well posedness of korteweg de vries benjamin bona mahony equation on a finite domain abstract in this paper we consider the korteweg de vries benjamin bona mahony equation on a finite domain with initial value and nonhomogeneous boundary conditions this particular problem arises from the phenomenon of long wave with small amplitude in fluid we get the global well posedness of this system

dispersive blowup of solutions of generalized korteweg de vries equations abstract the strong effect of dispersion on short wavelength disturbances featured by the korteweg de vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data which exhibit a specified loss of spatial smoothness at particular times the points in space time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half plane x t x r t 0 our results are related to recent work on smoothing of solutions of such equations some of which are sharpened here and they show that in certain aspects these earlier results are not far from being optimal the theory makes use of new results concerning well posedness of such equations in weighted sobolov spaces and some detailed analysis of the linear korteweg de vries equation

a non homogeneous boundary value problem for the korteweg de vries equation posed on a finite domain ii abstract studied here is an initial and boundary value problem for the korteweg de vries equation posed on a bounded interval with nonhomogeneous boundary conditions this particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed it is shown here that this initial boundary value problem is globally well posed in the l 2 based sobolev space h s 0 1 for any s 0 in addition the mapping that associates to appropriate initial and boundary data the corresponding solution is shown to be analytic as a function between appropriate banach spaces

on d type and p type ilc designs and anticipatory approach many schemes of iterative learning control ilc have been developed for continuous time non linear dynamic systems to improve tracking performance two schemes d type and p type have been the bases for many ilc designs recently the anticipatory ilc scheme has been introduced on the basis of a different approach wang 1998a 1999 in this paper these basic schemes are compared from both analysis and implementation view points the anticipatory ilc scheme is designed on the basis of a causal pair of the action taken and its resulting state variables this approach has the anticipatory characteristics of the d type ilc and the simplicity for implementation of p type ilcs a sampled data ilc scheme is presented as another form of this anticipatory ilc scheme furthermore control device saturation is taken into account and tracking error convergence results are established with proofs the convergence results are also provided in the presence of uncertainties disturbances and measurement noises ex

the weyl marchaud fractional derivative of a type of self affine functions abstract the linear relationship between the box dimension of certain self affine functions and orders of the weyl marchaud fractional derivative has been proved the corresponding relationship of the besicovitch function has been explored under different conditions

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

robust self learning fuzzy controller design for a class of nonlinear mimo systems a new learning paradigm that can be applied for the design of a fuzzy logic based learning controller that is robust to external signals such as disturbances and set point changes is proposed in the paper it is well known that the self organizing fuzzy controller proposed by procyk and mamdani is sensitive to external signals such a phenomenon may be observed in other types of direct fuzzy logic based learning controllers that utilize an adaptation scheme in which the locational information of the current error state vector determines the degree of adaptation that should be made in the learning controller to resolve the problem of sensitivity to external signals it is proposed that learning and modification of the controller be made in consideration of the motional trend of the error state vector as well as its locational information this paradigm is adopted in the design of a new robust self learning fuzzy controller for a class of nonlinear mimo systems the well known techniques of sliding mode control and the fuzzy decision making method are utilized to implement the proposed learning scheme in the fuzzy learning controller via simulation study and experimental results the proposed learning controller is verified to be robust in the presence of external disturbances

iterative learning of model reference adaptive controller for uncertain nonlinear systems with only output measurement in this paper a model reference adaptive control strategy is used to design an iterative learning controller for a class of repeatable nonlinear systems with uncertain parameters high relative degree initial output resetting error input disturbance and output noise the class of nonlinear systems should satisfy some differential geometric conditions such that the plant can be transformed via a state transformation into an output feedback canonical form a suitable error model is derived based on signals filtered from plant input and output the learning controller compensates for the unknown parameters uncertainties and nonlinearity via projection type adaptation laws which update control parameters along the iteration domain it is shown that the internal signals remain bounded for all iterations the output tracking error will converge to a profile which can be tuned by design parameters and the learning speed is improved if the learning gain is large

brief on the equivalence of causal lti iterative learning control and feedback control the goal of iterative learning control ilc is to improve the accuracy of a system that repeatedly follows a reference trajectory this paper proves that for each causal linear time invariant ilc there is an equivalent feedback that achieves the ultimate ilc error with no iterations remarkably this equivalent feedback depends only on the ilc operators and hence requires no plant knowledge this equivalence is obtained whether or not the ilc includes current cycle feedback if the ilc system is internally stable and converges to zero error there exists an internally stabilizing feedback that approaches zero error at high gain since conventional feedback requires no iterations there is no reason to use causal ilc

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

blow up of smooth solutions of the korteweg de vries equation abstract the paper is devoted both to some initial boundary value problems and to the cauchy problem for the kdv equation blow up results are proved for these problems by using the nonlinear capacity approach 9

fractal random series generated by poisson voronoi tessellations in this paper we construct a new family of random series defined on r d indexed by one scaling parameter and two hurst like exponents the model is close to takagi knopp functions save for the fact that the underlying partitions of r d are not the usual dyadic meshes but random voronoi tessellations generated by poisson point processes this approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and hausdorff dimensions the proof of this main result is based on several new distributional properties of the poisson voronoi tessellation on the one hand an estimate of the oscillations of the function coupled with an application of a frostman type lemma on the other hand finally we introduce two related models and provide in particular a box dimension calculation for a derived deterministic takagi knopp series with hexagonal bases

on initial conditions in iterative learning control initial conditions or initial resetting conditions play a fundamental role in all kinds of iterative learning control methods in this note we study five different initial conditions disclose the inherent relationship between each initial condition and corresponding learning convergence or boundedness property the iterative learning control method under consideration is based on lyapunov theory which is suitable for plants with time varying parametric uncertainties and local lipschitz nonlinearities

similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure some modern scaling and fractal approaches to the characterization of surface roughness are under consideration a new mathematical model of a multilevel profile suitable for flexible simulation of roughness is presented as a development from these approaches the iterative construction of the model is described it is shown that the profile is a hierarchical structure with self affine and fractal features its geometrical properties are studied both analytically and numerically and some of them are proved to be of the renormalization type the profile contour length can be finite or infinite depending on the chosen values of the structural parameters some characteristics including the bearing curve and the upper volume function of the profile and the corresponding prestructures are calculated

on the p type learning control sufficient conditions for the robustness and convergence of p type learning control algorithms for a class of time varying nonlinear systems are presented the authors prove the uniform boundedness of the system state and the input control with respect to the existence of errors of initialization measurement noises and fluctuations of system dynamics furthermore the system output converges uniformly to the desired one in absence of all disturbances finally specialization of the results to linear systems are presented

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

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

the relationship between the box dimension of the besicovitch functions and the orders of their fractional calculus the relationship between the box dimension of the besicovitch functions and the orders of their fractional calculus has been investigated on some special conditions the linear connection between them has been proved and the other case has also been discussed

global well posedness and exponential decay for fifth order korteweg de vries equation posed on the finite domain abstract this paper studies the fifth order korteweg de veries equation posed on finite interval 0 1 t u x 5 u e 1 x 4 u 1 x 3 u e 2 x 2 u 2 x u e 3 u u x u 0 x 0 1 t 0 u x 0 x x 0 1 u 0 t u 1 t x u 0 t x u 1 t 0 x 2 u 1 t x 2 u 0 t t 0 firstly we establish smoothing effect of kato type for the fifth order korteweg de veries equation posed on finite interval secondly we show that the closed loop system 0 1 is globally well posed in the space h s 0 1 for s 0 5 lastly we prove that the solution of the closed loop system 0 1 is exponentially decay in the space h s 0 1 for s 0 5

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

an output based adaptive iterative learning controller for high relative degree uncertain linear systems in this paper we derive an output tracking error model based on signals filtered from plant input and output and then present a new output based adaptive iterative learning controller for repeatable linear systems with unknown parameters high relative degree initial resetting error input disturbance and output noise the proposed controller solves the important robustness issues without assuming the bounds of uncertainties to be sufficiently small and can be applied to high relative degree plants without using output differentiation control parameters are updated between successive iterations so as to compensate for unknown system parameters and uncertainties it is shown that the internal signals inside closed loop learning system remain bounded and the output tracking error will asymptotically converge to a profile tunable by some design parameters furthermore the learning speed is easily improved if the learning gain is increased

brief iterative learning control with initial rectifying action this paper addresses the initial shift problem in iterative learning control with system relative degree the tracking error caused by nonzero initial shift is detected when applying a conventional learning algorithm finite initial rectifying action is introduced in the learning algorithm and is shown effective in the improvement of tracking performance in particular robustness with respect to variable initial shifts the uniform convergence of the output trajectory to a desired one jointed smoothly with a specified transient trajectory from the starting position is ensured in the presence of fixed initial shift

