interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

dilation method and smoothing effects of solutions to the benjamin ono equation in this paper we study smoothing effects of solutions to the benjamin ono equation where h is the hilbcrt transform defined by we prove that if h 2 and x x 4 then the solution u of bo belongs to where

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

well posedness and porosity in the calculus of variations without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in zaslavski nonlinear analysis 43 2001 339 a generic well posedness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in zaslavski communications in applied analysis to appear we extended this generic well posedness result to a class of variational problems in which the values at the end points are also subject to variations more precisely we established a generic well posedness result for a class of variational problems without convexity assumptions over functions with values in a banach space e which is identified with the corresponding complete metric space of pairs f 1 2 where f is an integrand satisfying the cesari growth condition and 1 2 e are the values at the end points denoted by a we showed that for a generic f 1 2 a the corresponding variational problem is well posed in this paper we study the set of all pairs f 1 2 a for which the corresponding variational problem is well posed we show that the complement of this set is not only of the first category but also a porous set

generic well posedness of minimization problems with mixed smooth constraints abstract in this paper we study mathematical programming problems with mixed smooth constraints in a banach space and show that most of the problems in the baire category sense are well posed our result is a generalization of a result of a d ioffe et al a variational principle for problems with functional constraints siam j optim 12 2001 461 478 obtained for finite dimensional banach spaces it is also an extension of our recent result a j zaslavski generic well posedness of minimization problems with mixed continuous constraints nonlinear anal doi 10 1016 j na 2005 10 032 which was obtained for mathematical programming problems with continuous constraints

generic well posedness of nonconvex optimal control problems the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic existence and uniqueness result to a class of optimal control problems in which the right hand side of differential equations is also subject to variations

complex factor analysis and extensions many subspace based array signal processing algorithms assume that the noise is spatially white in this case the noise covariance matrix is a multiple of the identity and the eigenvectors of the data covariance matrix are not affected by it if the noise covariance is an unknown arbitrary diagonal e g for an uncalibrated array the eigenvalue decomposition leads to incorrect subspace estimates and it has to be replaced by a more general factor analysis decomposition fa which then reveals all relevant information we consider this data model and several extensions where the noise covariance matrix has a more general structure such as banded sparse block diagonal and cases where we have multiple data covariance matrices that share the same noise covariance matrix starting from a nonlinear weighted least squares formulation we propose new estimation algorithms for both classical fa and its extensions the optimization is based on gauss newton gradient descent generally this leads to an iteration involving the inversion of a very large matrix using the structure of the problem we show how this can be reduced to the inversion of a matrix with dimension equal to the number of unknown noise covariance parameters this results in new algorithms that have faster numerical convergence and lower complexity compared to several maximum likelihood based algorithms that could be considered state of the art the new algorithms scale well to large dimensions and can replace eigenvalue decompositions in many applications even if the noise can be assumed to be white

almost every convex or quadratic programming problem is well posed we provide an abstract principle aimed at proving that classes of optimization problems aretypically well posed in the sense that the collection of ill posed problems within each class is s porous as a consequence we establish typical well posedness in the above sense for unconstrained minimization of certain classes of functions e g convex and quasi convex continuous as well as of convex programming with inequality constraints we conclude the paper by showing that the collection of consistent ill posed problems of quadratic programming of any fixed size has lebesgue measure zero in the corresponding euclidean space

convex optimization and the epi distance topology let 17 x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology t of uniform convergence of distance functions on bounded sets a function f in 17 x is called well posed provided it has a unique minimizer and each minimizing sequence converges to this minimizer we show that well posedness of f e r x is the minimal condition that guarantees strong convergence of approximate minima of t approximating functions to the minimum of f moreover we show that most functions in 17 x taw are well posed and that this fails if 17 x is topologized by the weaker topology of mosco convergence whenever x is infinite dimensional applications to metric projections are also given including a fundamental characterization of approximative compactness

a variational principle for problems with functional constraints in this paper we show that in several important classes of optimization problems like mathematical programming with k smooth data quadratic programming in a hilbert space convex programming in a banach space semi infinite programming and optimal control of linear systems with quadratic cost most of the problems in the baire category sense are well posed this is derived from a general variational principle for problems with functional constraints

neumann initial boundary value problem for benjamin ono equation abstract we consider the neumann initial boundary value problem for benjamin ono equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

a numerical approach to blow up issues for dispersive perturbations of burgers equation abstract we provide a detailed numerical study of various issues pertaining to the dynamics of the burgers equation perturbed by a weak dispersive term blow up in finite time versus global existence nature of the blow up existence for long times and the decomposition of the initial data into solitary waves plus radiation we numerically construct solitary waves for fractional korteweg de vries equations

density of the set of local minimizers for a generic cost function abstract in this work we study the minimization problem f x min x x where f x r 1 is a continuous function defined on a complete separable metric space x we show that for a generic function f the set of all local minimizers is everywhere dense in x

