generalized ornstein uhlenbeck process having a characteristic operator with polynomial coefficients let be a weighted schwartz s space of rapidly decreasing functions the dual space and t a perturbed diffusion operator with polynomial coefficients from into itself it is proven that t generates the kolmogorov evolution operator from into itself via stochastic method as applications we construct a unique solution of a langevin s equation on wherew t is a brownian motion and t is the adjoint of t and show a central limit theorem for interacting multiplicative diffusions

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

the well posedness of the korteweg de vries benjamin ono equation abstract the well posedness of the cauchy problem to the generalized korteweg de vries benjamin ono equation is considered local results for data in h s r s 1 8 and the global well posedness for data in l 2 r are obtained if l 2 moreover for l 3 the problem is locally well posed for data in hs s 1 4 the main idea is to use the fourier restriction norm method

tfpr a fuzzy and structual pattern recognition system of multi variate time dependent pattern classes based on sigmoidal functions abstract tfpr is a hybrid fuzzy and structural pattern recognition system that uses fuzzy sets to represent multi variate pattern classes that can be either static or dynamic depending on time or some other parameter space the membership functions of the fuzzy sets that represent pattern classes are modeled by sigmoidal functions the choice of sigmoidal functions was motivated by their ability to represent efficiently and concisely different multi variate pattern classes given a set of input data and a pattern class specification that is defined as a fuzzy set tfpr estimates how closely the data match the pattern class specification using the grade of membership of the data in the fuzzy set the input data may be a number of time dependent signals whose past values may influence the evaluation of the pattern class in that case structural pattern recognition methods in addition to fuzzy ones are employed in order to match multi dimensional curves rather than points in the input domain two different structural segmentation methods are presented the first is an efficient greedy algorithm that is linear to the number of input data points while the second is an algorithm that is optimal under certain conditions but is slower than the first method although efficiency is a very important consideration in tfpr the main issues are knowledge acquisition and knowledge representation in terms of pattern classes tfpr has been implemented in the bb1 blackboard architecture it is currently being applied in a system for medical monitoring potential applications in other domains are under investigation

an dimensional inhomogeneous langevin s equation abstract on a modified space from the space j of tempered distributions it is proven that a stochastic equation x t w t 0 t l s x s ds has a unique solution where w t is a valued brownian motion independent of a valued gaussian random variable and l s is an integro differential operator as an application a fluctuaton result or central limit theorem is shown for interacting diffusions

well posedness of stochastic kdv bo equation driven by fractional brownian motion the cauchy problem for the korteweg de vries benjamin ono equation driven by cylindrical fractional brownian motion is discussed in this paper fractional brownian motion is a family of processes b h it is known that the smaller the value of hurst parameter h is the worse of the regularity of fbm is using bourgain restriction method we obtain the lower bound of the hurst parameter h for the driving processes b h with h 38 we prove local existence results with initial value in classical sobolev spaces of negative indices i e h s with s 18

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

analyticity and smoothing effect for the korteweg de vries equation with a single point singularity we show that a solution of the cauchy problem for the kdv equation cases partial tv partial 3 x v partial x v 2 0 qquad t in t t x in bbb r v 0 x phi x endcases has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x 0 it is shown that for h s mathbb r s 3 4 data satisfying the condition sum k 0 infty frac a 0 k k x partial x k phi h s infty the solution is analytic in both space and time variable the above condition allows us to take as initial data the dirac delta measure or the cauchy principal value of 1 x the argument is based on the recent progress on the well posedness result by bourgain 2 and kenig ponce vega 20 and a systematic use of the dilation generator 3t partial t x partial x

exact boundary controllability of the korteweg de vries equation we consider boundary control of the distributed parameter system described by the korteweg de vries kdv equation posed on a finite interval alpha leq x leq beta left begin array l u t u x uu x u xxx 0 u alpha t h 1 t qquad u beta t h 2 t qquad u x beta t h 3 t u end array right qquad for t geq 0 it is shown that by choosing appropriate control inputs h j t j 1 2 3 one can always guide the system from a given initial state phi in h s alpha beta to a given terminal state psi in h s alpha beta in the time period 0 t so long as phi and psi satisfy phi cdot w cdot 0 h s alpha beta leq delta quad mbox and quad psi cdot w cdot t h s alpha beta leq delta for some delta 0 independent of phi and psi where s geq 0 and w equiv w x t is a given smooth solution of the kdv equation this exact boundary controllability is established by considering a related initial value control problem of the kdv equation posed on the whole line r various recently discovered smoothing properties of the kdv equation have played important roles in our approach

a korteweg de vries type of fifth order equations on a finite domain with point dissipation abstract the paper discusses initial value problem of a korteweg de vries type of fifth order equation w t w x x x w x x x x x j 1 n a j w j w x 0 w x 0 w 0 x posed on a periodic domain x 0 2 with periodic boundary conditions w i x 0 t w i x 2 t i 0 2 3 4 and an l 2 stabilizing feedback control law w x 2 t w x 0 t 1 w x x x 0 t where n is a fixed positive integer a j j 1 2 n are real constants and 1 it is shown that for w 0 x h 1 0 2 with the boundary conditions described above the problem is locally well posed for w c 0 t h 1 0 2 with a conserved volume of w w 0 2 w x t d x moreover the solution with small initial condition exists globally and approaches to w 0 x 2 as t

the stochastic damped kdv equation the author applies singular perturbation theory to the study of a damped kdv soliton under the influence of space and time dependent external noise he finds that asymptotically the shape of the averaged solution approaches that of a gaussian packet whose amplitude decays and width grows in the same way as in the case of purely time dependent noise

markov properties of the fluctuation limit of a particle system with death abstract consider a sequence of independent brownian motions in rd whose initial positions are distributed according to a probability measure on rd r and disappear after an exponentially distributed lifetime with parameter it is shown that the fluctuation process of the empirical distribution of the first n motions converges weakly in the skorokhod spaces d 0 t0 l rd and d t0 l rd as n where t0 inf t rd t 0 to a continuous centered gaussian process x conditions for x to be markovian are determined in terms of the cases 0 and 0 i e the motions go on forever exhibit different markovian properties

well posedness of the cauchy problem for ostrovsky stepanyams and tsimring equation with low regularity data abstract in this paper we prove that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 3 4 and globally well posed in h s r when s 0

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

l2global well posedness of the initial value problem associated to the benjamin equation abstract we study well posedness for the initial value problem associated to the benjamin equation and the periodic benjamin equation global results are established for data inl2 r andl2 t respectively we apply the recent theory developed by kenig ponce and vega and bourgain to deal with low regularity data for the initial value problem associated to the korteweg de vries equation

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

on the stochastic korteweg de vries equation abstract we consider a stochastic korteweg de vries equation forced by a random term of white noise type this can be a model of water waves on a fluid submitted to a random pressure we prove existence and uniqueness of solutions inh1 r in the case of additive noise and existence of martingales solutions inl2 r in the case of multiplicative noise copyright 1998 academic press nous etudions une equation de korteweg de vries stochastique comportant une force aleatoire de type bruit blanc cette equation peut decrire les ondes de surface sur un fluide soumis a un champ de pression aleatoire nous montrons l existence et l unicite de solutions dansh1 r lorsque le bruit est additif puis dans le cas du bruit multiplicatif nous etablissons l existence de solutions martingales dansl2 r

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

korteweg de vries solitons under additive stochastic perturbations the evolution of solitonic solutions of the korteweg de vries equation subject to additive noise is investigated using numerical techniques various types of additive white gaussian noise are considered the averaged solution amplitudes exhibit in all cases algebraic decay verifying wadati s universality conjecture if the noise is time dependent or position and time dependent algebraic decay is obtained for intermediate times too these intermediate time results agree well with the outcome of an experiment on ion acoustic soliton propagation in a noisy plasma the distribution of soliton first passage times in a noisy medium is also discussed

well posedness for the stochastic higher order modified camassa holm equation abstract in this article the stochastic higher order modified camassa holm equation with additive noise is considered using the fourier restriction norm method and bilinear estimates the local existence and uniqueness result is proved by the contraction mapping principle the global existence of the h 1 r solution of this problem is also obtained by energy inequality

fuzzy min max neural networks i classification a supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described each fuzzy set is an aggregate union of fuzzy set hyperboxes a fuzzy set hyperbox is an n dimensional box defined by a min point and a max point with a corresponding membership function the min max points are determined using the fuzzy min max learning algorithm an expansion contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining the use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher level decision making the relationship between fuzzy sets and pattern classification is described the fuzzy min max classifier neural network implementation is explained the learning and recall algorithms are outlined and several examples of operation demonstrate the strong qualities of this new neural network classifier

