the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the quermassintegral inequalities for k convex starshaped domains abstract we give a proof of the isoperimetric inequality for quermassintegrals of non convex starshaped domains using a result of gerhardt c gerhardt flow of nonconvex hypersurfaces into spheres j differential geometry 32 1990 299 314 and urbas j urbas on the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures math z 205 1990 355 372 on an expanding geometric curvature flow

prediction in lms type adaptive algorithms for smoothly time varying environments the aim of this correspondence is to improve the performance of the least mean square lms and normalized lms nlms adaptive algorithms in tracking of time varying models a new procedure for estimation of weight increments for including in the lms type adaptive algorithms is proposed this procedure applies a simple smoothing on the increment of the estimated weights to estimate the speed of weights the estimated speeds are then used to predict the weights for the next iteration the efficiency of the algorithm is confirmed by simulation results the algorithm has a very low order of arithmetic complexity moreover this procedure could be combined with a wide class of adaptive filters e g rls gradient lattice algorithm etc to improve their behaviors the proposed algorithm is obtained by simplifying a kalman filter to this end a markov model of second order is considered for the weight vector this model shows that the estimation of parameter increments inferred from the predicted parameters improves the tracking performance

matroid representations and free arrangements we show that terao s conjecture freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid holds over fields with at most four elements however an example demonstrates that the field characteristic has to be fixed for this

concavity properties for elliptic free boundary problems abstract we prove some concavity properties connected to nonlinear bernoulli type free boundary problems in particular we prove a brunn minkowski inequality and an urysohn s type inequality for the bernoulli constant and we study the behavior of the free boundary with respect to the given boundary data moreover we prove a uniqueness result regarding the interior problem

on the case of equality in the brunn minkowski inequality for capacity suppose that and 1 are convex open subsets of rn denote their convex combination by the brunn minkowski inequality says that vol t 1 t vol 0 1 n t vol for 0 t l moreover if there is equality for some t other than an endpoint then the domains 1 and 0 are translates and dilates of each other borell proved an analogue of the brunn minkowski inequality with capacity defined below in place of volume borel s theorem b says theorem a let t t 1 1 t 0 be a convex combination of two convex subsets of rn n 3 then cap the main purpose of this note is to prove

serrin type overdetermined problems an alternative proof we prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations namely the hessian equations in the case of the poisson equation our proof is alternative to the proofs proposed by serrin moving planes and by weinberger moreover our proof makes no direct use of the maximum principle while it sheds light on a relation between the serrin problem and the isoperimetric inequality

changing spectrum estimation abstract an autoregressive ar model with time varying coefficients is used for the modeling of non stationary covariance time series a stochastically perturbed linear constraint model is presented as a model of time varying ar coefficients the overall model for the time series is fitted by using the kalman filter and akaike s aic criterion an estimate of a changing spectrum is obtained by the fitted model and the fixed interval smoother examples are given to illustrate the procedure

evolutionary periodogram for nonstationary signals presents a novel estimator for the time dependent spectrum of a nonstationary signal by modeling the signal at any given frequency as having a time varying amplitude accurately represented by an orthonormal basis expansion the authors are able to compute a minimum mean squared error estimate of this time varying amplitude repeating the process over all frequencies they obtain a power distribution as a function of time and frequency that is consistent with the wold cramer evolutionary spectrum based on the model assumptions the authors develop the evolutionary periodogram ep for nonstationary signals an estimator analogous to the periodogram used in the stationary case they also derive the time frequency resolution of the new estimator the approach is free of some of the drawbacks of the bilinear distributions and of the short time fourier transform spectral estimates it is guaranteed to produce nonnegative spectra without the cross term behavior of the bilinear distributions and it does not require windowing of data in the time domain examples illustrating the new estimator are given

a formula for the characteristic polynomial of an arrangement on etablit une formule permettant de calculer le polynome caracteristique d un arrangement constitue d un ensemble fini d hyperplans en termes de serie de poincarre de modules gradues

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

convexity of solutions and brunn minkowski inequalities for hessian equations in r3 by using minkowski addition of convex functions we prove convexity and rearrangement properties of solutions to some hessian equations in r 3 and brunn minkowski and isoperimetric inequalities for related functionals

variational formula and overdetermined problem for the principle eigenvalue of k hessian operator abstract we first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k hessian operator and then prove serrin type symmetry result for our overdetermined problems

adaptive control of a simple time varying system adaptive control of a first order randomly varying stochastic process is considered several authors have treated this problem using either ergodic theory for markov processes or martingale limit theorems in both cases some initial variance calculations and bounds evaluations must be performed in order to apply these techniques and establish the sample path results generally considered in adaptive control these prior calculations are performed in the more general case where the single parameter follows a random walk and for an identification algorithm requiring no a priori knowledge of noise characteristics

a microscopic convexity principle for nonlinear partial differential equations we study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations under certain general structure condition we establish that the rank of hessian 2u is of constant rank for any convex solution u of equation f 2u u u x 0 the similar result is also proved for parabolic equations some of geometric applications are also discussed

an ilc based adaptive control for general stochastic systems with strictly decreasing entropy in this paper a new method for adaptive control of general nonlinear and non gaussian unknown stochastic systems has been proposed the method applies the minimum entropy control scheme to decrease the closed loop randomness of the output under an iterative learning control ilc basis both modeling and control of the plant are performed using dynamic neural networks for this purpose the whole control horizon is divided into a certain number of time domain subintervals called batches and a pseudo d type ilc law is employed to train the plant model and controller parameters so that the entropy of the closed loop tracking error is made to decrease batch by batch the method has the advantage of decreasing the output uncertainty versus the advances of batches along the time horizon the analysis on the proposed ilc convergence is made and a set of demonstrable experiment results is also provided to show the effectiveness of the obtained control algorithm where encouraging results have been obtained

estimation of stationary structural system parameters from non stationary random vibration data a locally stationary model method abstract stationary structural system building natural frequency and damping parameters are estimated from a long non stationary ambient vibration time series record the non stationary vibration record is decomposed into continuous not necessarily equal duration stationary time series segments by a locally stationary autoregressive ar model method estimates of the stationary structural system parameters are extracted from each of the locally stationary ar models the final estimate of the structural system parameters is a time duration weighted average of the estimates in each stationary segment under reasonable conditions the theoretical variance of the structural system parameter estimates decreases with the duration of the long non stationary record the results of the analysis of a long duration non stationary record are compared with the results of an earlier approach in which the long duration non stationary record was decomposed into relatively short fixed duration vibration records and analyzed as stationary time series 1 the variances of the damping parameter estimates achieved by the current method are one half to one quarter of those obtained by the earlier method the sample variance of the structural system parameter estimates decreases with the reciprocal of the duration of the overall non stationary record

convexity estimates for the solutions of a class of semi linear elliptic equations abstract in this paper we are concerned with convexity estimates for solutions of a class of semi linear elliptic equations involving the laplacian with power type nonlinearities we consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions then we give two applications of these convexity estimates we use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi linear elliptic equations finally we give a sharp lower bound estimate of the gaussian curvature for the solution surface of some specific equation by the curvatures of the domain s boundary

regularized fast recursive least squares algorithms for adaptive filtering fast recursive least squares frls algorithms are developed by using factorization techniques which represent an alternative way to the geometrical projections approach or the matrix partitioning based derivations the estimation problem is formulated within a regularization approach and priors are used to achieve a regularized solution which presents better numerical stability properties than the conventional least squares one the numerical complexity of the presented algorithms is explicitly related to the displacement rank of the a priori covariance matrix of the solution it then varies between o 5m and that of the slow rls algorithms to update the kalman gain vector m being the order of the solution an important advantage of the algorithms is that they admit a unified formulation such that the same equations may equally treat the prewindowed and the covariance cases independently from the used priors the difference lies only in the involved numerical complexity which is modified through a change of the dimensions of the intervening variables simulation results are given to illustrate the performances of these algorithms

a smoothness priors long ar model method for spectral estimation a new smoothness priors long ar model method approach is taken to the short data span spectral estimation problem an autoregressive ar model that is relatively long compared to the data length is considered the smoothness priors are in the form of the integrated squared derivatives of the ar model whitening filter a smoothness tradeoff parameter or bayesian hyperparameter balances the tradeoff between the infidelity of the ar model to the data and the infidelity of the model to the smoothness constraint the critical computation of the likelihood of the hyperparameters of the bayesian model is realized by a constrained least squares computation numerical examples are shown the results of simulation studies using entropy comparison evaluations of the bayesian and minimum aic ar methods of spectral estimation are also shown

shaping frequency dependent time resolution when estimating spectral properties with parametric methods the problem of tracking time varying properties of a signal is studied the somewhat contradictory notion of time varying spectrum and how to estimate the current spectrum in an on line fashion is discussed the traditional concepts and relations between time and frequency resolution are crucial for this problem we introduce two definitions for the time resolution of filters essentially measuring the effective number of past data that are used to form the estimate in for example wavelet transform techniques frequency dependent time resolutions are used so that fewer data are used at higher frequencies thus enabling faster tracking of high frequency components at the price of worse frequency resolution the main contribution of the paper is to show how this same feature can be introduced when estimating spectra via a time varying autoregressive model of the signal this is achieved by a special choice of nominal covariance matrix for the underlying parameter changes

existence and convexity of local solutions to degenerate hessian equations abstract in this work we prove the existence of convex solutions to the following k hessian equation s k u k y g y u d u in the neighborhood of a point y 0 u 0 p 0 r n r r n where g c g y 0 u 0 p 0 0 k c is nonnegative near y 0 k y 0 0 and rank d y 2 k y 0 n k 1

the convexity of solution of a class hessian equation in bounded convex domain in r 3 we use the deformation methods to obtain the convexity of solution of a class hessian equation in bounded convex domain in r 3 c 2008 elsevier inc all rights reserved

a smoothness priors time varying ar coefficient modeling of nonstationary covariance time series a smoothness priors time varying ar coefficient model approach for the modeling of nonstationary in the covariance time series is shown smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each ar coefficient the unknown white noise variances are hyperparameters of the ar coefficient distribution the critical computation is of the likelihood of the hyperparameters of the bayesian model this computation is facilitated by a state space representation kalman filter implementation the best difference equation order best ar model order best hyperparameter model locally in time is selected using the minimum aic method also an instantaneous spectral density is defined in terms of the instantaneous ar model coefficients and a smoothed estimate of the instantaneous time series variance an earthquake record is analyzed the changing spectral analysis of the original data and of simulations from a time varying ar coefficient model of that data are shown

ilc based fixed structure controller design for output pdf shaping in stochastic systems using lmi techniques in this paper a generalized state space controller design for the shaping of the output probability density function pdf is presented for non gaussian dynamical stochastic systems a radial basis function rbf neural network is used to approximate the output pdf of the system such a neural network consists of a number of weights and corresponding basis functions using such an approximation the dynamics of the original stochastic system can be expressed as the dynamics between the control input and the weights of the rbf neural network the task of output pdf control can therefore be reduced to a rbf weight control together with an adaptive tuning of the basis function parameters i e the centers and widths of the basis functions to achieve this aim the control horizon is divided into certain intervals hereinafter called batches using these definitions the whole control strategy consists of three stages namely a sub space parameter identification of the dynamic nonlinear model that relates the control signal to the weights of the rbf neural network b weight tracking controller design using an lmi based convex optimization technique and c rbf basis functions shape tuning in terms of their centers and widths using an iterative learning control ilc framework among the above stages the first two are performed within each batch while stage c is carried out between any two adjacent batches such an algorithm has the advantage of the batch by batch improvement of the closed loop output pdf tracking performance moreover the controller mentioned in stage b is a general controller in a state space form stability analysis has been performed and simulation results are included to show the effectiveness of the proposed method where encouraging results have been made

a brunn minkowski inequality for the monge ampere eigenvalue abstract we prove a brunn minkowski type inequality for the eigenvalue of the monge ampere operator 1 2 n is concave in the class of c 2 domains in r n endowed with minkowski addition the equality case is explicitly described too the main device of the proof is a notion of addition for convex functions called infimal convolution which corresponds to the minkowski addition of the graphs of the involved functions

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

estimating time varying parameters by the kalman filter based algorithm stability and convergence convergence and stability properties of the kalman filter based parameter estimator are established for linear stochastic time varying regression models the main features are both the variances and sample path averages of the parameter tracking error are shown to be bounded the regression vector includes both stochastic and deterministic signals and no assumptions of stationarity or independence are requires and the unknown parameters are only assumed to have bounded variations in an average sense

a new approach to stochastic adaptive control the principal techniques used up to now for the analysis of stochastic adaptive control systems have been 1 super martingale often called stochastic lyapunov methods and 2 methods relying upon the strong consistency of some parameter estimation scheme optimal stochastic control and filtering methods have also been employed although there have been some successes the extension of these techniques to a broader class of adaptive control problems including the case of time varying parameters has been difficult in this paper a new approach is adopted if an underlying markovian state space system for the controlled process is available and if this process possesses stationary transition probabilities then the powerful ergodic theory of markov processes may be applied subject to technical conditions such as stability one may deduce 1 the existence of an invariant measure for the process and 2 the convergence almost surely of the sample averages of a function of the state process and of its expectation to its conditional expectation the technique is illustrated by an application to a previously unsolved problem involving a linear system with unbounded random time varying parameters

free arrangements of hyperplanes and supersolvable lattices a finite family of hyperplanes through the origin in c is called an iarrangement of hyperplanes or simply an l arrangement an l arrangement is said to be free if the module of logarithmic vector fields holomorphic vector fields tangent to all hyperplanes in the arrangement is a free module the accurate definition will be given in 2 1 let a be a free i arrangement then a set do d of i 1 nonnegative integers can be defined 2 3 these d d are called the exponents of a on the other hand one can associate a geometric lattice l a with any arrangement a 3 1 the characteristic polynomial xlca t e z t is defined by using the mobius function 3 4 the second author 7 proved that

stochastic adaptive control and martingale limit theory recently s p meyn and p e caines ibid vol ac 32 p 220 6 1987 have used ergodic theory for markov processes to give the first asymptotic stability analysis of a nontrivial stochastic adaptive control problem by nontrivial is meant a stochastic adaptive control problem whose parameter variation has finite nonzero power they correctly observed that the stochastic lyapunov function methods fail here because there is no almost sure parameter convergence it is shown here how martingale asymptotics can be used to produce many results close to those of meyn and caines as well as to supply some new observations strengths and weaknesses of both approaches are discussed

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

a brunn minkowski inequality for the hessian eigenvalue in three dimensional convex domain abstract we use the deformation methods to obtain the strictly log concavity of solution of a class hessian equation in bounded convex domain in r 3 as an application we get the brunn minkowski inequality for the hessian eigenvalue and characterize the equality case in bounded strictly convex domain in r 3

the brunn minkowski inequality for p capacity of convex bodies in this paper we prove the brunn minkowski inequality for the p capacity of convex bodies i e convex compact sets with non empty interior in r n for every p 1 n moreover we prove that the equality holds in such inequality if and only if the involved bodies coincide up to a translation and a dilatation

the brunn minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators abstract we prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a brunn minkowski type inequality on the class of open bounded sets in r n satisfying a uniform exterior sphere condition in particular the result applies to the possibly normalized p laplacian and to the minimal pucci operator the proof is inspired by the approach introduced by colesanti for the principal frequency of the laplacian within the class of convex domains and relies on a generalization of the convex envelope method by alvarez lasry lions we also deal with the existence and log concavity of positive viscosity eigenfunctions

the hadamard variational formula and the minkowski problem for p capacity abstract a hadamard variational formula for p capacity of convex bodies in r n is established when 1 p n the formula is applied to solve the minkowski problem for p capacity which involves a degenerate monge ampere type equation uniqueness for the minkowski problem for p capacity is established when 1 p n and existence and regularity when 1 p 2 these results are non linear extensions of the now classical solution of jerison of the minkowski problem for electrostatic capacity p 2

capacities surface area and radial sums a dual capacitary brunn minkowski inequality is established for the n 1 capacity of radial sums of star bodies in rn this inequality is a counterpart to the capacitary brunn minkowski inequality for the p capacity of minkowski sums of convex bodies in rn 1 p n proved by borell colesanti and salani when n 3 the dual capacitary brunn minkowski inequality follows from an inequality of bandle and marcus but here a new proof is given that provides an equality condition note that when n 3 the n 1 capacity is the classical electrostatic capacity a proof is also given of both the inequality and a different equality condition when n 2 the latter case requires completely different techniques and an understanding of the behavior of surface area perimeter under the operation of radial sum these results can be viewed as showing that in a sense n 1 capacity has the same status as volume in that it plays the role of its own dual set function in the brunn minkowski and dual brunn minkowski theories

combinatorial construction of logarithmic differential forms abstract h terao has shown that the structure of the module of rational differential forms with at most logarithmic poles at an arrangement of hyperplanes as defined by k saito is very strongly controlled by the combinatorial structure of the arrangement in this paper we demonstrate how the existence of rational logarithmic forms with poles of high order depends on the existence of highly degenerate special position subarrangements the associated combinatorial structures are studied first a strong version of k saito s preparation lemma for logarithmic forms leads to a new simple proof of terao s celebrated addition deletion theorem for hyperplane arrangements with free module of logarithmic 1 forms free arrangements then structural characterizations are developed in two extreme cases for the generation of the module of logarithmic differential 1 forms this module has a triangular basis iff the arrangement is supersolvable strictly linearly fibered and it is generated by forms of degree 1 iff the arrangement is generic in codimension 3 both conditions on the geometry of the arrangement are combinatorial in a very strong sense determined by restricted data on the lattice of intersections of the hyperplanes however examples show that the cardinality and degree sequence of a minimal set of generators for the module of logarithmic 1 forms are not in general determined by this intersection lattice

capacitary inequalities of the brunn minkowski type the main purpose of this paper is to show how so called a concave capacities may be built up from newton capacity c in lr n 3 in much the same way as concave measures from lebesgue measure m n in lr n 1 throughout o f2 o stand for non empty open and convex subsets of ir and j o denotes the class of all convex bodies contained in f2 recall that a positive radon measure in f2 is said to be concave if a and

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

submanifolds that are level sets of solutions to a second order elliptic pde abstract we consider the problem of characterizing which noncompact hypersurfaces in r n can be regular level sets of a harmonic function modulo a c diffeomorphism as well as certain generalizations to other pdes we prove a versatile sufficient condition that shows in particular that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of r n the technique we use combines robust but not explicit local constructions with appropriate global approximation theorems in view of applications to a problem posed by berry and dennis intersections of level sets are also studied

some almost exceptional arrangements abstract we introduce and study a family of real hyperplane arrangements that includes the reflection arrangements of types a and d as well as an infinite family of arrangements e n constructed by orlik solomon and terao related to the exceptional arrangements of type e n n 8 we investigate freeness in the sense of terao of these arrangements disproving the conjecture that the arrangements e n are free in general

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

brunn minkowski inequalities for variational functionals and related problems abstract recently several inequalities of brunn minkowski type have been proved for well known functionals in the calculus of variations e g the first eigenvalue of the laplacian the newton capacity the torsional rigidity and generalizations of these examples in this paper we add new contributions to this topic in particular we establish equality conditions in the case of the first eigenvalue of the laplacian and of the torsional rigidity and we prove a brunn minkowski inequality for another class of variational functionals moreover we describe the links between brunn minkowski type inequalities and the resolution of minkowski type problems

hyperplane arrangements and linear strands in resolutions the cohomology ring of the complement of a central complex hyperplane arrangement is the well studied orlik solomon algebra the homotopy group of the complement is interesting complicated and few results are known about it we study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field c over the orlik solomon algebra

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

adaptive scheme for elimination of broadband noise and impulsive disturbances from ar and arma signals the problem of elimination of measurement noise and impulsive disturbances from autoregressive and autoregressive moving average signals is considered it is shown that the task of simultaneous detection tracking restoration can be stated as a nonlinear filtering problem and solved using the theory of extended kalman filter numerical tests carried out for real audio signals corrupted by both real and artificially generated disturbances confirm very good properties of the proposed algorithm

power convexity of a class of elliptic equations involving the hessian operator in a 3 dimensional bounded convex domain abstract in this paper we are concerned with power convexity of a class of fully nonlinear elliptic equations involving the hessian operator in a bounded and strictly convex domain in r 3 we use the constant rank theorem and deformation technique to prove that some power of the smooth admissible solution is strictly convex as a corollary we deduce that all the level sets of the solution are strictly convex