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

hausdorff dimension of the graphs of the classical weierstrass functions we show that the graph of the classical weierstrass function sum n 0 infty lambda n cos 2 pi b n x has hausdorff dimension 2 log lambda log b for every integer b ge 2 and every lambda in 1 b 1 replacing cos 2 pi x by a general non constant c 2 periodic function we obtain the same result under a further assumption that lambda b is close to 1

brief paper analysis of two particular iterative learning control schemes in frequency and time domains this paper deals with iterative learning control design for multiple input multiple output mimo linear time invariant lti systems two particular ilc schemes are considered and analyzed in both frequency and time domains some remarks on the convergence implementation robustness with respect to disturbances and reinitialization errors as well as positive realness issues related to both schemes are provided

an iterative learning controller with initial state learning in iterative learning control ilc a common assumption is that the initial states in each repetitive operation should be inside a given ball centred at the desired initial states which may be unknown this assumption is critical to the stability analysis and the size of the ball will directly affect the final output trajectory tracking errors in this paper this assumption is removed by using an initial state learning scheme together with the traditional d type ilc updating law both linear and nonlinear time varying uncertain systems are investigated uniform bounds for the final tracking errors are obtained and these bounds are only dependent on the system uncertainties and disturbances yet independent of the initial errors furthermore the desired initial states can be identified through learning iterations

on a linearity between fractal dimension and order of fractional calculus in holder space abstract in this paper the linear relationship between fractal dimensions and the order of fractional calculus of functions in holder space has been mainly investigated under specific holder condition the linear connection between box dimension and the order of riemann liouville fractional integral and derivative has been proved this linear connection is also established with k dimension and packing dimension some function examples have been given in the end

non homogeneous boundary value problems of the fifth order kdv equations on a bounded interval abstract we investigate well posedness of initial boundary value problem for the fifth order kdv equation or kawahara equation posed on a finite interval t u x 5 u u x u 0 0 x 1 t 0 with general non homogeneous boundary conditions firstly all possible boundary conditions are found while searching enough dissipative effects to the initial boundary value problem then boundary smoothing effect is established through estimating explicit solution established by using laplacian transformation finally based on smoothing effect local well posedness of the initial boundary value problem was proved by applying banach s fixed point theorem

a robust high order p type iterative learning controller using current iteration tracking error a high order p type updating law in which the current iteration tracking error cite is employed is proposed for iterative learning control ilc of a class of uncertain nonlinear repetitive systems uniform boundedness of the tracking error is established in the presence of uncertainty disturbance and initialization error the bound and the ilc convergence rate can be adjusted by tuning the learning gain of the cite

selection of the learning gain matrix of an iterative learning control algorithm in presence of measurement noise arbitrary high precision output tracking is one of the most desirable control objectives found in industrial applications regardless of measurement errors the main purpose of this paper is to supply to the iterative learning control ilc designer guidelines to select the corresponding learning gain in order to achieve this control objective for example if certain conditions are met then it is necessary for the learning gain to converge to zero in the learning iterative domain in particular this paper presents necessary and sufficient conditions for boundedness of trajectories and uniform tracking in presence of measurement noise and a class of random reinitialization errors for a simple ilc algorithm the system under consideration is a class of discrete time affine nonlinear systems with arbitrary relative degree and arbitrary number of system inputs and outputs the state function does not need to satisfy a lipschitz condition this work also provides a recursive algorithm that generates the appropriate learning gain functions that meet the arbitrary high precision output tracking objective the resulting tracking output error is shown to converge to zero at a rate inversely proportional to square root of the number of learning iterations in presence of measurement noise and a class of reinitialization errors two illustrative numerical examples are presented

the korteweg de vries equation on a periodic domain with singular point dissipation this paper considers the korteweg de vries kdv equation eqalignno u t uu x u xxx 0 0 0 quad u 0 x u 0 x and the periodic boundary conditions u t 1 u t 0 u xx t 0 u xx t 1 with an l 2 stabilizing control input implemented by a feedback mechanism u x t 1 alpha u x t 0 and alpha 1 it can be shown that the solutions conserve the volume u int 1 0u t x dx and the constant state u 0 possesses the smallest energy among solutions with same volume it has been proved that the solution of the system exists and approaches u 0 as t rightarrow infty when alpha not 1 2 this paper studies the case for alpha 1 2 and gives a proof of the existence and exponential decay of the solutions by deriving estimates of the corresponding green s function and using semigroup theory the method used here also works for the other cases with alpha 1

a discrete time learning control algorithm for a class of linear time invariant systems a discretized version of the d type learning control algorithm is presented for a mimo linear discrete time system a necessary and sufficient condition for uniform convergence of the proposed learning algorithm is presented then we prove that the same condition is sufficient for the global robustness of the proposed learning algorithm to state disturbances measurement noise at the output and reinitialization error are present at each iteration a numerical example is given to illustrate the results

fractal dimensions and singularities of the weierstrass type functions a new type of fractal measures k s 1 s 2 defined on the subsets of the graph of a continuous function is introduced the k dimension defined by this measure is closer to the hausdorff dimension than the other fractal dimensions in recent literatures for the weierstrass type functions defined by w x 0 i g i x where 1 0 1 and g is an almost periodic lipschitz function of order greater than it is shown that the k dimension of the graph of w equals to 2 this conclusion is also equivalent to certain rate of the local oscillation of the function some problems on the knot points and the nondifferentiability of w are also discussed