generic well posedness of optimal control problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in a j zaslavski nonlinear anal to appear a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend the generic existence and uniqueness result in a j zaslavski nonlinear anal to appear to a class of optimal control problems in which constraint maps are also subject to variations the main result of the paper is obtained as a realization of a variational principle extending the variational principle introduced in a d ioffe and a j zaslavski siam j control optim 38 2000 pp 566 581

generic well posedness of variational problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic well posedness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic well posedness result to two classes of variational problems in which the values at the end points are also subject to variations the main results of the paper are obtained as realizations of a general variational principle

the benjamin ono equation in weighted sobolev spaces abstract in this paper we continue the study of the initial value problem t u x u 2 2 x u u 0 in the spaces y s h s r l 2 r where 0 s 3 2 and l r 2 r denotes the set of all complex valued measurable functions such that f r 2 r 1 x 2 r f x 2 dx our main result is the proof of well posedness of the aforementioned problem in y 2 0 1 the proof involves parabolic regularization the riesz thorin interpolation theorem and kato s theory of linear evolution equations of hyperbolic type

generic well posedness of minimization problems with mixed continuous constraints in this paper we study mathematical programming problems with mixed constraints in a banach space and show that most of the problems in the baire category sense are well posed our result is a generalization of a result of ioffe et al siam j optim 12 2001 461 478 obtained for finite dimensional banach spaces

mixed initial boundary value problem for the benjamin ono equation abstract we consider the mixed initial boundary value problem for the benjamin ono equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

variational principles and well posedness in optimization and calculus of variations the concluding result of the paper states that variational problems are generically solvable and even well posed in a strong sense without the convexity and growth conditions always present in individual existence theorems this and some other generic well posedness theorems are obtained as realizations of a general variational principle extending the variational principle of deville godefroy zizler

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

smoothing properties and existence of solutions for the generalized benjamin ono equation abstract we study the cauchy problem for the generalized benjamin ono equation tu d d 2 u dv u where d d dx u 0 and v e 1 r r with v 0 v 0 0 using commutator identities and estimates we prove that the equation exhibits smoothing properties similar to those of the generalized korteweg de vries equation gkdv which is the special case 1 of if v satisfies suitable estimates at zero and at infinity we prove that the cauchy problem for the equation with initial data u 0 u0 has a solution u l r l2 l2loc r h loc if u0 l2 and has a solution u l r h l2loc r h2 loc if u0 h finite energy solution in addition we prove that the usual benjamin ono equation 1 2 v u u 2 has a solution u l r h1 l2loc r h 3 2 loc for u0 h1 those results are easily extended to related equations such as the intermediate long wave equation or the smith equation

large time asymptotics of solutions to the generalized benjamin ono equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized benjamin ono bo equation ut u 1u x huxx 0 where h is the hilbert transform x t r when the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time in the case 3 critical for the asymptotic behavior i e for the modified benjamin ono equation we prove that the solutions have the same l time decay as in the corresponding linear bo equation also we find the asymptotics for large time of the solutions of the cauchy problem for the bo equation in the critical and noncritical cases for the critical case we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted sobolev spaces these modified scattering states differ from the free scattering states by a rapidly oscillating factor

on the stochastic benjamin ono equation abstract we discuss the cauchy problem for the stochastic benjamin ono equation in the function class h s r s 3 2 when there is a zero order dissipation we also establish the existence of an invariant measure with support in h 2 r many authors have discussed the cauchy problem for the deterministic benjamin ono equation but our results are new for the stochastic benjamin ono equation our goal is to extend known results for the deterministic equation to the stochastic equation

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

on the propagation of regularity for solutions of the fractional korteweg de vries equation abstract we consider the initial value problem ivp for the fractional korteweg de vries equation fkdv 0 1 t u d x x u u x u 0 x t r 0 1 u x 0 u 0 x it has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena more precisely it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution this property was found originally in solutions of korteweg de vries kdv equation and it has been also verified in other dispersive equations as the benjamin ono bo equation recently it has been shown that the solutions of the dispersive generalized benjamin ono dgbo equation this is 2 3 in 0 1 also satisfy the propagation of regularity phenomena this is achieved by introducing a commutator decomposition to handle the dispersive part in the equation following the approach used in the dgbo we prove that the solutions of the fkdv also satisfies the propagation of regularity phenomena consequently this type of regularity travels with infinite speed to its left as time evolves

typical convex program is very well posed in this paper we consider the collection of convex programming problems with inequality and equality constraints in which every problem of the collection is obtained by linear perturbations of the cost function and right hand side perturbation of the constraints while the core cost function and the left hand side constraint functions are kept fixed the main result shows that the set of the problems which are not well posed is porous in a certain strong sense our results concern both the infinite and finite dimensional case in the last case the conclusions are significantly sharper

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies