplified entropy condition div f u s l t and when this condition guarantees uniqueness for given lx cauchy data we show that when f is c1 our condition guarantees uniqueness iff f is isotropic and that for such f the volpert kruzkov solution always satisfies our condition

exact boundary controllability for 1 d quasilinear hyperbolic systems with a vanishing characteristic speed the general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of the system do not vanish this paper deals with exact boundary controllability when this is not the case some important models are also shown as applications of the main result the strategy uses the return method which allows one in certain situations to recover nonzero characteristic speeds

energy decay rate of wave equations with indefinite damping abstract we consider the one dimensional wave equation with an indefinite sign damping and a zero order potential term using a shooting method we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients in addition if the damping coefficient is more positive than negative we prove that the energy of system decays uniformly exponentially to zero this sharp result generalizes a previous work of freitas and zuazua 1996

controllability and stabilization of a conservation law modeling a highly re entrant manufacturing system abstract in this paper we study the controllability and stabilization for a scalar conservation law modeling a highly re entrant manufacturing system with local and nonlocal velocity we prove a local state controllability result i e there exists a control that drives the solution from any given initial condition to any desired final condition in a certain time period provided that the initial and final data are both close to the origin a local result on nodal profile controllability is also given i e for any initial data and any given out flux in a neighborhood of the origin there exists a control under which the solution starts from any initial data reaches exactly any desired out flux over a fixed time period besides using a lyapunov function approach we can stabilize the system to the origin exponentially by output feedback control

impact of redundant checks on the lp decoding thresholds of ldpc codes feldman et al 11 asked whether the performance of the linear programming lp decoder can be improved by adding redundant parity checks to tighten the lp relaxation we prove in this paper that for low density parity check codes even if we include all redundant parity checks asymptotically there is no gain in the lp decoder threshold on the binary symmetric channel under certain conditions on the base tanner graph first we show that if the tanner graph has bounded check degree and satisfies a condition which we call asymptotic strength then including high degree redundant parity checks in the lp does not significantly improve the threshold of the lp decoder in the following sense for each constant delta 0 there is a constant k 0 such that the threshold of the lp decoder containing all redundant checks of degree at most k improves by at most delta upon adding to the lp all redundant checks of degree larger than k we conclude that if the graph satisfies an additional condition which we call rigidity then including all redundant checks does not improve the threshold of the base lp we call the graph asymptotically strong if the lp decoder corrects a constant fraction of errors even if the log likelihood ratios of the correct variables are arbitrarily small by building on a construction due feldman et al 9 and its recent improvement by viderman 24 we show that asymptotic strength follows from sufficiently large variable to check expansion we also give a geometric interpretation of asymptotic strength in terms pseudocodewords we call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity under the assumptions that the graph girth is logarithmic and the minimum check degree is at least 3 rigidity is equivalent to the nondegeneracy property that adding at least logarithmically many checks does not give a constant weight check we argue that nondegeneracy is a typical property of random check regular tanner graphs

equivalence of smoothing parameter selectors in density and intensity estimation abstract kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary poisson process in each case the amount of smoothing controlled by the bandwidth that is smoothing parameter is crucial to the performance of the estimator bandwidth selection by cross validation has been widely studied in the context of density estimation a bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary cox process this article shows that these two methods each select the same bandwidth even though they are motivated in much different ways in addition to providing further justification of each method this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation a benefit for intensity estimation is that this equivalence makes it clear how

sufficient conditions for local existence via glimm s scheme for large bv data abstract using glimm s scheme sufficient conditions are derived for the global existence of a weak solution to a strictly hyperbolic genuinely nonlinear system of partial differential equations in one space dimension when the initial data is a small bv perturbation of a solvable riemann problem by using the finite propagation speed of the system this yields a local existence theorem for arbitrary bv initial data that satisfies the above mentioned conditions at all large jumps the case of linearly degenerate fields is also treated and the results are applied to the p system and to the 1 d nonisentropic gas law euler equations