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

fuzzy artmap a neural network architecture for incremental supervised learning of analog multidimensional maps a neural network architecture is introduced for incremental supervised learning of recognition categories and multidimensional maps in response to arbitrary sequences of analog or binary input vectors which may represent fuzzy or crisp sets of features the architecture called fuzzy artmap achieves a synthesis of fuzzy logic and adaptive resonance theory art neural networks by exploiting a close formal similarity between the computations of fuzzy subsethood and art category choice resonance and learning four classes of simulation illustrated fuzzy artmap performance in relation to benchmark backpropagation and generic algorithm systems these simulations include finding points inside versus outside a circle learning to tell two spirals apart incremental approximation of a piecewise continuous function and a letter recognition database the fuzzy artmap system is also compared with salzberg s nge systems and with simpson s fmmc system

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

well posedness for stochastic camassa holm equation abstract the camassa holm equation describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity due to uncertainty in the modelling and external environment this modelling could be subject to random fluctuations in this article the stochastic camassa holm equation with additive noise is considered using regularization a local existence and uniqueness result in the sobolev space h s r with s 3 2 of stochastic camassa holm equation is obtained with the help of priori estimates the local solution will blow up in h q r in finite time for any q 3 2 when initial value satisfies some conditions

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

tightness of probabilities on c lbrack 0 1 rbrack mathscr y and d lbrack 0 1 rbrack mathscr y let c mathscr i c lbrack 0 1 rbrack mathscr i be the space of all continuous mappings of lbrack 0 1 rbrack to mathscr i where mathscr l is the topological dual of the schwartz space mathscr i of all rapidly decreasing functions let c be the banach space of all continuous functions on lbrack 0 1 rbrack for each varphi in mathscr i pi varphi is defined by pi varphi x in c mathscr i rightarrow x varphi in c given a sequence of probability measures p n on c mathscr i such that for each varphi in mathscr i p n pi 1 varphi is tight in c we prove that p n itself is tight in c mathscr i a similar result is proved for the space of all right continuous mappings of lbrack 0 1 rbrack to mathscr i

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

stochastic evolution equations with fractional brownian motion in this paper linear stochastic evolution equations driven by infinite dimensional fractional brownian motion are studied a necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed separate proofs are required for the cases of hurst parameter above and below 1 2 the particular case of the laplacian on the circle is discussed in detail

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

stochastic burgers equation driven by fractional brownian motion abstract in this paper we consider the stochastic burgers equation driven by a genuine cylindrical fractional brownian motion with hurst parameter h 1 4 we first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic burgers equation then we obtain the local and global existence and uniqueness results for the stochastic burgers equation

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

refined energy inequality with application to well posedness for the fourth order nonlinear schrodinger type equation on torus we consider the time local and global well posedness for the fourth order nonlinear schrodinger type equation 4nls on the torus the nonlinear term of 4nls contains the derivatives of unknown function and this prevents us to apply the classical energy method to overcome this difficulty we introduce the modified energy and derive an a priori estimate for the solution to 4nls

fuzzy constraint networks for signal pattern recognition this paper deals with representation and reasoning on information concerning the evolution of a physical parameter by means of a model based on the fuzzy constraint satisfaction problem formalism and with which it is possible to define what we call fuzzy temporal profiles ftp based on fundamentally linguistic information this model allows the integration of knowledge on the evolution of a set of parameters into a knowledge representation scheme in which time plays a fundamental role the ftp model describes the behaviour of a physical parameter on the basis of a set of signal events and which allows the evolution of the parameter between each pair of events to be modelled as signal episodes given the fundamentally linguistic nature of the information represented the consistency analysis of this information is an essential task nevertheless the obtention of the minimal representation of the network that defines an ftp is an np hard problem in spite of this we supply algorithms guaranteeing local levels of consistency that allow to correct a large proportion of the errors committed by a human expert in the linguistic description of the profile furthermore we propose a new topology whose consistency can be guaranteed in polynomial time we also study the applicability of this model in the recognition of signal patterns

global solutions of the klein gordon schrodinger system with rough data in r 2 1 abstract in this paper we consider the klein gordon schrodinger system with quadratic yakuwa coupling and cubic autointeractions in r 2 1 and prove the existence and uniqueness of global solution for rough data the techniques to be used are adapted from a general scheme originally introduced by j bourgain to split the data into the low and high frequency parts

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation