stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

traveling solutions and evolution properties of the higher order camassa holm equation abstract this paper investigates general mild traveling solutions and the stationary solution of the higher order camassa holm equation by argument of regularity and conservation we construct admitted traveling solutions of the second order and the third order camassa holm equations furthermore evolution procedure and evolution properties of a traveling solution are discussed the discussion reveals the periodic behavior of the traveling solution curve itself and its curvature these investigations are valuable to understand forming and evolution of a pattern

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

on finity countability cardinalities and cylindric extensions of type 2 fuzzy sets in linguistic summarization of databases the discussion in this paper is closely related to our idea of a type 2 linguistic summary of a database presented by niewiadomski a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008 which is intended to be an efficient tool of knowledge discovery from large databases in that approach we put emphasis on algorithms and applications of type 2 fuzzy sets in summarizing databases hence we have implicitly assumed and considered linguistic expressions represented by sets in discrete or even finite universes of discourse now in this paper we discuss more thoroughly some of the properties and formal aspects of both discrete and continuous type 2 fuzzy sets that represent the elements of linguistic summaries i e quantifiers summarizers and or qualifiers we underline differences between type 2 fuzzy sets in finite and in infinite usually continuous universes of discourse and henceforth between their cardinalities and similar measures which are used in evaluating the goodness quality of type 2 linguistic summaries in addition we define new imprecision measures of linguistic expressions represented by type 2 fuzzy sets and propose a definition of the cylindric extension of a type 2 fuzzy set finally we apply these new concepts to generalize algorithms of type 2 linguistic summarization and propose some new features in comparison to our previous approach a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

global existence for the higher order camassa holm shallow water equation abstract in this paper we investigate the global existence of the higher order camassa holm equation in the case of k 2 we prove the local well posedness of this equation and find a conservation law then a global existence result is obtained

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

well posedness of higher order camassa holm equations abstract we consider higher order camassa holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane we establish the existence of global weak solutions we also present some invariant spaces under the action of the equation moreover we prove a weak equals strong uniqueness result

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

functional deconvolution in a periodic setting uniform case we extend deconvolution in a periodic setting to deal with functional data the resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation in the case when it is observed at a finite number of distinct points the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model we derive minimax lower bounds for the l 2 risk in the proposed functional deconvolution model when f is assumed to belong to a besov ball and the blurring function is assumed to possess some smoothness properties including both regular smooth and super smooth convolutions furthermore we propose an adaptive wavelet estimator of f that is asymptotically optimal in the minimax sense or near optimal within a logarithmic factor in a wide range of besov balls in addition we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points as an illustration we discuss particular examples for both continuous and discrete settings

adaptive sup norm estimation of the wigner function in noisy quantum homodyne tomography in quantum optics the quantum state of a light beam is represented through the wigner function a density on r 2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics in the framework of noisy quantum homodyne tomography with efficiency parameter 1 2 1 we study the theoretical performance of a kernel estimator of the wigner function we prove that it is minimax efficient up to a logarithmic factor in the sample size for the l risk over a class of infinitely differentiable functions we also compute the lower bound for the l 2 risk we construct an adaptive estimator i e which does not depend on the smoothness parameters and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size finite sample behaviour of our adaptive procedure is explored through numerical experiments

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

adaptive wavelet estimator for nonparametric density deconvolution 1999 institute of mathematical statistics the electronic version of this article is the complete one and can be found online at http projecteuclid org euclid aos 1017939249

global uniform risk bounds for wavelet deconvolution estimators we consider the statistical deconvolution problem where one observes n replications from the model y x where x is the unobserved random signal of interest and is an independent random error with distribution under weak assumptions on the decay of the fourier transform of we derive upper bounds for the finite sample sup norm risk of wavelet deconvolution density estimators f n for the density f of x where f r r is assumed to be bounded we then derive lower bounds for the minimax sup norm risk over besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds we further show that linear estimators adapt to the unknown smoothness of f if the fourier transform of decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if decays polynomially we also analyze the case where f is a supersmooth analytic density we finally show how our results and recent techniques from rademacher processes can be applied to construct global confidence bands for the density f

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

a framework for discrete integral transformations ii the 2d discrete radon transform although naturally at the heart of many fundamental physical computations and potentially useful in many important image processing tasks the radon transform lacks a coherent discrete definition for two dimensional 2d discrete images which is algebraically exact invertible and rapidly computable we define a notion of 2d discrete radon transforms for 2d discrete images which is based on summation along lines of absolute slope less than 1 values at nongrid locations are defined using trigonometric interpolation on a zero padded grid our definition is shown to be geometrically faithful the summation avoids wrap around effects our proposal uses a special collection of lines in mathbb r 2 for which the transform is rapidly computable and invertible we describe a fast algorithm using o n log n operations where n n 2 is the number of pixels in the image the fast algorithm exploits a discrete projection slice theorem which associates the discrete radon transform with the pseudopolar fourier transform a averbuch et al siam j sci comput 30 2008 pp 764 784 our definition for discrete images converges to a natural continuous counterpart with increasing refinement

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

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

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

conservative solutions for higher order camassa holm equations this paper studies the existence of global solutions to higher order camassa holm equations global solution is constructed by the small viscosity method for the frequency localized equation especially global solution is energy conservative for given finite band initial data

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

empirical risk minimization in inverse problems we study estimation of a multivariate function f r d r when the observations are available from the function af where a is a known linear operator both the gaussian white noise model and density estimation are studied we define an l 2 empirical risk functional which is used to define a net minimizer and a dense empirical risk minimizer upper bounds for the mean integrated squared error of the estimators are given the upper bounds show how the difficulty of the estimation depends on the operator through the norm of the adjoint of the inverse of the operator and on the underlying function class through the entropy of the class corresponding lower bounds are also derived as examples we consider convolution operators and the radon transform in these examples the estimators achieve the optimal rates of convergence furthermore a new type of oracle inequality is given for inverse problems in additive models

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

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

cardinality quantifiers and the aggregation of fuzzy criteria abstract in this paper we are interested in counting the number of elements of a fuzzy set we study two concepts of fuzzy cardinality one which gives the answer as a fuzzy set or fuzzy number and the other which gives an ordinary integer relationships between these concepts and others which have been studied in the literature e g sigma count are also investigated some properties of the new concepts are studied such as fuzzy convexity monotonicity and additivity the concepts of fuzzy cardinality are important for a better understanding of fuzzy quantified rules of the form qx s are a or qa s are b where q is a fuzzy quantifier and a b are fuzzy sets we study such rules as well as aggregation rules of the form q of a1 ap contain x

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

cardinality concepts for type two fuzzy sets in this paper we define generalized concepts of cardinality of a fuzzy valued function and obtained some properties of these new concepts also we investigate examples for the calculation of two generalized cardinality of fuzzy valued functions and compared with concepts of cardinality of a fuzzy set and a fuzzy valued function

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

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

correlation of fuzzy numbers abstract in this paper we introduce the concepts of correlation and correlation coefficient of fuzzy numbers which could be used to measure the interrelation of fuzzy numbers we also give a brief discussion about the so called information energy of fuzzy numbers

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

extensions of fuzzy aggregation abstract we discuss the equivalence between aggregation of fuzzy sets and integration with respect to a special class of non additive set functions both fuzzy integral and choquet integral are considered first we study aggregation of a finite family of fuzzy sets and then we extend our results to aggregation of an infinite family the concepts of comonotonic maxitivity and additivity play a central role we argue that for the purpose of aggregating fuzzy sets comonotonic maxitivity is a more desirable requirement than comonotonic additivity in the absence of any such requirement we explore a wider class of aggregation procedures another important subject that we study is a comparison between two aggregation functions and we obtain a simple characterization in both the finite and infinite case our results are related to concepts that were studied in the three areas where non additive set functions and integrals were found particularly useful fuzzy sets theory mathematical economics and mathematical statistics

decay property of solutions near the traveling wave solutions for the second order camassa holm equation abstract this paper studies the decay properties of solutions around the traveling wave solutions for cauchy problem of the second order camassa holm equation applying the extended pseudo conformal transformation methods appearing the relevant works on the generalized korteweg de vries equation kdv and nonlinear schrodinger equation nls from martel and merle the solution is controlled by the decaying function with exponential speed corresponding to the initial data and its second derivative with exponential decay

uniform limit theorems for wavelet density estimators let pn y k y k jn 1 l 0 k lk 2l 2 2 l y k be the linear wavelet density estimator where are a father and a mother wavelet with compact support are the empirical wavelet coefficients based on an i i d sample of random variables distributed according to a density p 0 on r and j n z j n several uniform limit theorems are proved first the almost sure rate of convergence of sup y r pn y ep n y is obtained and a law of the logarithm for a suitably scaled version of this quantity is established this implies that sup y r p n y po y attains the optimal almost sure rate of convergence for estimating p 0 if j n is suitably chosen second a uniform central limit theorem as well as strong invariance principles for the distribution function of p n that is for the stochastic processes n f w n s f s n s pn p 0 s r are proved and more generally uniform central limit theorems for the processes n p n p 0 f f for other donsker classes f of interest are considered as a statistical application it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by donoho et al ann statist 24 1996 508 539

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

a type 2 fuzzy approach to linguistic summarization of data this paper introduces an application of type 2 fuzzy sets in data linguistic summarization the original approach by yager 1982 based on representing natural language statements via type 1 i e the zadeh fuzzy sets is generalized with type 2 fuzzy sets applied as models of linguistically expressed quantities and or properties of objects type 2 sets extend the known summarization procedures by handling fuzzy values stored in databases and allow to represent a linguistic term via a few different membership functions e g provided by different experts which makes the method more general and human consistent furthermore quality measures for type 2 summaries are discussed in order to evaluate the informativeness of the messages generated finally two prototype applications are presented and the success of the new method is discussed