controllability and stabilizability of the third order linear dispersion equation on a periodic domain in this paper solutions of the third order linear dispersion equations frac partial w partial t frac partial 3 w partial x 3 f x t qquad text and qquad frac partial w partial t frac partial 3 w partial x 3 0 are studied for t geqq 0 0 leqq x leqq 2 pi in the first case periodic boundary conditions are imposed at 0 and 2 pi and the distributed control f which may however have support smaller than 0 2 pi is assumed to be generated by a linear feedback law conserving the volume int 0 2 pi w x t dx while monotonically reducing int 0 2 pi w x t 2 dx for the second equation a feedback boundary control having the same properties is applied in both cases uniform exponential decay to a constant state is obtained related exact controllability questions are also studied and affirmative results are obtained

cantor boundary behavior of analytic functions let mathbb d be the open unit disc and let partial mathbb d be the boundary of mathbb d for f z analytic in mathbb d and continuous on overline mathbb d it follows from the open mapping theorem that partial f mathbb d subset f partial mathbb d these two sets have very rich and intrigue geometric properties when f z is univalent then they are equal and there is a large literature to study their boundary behaviors our interest is on the class of analytic functions f z for which the image curves f partial mathbb d form infinitely many loops everywhere they are not univalent of course we formulate this as the cantor boundary behavior we give sufficient conditions for such property making use of the distribution of the zeros of f and the mean growth rate of f examples includes the complex weierstrass functions and the cauchy transform of the canonical hausdorff measure on the sierpiski gasket

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

exact controllability and stabilizability of the korteweg de vries equation in this paper we consider distributed control of the system described by the korteweg de vries equation i atu uaxu e3 u f on the interval 0 0 with periodic boundary conditions ii au 27r t ku o t k 0 1 2 where the distributed control f f x t is restricted so that the volume f u x t dx of the solution is conserved both exact controllability and stabilizibility questions are studied for the system in the case of open loop control if the control f is allowed to act on the whole spatial domain 0 27r it is shown that the system is globally exactly controllable i e for given t 0 and functions q x x with the same volume one can alway find a control f so that the system i ii has a solution u x t satisfying u x 0 0 x u x t x if the control f is allowed to act on only a small subset of the domain 0 27r then the same result still holds if the initial and terminal states b and 0 have small amplitude in a certain sense in the case of closed loop control the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u x t 2dx the solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state as in the open loop control case a small amplitude assumption is needed if the control is allowed to act on only a small subdomain the smoothing property of the periodic linear kdv equation discovered recently by bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

on the dimension of the graph of the classical weierstrass function abstract this paper examines dimension of the graph of the famous weierstrass non differentiable function w b x n 0 n cos 2 b n x for an integer b 2 and 1 b 1 we prove that for every b there exists explicitly given b 1 b 1 such that the hausdorff dimension of the graph of w b is equal to d 2 log log b for every b 1 we also show that the dimension is equal to d for almost every on some larger interval this partially solves a well known thirty year old conjecture furthermore we prove that the hausdorff dimension of the graph of the function f x n 0 n b n x for an integer b 2 and 1 b 1 is equal to d for a typical z periodic c 3 function

fractional derivatives of weierstrass type functions for a special type of fractional differentiation formulas for the weierstrass and the weierstrass mandelbrot functions are shown for integer valued parameters a well conditioned numerical procedure for computing the derivatives in the mean of these functions is derived and used to compute some values

brief sampled data iterative learning control for nonlinear systems with arbitrary relative degree in this paper a sampled daata iterative learning control method is proposed for nonlinear systems without restriction on system relative degree the learning algorithm does not require numerical differentiations of any order from the tracking error a sufficient condition is derived to guarantee the convergence of the system output at each sampling instant to the desired trajectory numerical simulation is conducted to demonstrate the theoretical result

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

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

iterative learning control design based on composite energy function with input saturation in this work an iterative learning control scheme is designed for a class of nonlinear uncertain systems with input saturation the analysis of convergence in the iteration domain is based on composite energy function which consists of both input and state information along the time and iteration axes through rigorous analysis the learning convergence in the iteration domain can be guaranteed under the input saturation provided the desired trajectory is realizable within the saturation bound

weierstrass functions with random phases consider the function f te x sum n 0 infty b n al g b nx te n where b 1 0 al 1 and g is a non constant 1 periodic lipschitz function the phases te n are chosen independently with respect to the uniform probability measure on 0 1 we prove that with probability one we can choose a sequence of scales delta k searrow 0 such that for every interval i of length abs i delta k the oscillation of f te satisfies osc f te i geq c abs i al moreover the inequality osc f te i geq c abs i al varepsilon is almost surely true at every scale when b is a transcendental number these results can be improved the minoration osc f te i geq c abs i al is true for every choice of the phases te n and at every scale

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