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

guessing facets polytope structure and improved lp decoder we investigate the structure of the polytope underlying the linear programming lp decoder introduced by feldman karger and wainwright we first show that for expander codes every fractional pseudocodeword always has at least a constant fraction of nonintegral bits we then prove that for expander codes the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword we further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as lp decoding that provably performs better the method is very simple it first applies ordinary lp decoding and when it fails it proceeds by guessing facets of the polytope and then resolving the linear program on these facets while the lp decoder succeeds only if the ml codeword has the highest likelihood over all pseudocodewords we prove that the proposed algorithm when applied to suitable expander codes succeeds unless there exists a certain number of pseudocodewords all adjacent to the ml codeword on the lp decoding polytope and with higher likelihood than the ml codeword we then describe an extended algorithm still with polynomial complexity that succeeds as long as there are at most polynomially many pseudocodewords above the ml codeword

on the optimization of the initial boundary value problem for a conservation law abstract in the present note the theory of shift differentiability for the cauchy problem is extended to the case of an initial boundary value problem for a conservation law this result allows to exhibit an euler lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

the approximate maximum likelihood certificate the confidence in the reliability of a codeword output by some not necessarily optimal decoding algorithm is discussed a new property which relies on the linear programming lp decoder the approximate maximum likelihood certificate amlc is introduced to address this issue as follows first the channel output vector is decoded by some symmetric decoder d e g belief propagation or min sum algorithm decoding second the channel output vector is decoded by lp decoding third if the decoding result of d is a codeword its lp value is compared to the lp value of the lp decoding result the latter need not be a codeword if these two values are close the amlc holds using upper bounding techniques we show that the conditional frame error probability given that the amlc holds is with some degree of confidence below a threshold in channels with low noise this threshold is orders of magnitude lower than the simulated frame error rate and our bound holds with a very high degree of confidence this is in stark contrast with standard monte carlo simulation which would require excessively long runs to demonstrate like performance when the amlc holds our approach thus provides the decoder with extra error detection capability which is especially important in applications requiring high data integrity

initial boundary value problem and asymptotic stabilization of the camassa holm equation on an interval abstract we investigate the nonhomogeneous initial boundary value problem for the camassa holm equation on an interval we provide a local in time existence theorem and a weak strong uniqueness result next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law

null controllability of the burgers system with distributed controls this paper is devoted to present some positive and negative controllability results for the viscous burgers equation more precisely in the context of null controllability with distributed controls we present sharp estimates of the minimal time of controllability t r of initial data of l 2 norm less or equal to r in particular we see that global null controllability does not hold in general unless the control is exerted

scalar non linear conservation laws with integrable boundary data where u u t x is the state variable u are integrable possibly unbounded initial and boundary data and f is assumed to be a superlinear strictly convex function for problems of this type since classical solutions develop discontinuities in nite time no matter how smooth their initial and boundary data it is natural to consider weak solutions satisfying the usual entropy conditions 13 15 u t x u t x t x 0 1 4

iterative approximate linear programming decoding of ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of low density parity check ldpc codes is considered an iterative algorithm similar to min sum and belief propagation for efficient approximate solution of this problem was proposed by vontobel and koetter in this paper the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose in particular we are interested in obtaining a feasible vector in the lp decoding problem that is close to optimal in the following sense the distance normalized by the block length between the minimum and the objective function value of this approximate solution can be made arbitrarily small it is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length combined with previous results that have shown that the lp decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity this is achieved by first applying the iterative lp decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits and then correcting the remaining errors using some standard method these conclusions are also extended to generalized ldpc codes

optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data we consider the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic conservation laws in particular we consider initial boundary value problems where the initial and boundary data switch between different c 1 functions at certain switching points and both the functions and the switching points are controlled we show that the control to state mapping is differentiable in a certain generalized sense which implies frechet differentiability with respect to the control functions and the switching points for the composition with a tracking type functional even in the presence of shocks we also present an adjoint based formula for the gradient of the reduced objective functional

a complex analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit abstract we revisit a result by coron and guerrero stating that the one dimensional transport diffusion equation u t m u x e u x x 0 in 0 t 0 l controlled by the left dirichlet boundary value is zero controllable at a bounded cost as e 0 when t 4 3 l m if m 0 and when t 57 2 l m if m 0 by a completely different method relying on complex analysis we prove that this still holds when t 4 2 l m if m 0 and when t 6 1 l m if m 0

ordering and selecting components in multivariate or functional data linear prediction the problem of component choice in regression based prediction has a long history the main cases where important choices must be made are functional data analysis and problems in which the explanatory variables are relatively high dimensional vectors indeed principal component analysis has become the basis for methods for functional linear regression in this context the number of components can also be interpreted as a smoothing parameter and so the viewpoint is a little different from that for standard linear regression however arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention we give a theoretical argument which is applicable in a wide variety of settings justifying the conventional approach although our result is of minimax type it is not asymptotic in nature it holds for each sample size motivated by the insight that is gained from this analysis we give theoretical and numerical justification for cross validation choice of the number of components that is used for prediction in particular we show that cross validation leads to asymptotic minimization of mean summed squared error in settings which include functional data analysis copyright c 2010 royal statistical society

on the attainable set for a scalar nonconvex conservation law we deal with a cauchy problem for a scalar conservation law in one space dimension the flux function is assumed to be nonconvex in particular to have a single inflection point we consider a compactly supported initial datum and regard it as a control the main result of the paper states sufficient conditions for a function v to be attained at a fixed time t by a trajectory of the conservation law

on the attainable set for temple class systems with boundary controls consider the initial boundary value problem for a strictly hyperbolic genuinely nonlinear temple class system of conservation laws begin equation label eq 1 u t f u x 0 qquad u 0 x ov u x qquad left begin array ll u t a widetilde u a t 2pt u t b widetilde u b t end array right end equation on the domain omega t x in r 2 t geq 0 a le x leq b we study the mixed problem ref eq 1 from the point of view of control theory taking the initial data overline u fixed and regarding the boundary data widetilde u a widetilde u b as control functions that vary in prescribed sets u a u b of li boundary controls in particular we consider the family of configurations a t doteq u t cdot u mbox is a sol to ref eq 1 widetilde u a in u a widetilde u b in u b that can be attained by the system at a given time t 0 and we give a description of the attainable set a t in terms of suitable oleinik type conditions we also establish closure and compactness of the set a t in the lu topology

global controllability of nonviscous and viscous burgers type equations in this paper we deal with both nonviscous and viscous burgers type equations on a bounded interval we study the global exact controllability of these equations when we have three controls one control is the right member of the equation and is constant with respect to the space variable the two others are the boundary values for a first time we are interested in nonviscous burgers type equations and we prove their global exact controllability for every time t 0 with such controls thanks to the return method then we prove the global exact controllability of the viscous burgers equation for every time t 0

controllability for a scalar conservation law with nonlocal velocity abstract in this paper we study the state controllability and nodal profile controllability for a scalar conservation law with a nonlocal velocity that models a highly re entrant manufacturing system as encountered in semi conductor production we first prove a local state controllability result i e there exists a control that drives the solution from any given initial data to any desired final data in a certain time period provided that the initial and final data are both close to a given equilibrium 0 we also obtain a global state controllability result for the same system where there is no limitation on the distance between the initial and final data finally we prove a nodal profile controllability result i e there exists a control under which the solution starts from any initial data reaches exactly any given out flux over a fixed time period

controllability and asymptotic stabilization of the camassa holm equation we investigate the problems of exact controllability and asymptotic stabilization of the camassa holm equation on the circle by means of a distributed control the results are global and in particular the control prevents the solution from blowing up

comparison of data driven bandwidth selectors abstract this article compares several promising data driven methods for selecting the bandwidth of a kernel density estimator the methods compared are least squares cross validation biased cross validation and a plug in rule the comparison is done by asymptotic rate of convergence to the optimum and a simulation study it is seen that the plug in bandwidth is usually most efficient when the underlying density is sufficiently smooth but is less robust when there is not enough smoothness present we believe the plug in rule is the best of those currently available but there is still room for improvement

a new linear programming approach to decoding linear block codes in this paper we propose a new linear programming formulation for the decoding of general linear block codes different from the original formulation given by feldman the number of total variables to characterize a parity check constraint in our formulation is less than twice the degree of the corresponding check node the equivalence between our new formulation and the original formulation is proven the new formulation facilitates to characterize the structure of linear block codes and leads to new decoding algorithms in particular we show that any fundamental polytope is simply the intersection of a group of the so called minimum polytopes and this simplified formulation allows us to formulate the problem of calculating the minimum hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables we then propose a branch and bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem in addition we prove that for the family of single parity check spc product codes the fractional distance and the pseudodistance are both equal to the minimum distance finally we propose an efficient algorithm for decoding spc product codes with low complexity and maximum likelihood ml decoding performance

extent to which least squares cross validation minimises integrated square error in nonparametric density estimation let ho o and c be the windows which minimise mean integrated square error integrated square error and the least squares cross validatory criterion respectively for kernel density estimates it is argued that o not ho should be the benchmark for comparing different data driven approaches to the determination of window size asymptotic properties of ho o and c o and of differences between integrated square errors evaluated at these windows are derived it is shown that in comparison to the benchmark o the observable window c performs as well as the so called optimal but unattainable window ho to both first and second order

uniform controllability of scalar conservation laws in the vanishing viscosity limit we deal with viscous perturbations of scalar conservation laws on a bounded interval with a general flux function f and a small dissipation coefficient varepsilon acting on this system on both endpoints of the interval we prove global exact controllability to constant states with nonzero speed more precisely we construct boundary controls so that the solution is driven to the targeted constant state and we moreover require these controls to be uniformly bounded as varepsilon rightarrow 0 in an appropriate space for general nonconvex flux functions this can be done for sufficiently large time and for convex fluxes f we have a precise estimate on the minimal time needed to control

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

decomposition methods for large scale lp decoding when binary linear error correcting codes are used over symmetric channels a relaxed version of the maximum likelihood decoding problem can be stated as a linear program lp this lp decoder can be used to decode error correcting codes at bit error rates comparable to state of the art belief propagation bp decoders but with significantly stronger theoretical guarantees however lp decoding when implemented with standard lp solvers does not easily scale to the block lengths of modern error correcting codes in this paper we draw on decomposition methods from optimization theory specifically the alternating direction method of multipliers admm to develop efficient distributed algorithms for lp decoding the key enabling technical result is a two slice characterization of the parity polytope the polytope formed by taking the convex hull of all codewords of the single parity check code this new characterization simplifies the representation of points in the polytope using this simplification we develop an efficient algorithm for euclidean norm projection onto the parity polytope this projection is required by the admm decoder and its solution allows us to use lp decoding with all its theoretical guarantees to decode large scale error correcting codes efficiently we present numerical results for ldpc codes of lengths more than 1000 the waterfall region of lp decoding is seen to initiate at a slightly higher snr than for sum product bp however an error floor is not observed for lp decoding which is not the case for bp our implementation of lp decoding using the admm executes as fast as our baseline sum product bp decoder is fully parallelizable and can be seen to implement a type of message passing with a particularly simple schedule

improved linear programming decoding of ldpc codes and bounds on the minimum and fractional distance we examine ldpc codes decoded using linear programming lp four contributions to the lp framework are presented first a new method of tightening the lp relaxation and thus improving the lp decoder is proposed second we present an algorithm which calculates a lower bound on the minimum distance of a specific code this algorithm exhibits complexity which scales quadratically with the block length third we propose a method to obtain a tight lower bound on the fractional distance also with quadratic complexity and thus less than previously existing methods finally we show how the fundamental lp polytope for generalized ldpc codes and nonbinary ldpc codes can be obtained

on the uniform controllability of the burgers equation in this paper we deal with the viscous burgers equation with a small dissipation coefficient nu we prove the global exact controllability property to nonzero constant states that is to say the possibility of finding boundary values such that the solution of the associated burgers equation is driven to a constant state the main objective of this paper is to do so with control functions whose norms in an appropriate space are bounded independently of nu which belongs to a suitably small interval this result is obtained for a sufficiently large time

convergence of the godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices convergence of approximate solutions derived by the godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices is established by using the compensated compactness method a global existence theorem is shown and a numerical method for the computation of the physical global solution of this model is provided by this approach

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

exact controllability of scalar conservation laws with an additional control in the context of entropy solutions in this paper we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions with the boundary data and a source term depending only on the time as controls we provide sufficient conditions for a state to be reachable in arbitrary small time to do so we introduce a slightly modified wave front tracking algorithm

on the global null controllability of a navier stokes system with navier slip boundary conditions abstract in this paper we deal with a two dimensional navier stokes system in a rectangle with navier slip boundary conditions on the horizontal sides we establish the global null controllability of the system by controlling the normal component and the vorticity of the velocity on the vertical sides the linearized control system around zero is controllable but one does not know how to deduce global controllability results for the nonlinear system our proof uses the return method together with a local exact controllability result by fursikov and imanuvilov

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

sharp estimates of the one dimensional boundary control cost for parabolic systems and application to the n dimensional boundary null controllability in cylindrical domains in this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form omega 0 pi times omega 2 with omega 2 a smooth domain of mathbb r n 1 n 1 when the control is exerted on 0 times omega 2 with omega 2 subset omega 2 we obtain a necessary and sufficient condition that completely characterizes the null controllability this result is obtained through the lebeau robbiano strategy and requires an upper bound of the cost of the one dimensional boundary null control on 0 pi the latter is obtained using the moment method and it is shown to be bounded by ce c t when t goes to 0

a comparison of cross validation techniques in density estimation in the setting of nonparametric multivariate density estimation theorems are established which allow a comparison of the kullback leibler and the least squares cross validation methods of smoothing parameter selection the family of delta sequence estimators including kemel orthogonal series histogram and histospline estimators is considered these theorems also show that either type of cross validation can be used to compare different estimators e g kernel versus orthogonal series 1 introduction consider the problem of trying to estimate a d dimensional probability density function f x using a random sample xl xn from f most proposed estimators of f depend on a smoothing parameter say x e dr whose selection is crucial to the performance of the estimator in this paper for the large class of delta sequence estimators theorems are obtained which allow comparison of two smoothing parameter selectors which are known to be asymptotically optimal an important consequence of these results is that either smoothing parameter selector may be used for a data based comparison of two density estimators for example kernel versus orthogonal series another attractive feature of these results is that they are set in a quite general framework special cases of which provide simpler proofs of several recent asymptotic optimality results in sections 2 and 3 the family of delta sequence estimators and the smoothing parameter selectors are given the theorems are stated in section 4 with some remarks in section 5 the rest of the paper consists of proofs

an efficient pseudocodeword search algorithm for linear programming decoding of ldpc codes in linear programming lp decoding of a low density parity check ldpc code one minimizes a linear functional with coefficients related to log likelihood ratios over a relaxation of the polytope spanned by the codewords in order to quantify lp decoding it is important to study vertexes of the relaxed polytope so called pseudocodewords we propose a technique to heuristcally create a list of pseudocodewords close to the zero codeword and their distances our pseudocodeword search algorithm starts by randomly choosing configuration of the noise the configuration is modified through a discrete number of steps each step consists of two substeps one applies an lp decoder to the noise configuration deriving a pseudocodeword and then finds configuration of the noise equidistant from the pseudocodeword and the zero codeword the resulting noise configuration is used as an entry for the next step the iterations converge rapidly to a pseudocodeword neighboring the zero codeword repeated many times this procedure is characterized by the distribution function of the pseudocodeword effective distance the efficiency of the procedure is demonstrated on examples of the tanner code and margulis codes operating over an additive white gaussian noise awgn channel

exact boundary controllability for quasi linear hyperbolic systems using a result on the existence and uniqueness of the semiglobal c1 solution to the mixed initial boundary value problem for first order quasi linear hyperbolic systems with general nonlinear boundary conditions we establish the exact boundary controllability for quasi linear hyperbolic systems if the c1 norm of initial and final states is small enough

quantifying epidemiologic risk factors using non parametric regression model selection remains the greatest challenge logistic regression is widely used to estimate relative risks odds ratios from case control studies but when the study exposure is continuous standard parametric models may not accurately characterize the exposure response curve semi parametric generalized linear models provide a useful extension in these models the exposure of interest is modelled flexibly using a regression spline or a smoothing spline while other variables are modelled using conventional methods when coupled with a model selection procedure based on minimizing a cross validation score this approach provides a non parametric objective and reproducible method to characterize the exposure response curve by one or several models with a favourable bias variance trade off we applied this approach to case control data to estimate the dose response relationship between alcohol consumption and risk of oral cancer among african americans we did not find a uniquely best model but results using linear cubic and smoothing splines were consistent there does not appear to be a risk free threshold for alcohol consumption vis a vis the development of oral cancer this finding was not apparent using a standard step function model in our analysis the cross validation curve had a global minimum and also a local minimum in general the phenomenon of multiple local minima makes it more difficult to interpret the results and may present a computational roadblock to non parametric generalized additive models of multiple continuous exposures nonetheless the semi parametric approach appears to be a practical advance

local controllability to stationary trajectories of a burgers equation with nonlocal viscosity abstract this article studies the local controllability to trajectories of a burgers equation with nonlocal viscosity by linearization we are led to an equation with a non local term whose controllability properties are analyzed by using fourier decomposition and biorthogonal techniques once the existence of controls is proved and the dependence of their norms with respect to the time is established for the linearized model a fixed point method allows us to deduce the result for the nonlinear initial problem

achieving near perfect classification for functional data summary we show that in functional data classification problems perfect asymptotic classification is often possible making use of the intrinsic very high dimensional nature of functional data this performance is often achieved by linear methods which are optimal in important cases these results point to a marked contrast between classification for functional data and its counterpart in conventional multivariate analysis where the dimension is kept fixed as the sample size diverges in the latter setting linear methods can sometimes be quite inefficient and there are no prospects for asymptotically perfect classification except in pathological cases where for example a variance vanishes by way of contrast in finite samples of functional data good performance can be achieved by truncated versions of linear methods truncation can be implemented by partial least squares or projection onto a finite number of principal components using in both cases cross validation to determine the truncation point we establish consistency of the cross validation procedure

on the boundary control of systems of conservation laws this paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws we prove a general result on the asymptotic stabilization of a system near a constant state on the other hand we give an example showing that exact controllability in finite time cannot be achieved in general

iterative linear programming decoding of nonbinary ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of nonbinary low density parity check ldpc codes is considered and an iterative lp decoding algorithm is presented results that were previously derived for binary ldpc codes are extended to the nonbinary case both simple and generalized nonbinary ldpc codes are considered it is shown how the algorithm can be implemented efficiently using a finite field fast fourier transform then the convergence rate of the algorithm is analyzed the complexity of the algorithm scales linearly in the block length and it can approximate up to an arbitrarily small relative error the objective function of the exact lp solution when applied to a typical code from an appropriate nonbinary ldpc code ensemble the algorithm can correct a constant fraction of errors in linear in the block length computational complexity computer experiments with the new iterative lp decoding algorithm show that in the error floor region it can have better performance compared to belief propagation decoding with similar computational requirements

on the attainable set for scalar nonlinear conservation laws with boundary control we consider the initial value problem with boundary control for a scalar nonlinear conservation law begin equation u t f u x 0 qquad qquad u 0 x 0 qquad u cdot 0 tilde u in cal u tag ast end equation on the domain omega t x in real 2 t geq 0 x geq 0 here u u t x is the state variable cal u is a set of bounded boundary data regarded as controls and f is assumed to be strictly convex we give a characterization of the set of attainable profiles at a fixed time t 0 and at a fixed point bar x 0 begin equation begin aligned rag u t cdot u hbox is a solution of ast ragx u cdot bar x u hbox is a solution of ast end aligned qquad quad cal u ellein bbb r end equation moreover we prove that rag and ragx are compact subsets of elleuno and elleuno loc respectively whenever cal u is a set of controls which pointwise satisfy closed convex constraints together with some additional integral inequalities

global solutions to the isothermal euler poisson system with arbitrarily large data abstract we prove the global existence of a solution to the euler poisson system with arbitrarily large data in a one dimensional geometry the pressure law we consider is deduced from an isothermal assumption for the electrons gas in this case nishida has already pointed out that the linear part of the glimm functional is decreasing with respect to time using a glimm scheme he used this property to construct globally defined weak solutions for the euler system with arbitrary large data we follow his outline of proof here a new difficulty arises with the source term due to the electric field a key point is that the glimm scheme is almost conservative this quasi conservation of charge leads to a uniform estimate of the total variation of the electric field this estimate allows to prove the convergence of the scheme

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

null controllability of linear heat and wave equations with nonlocal spatial terms in this paper we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms under some analyticity assumptions on the corresponding kernel we show that the equations are controllable we employ compactness uniqueness arguments in a suitable functional setting an argument that is harder to apply for heat equations because of its very strong time irreversibility some possible extensions and open problems concerning other nonlocal systems are presented

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme