a variational method in image recovery this paper is concerned with a classical denoising and deblurring problem in image recovery our approach is based on a variational method by using the legendre fenchel transform we show how the nonquadratic criterion to be minimized can be split into a sequence of half quadratic problems easier to solve numerically first we prove an existence and uniqueness result and then we describe the algorithm for computing the solution and we give a proof of convergence finally we present some experimental results for synthetic and real images

a partial regularity result for an anisotropic smoothing functional for image restoration in bv space abstract here we examine the partial regularity of minimizers of a functional used for image restoration in bv space this functional is a combination of a regularized p laplacian for the part of the image with small gradient and a total variation functional for the part with large gradient this model was originally introduced in chambolle and lions using the laplacian due to the singular nature of the p laplacian we study a regularized p laplacian we show that where the gradient is small the regularized p laplacian smooths the image u in the sense that u c 1 for some 0 1 this functional thus anisotropically smooths the image where the gradient is small and preserves edges via total variation where the gradient is large

variable exponent bounded variation spaces in the riesz sense abstract in this paper we introduce variable exponent bounded variation spaces in the riesz sense prove some embedding results and finally we show that a type of riesz representation lemma is valid in the newly introduced spaces

cohomological study on variants of the mumford system and integrability of the noumi yamada system the purpose of this paper is twofold the first is to apply the method introduced in the works of nakayashiki and smirnov 11 12 on the mumford system to its variants the other is to establish a relation between the mumford system and the isospectral limit mathcal q g i and mathcal q g ii of the noumi yamada system 15 as a consequence we prove the algebraically completely integrability of the systems mathcal q g i and mathcal q g ii and get explicit descriptions of their solutions

testing in normal mixture models when the proportions are known summary the aim of this paper is to study the behaviour of the likelihood ratio test statistic in mixture models when the proportions of the distributions are known a priori the results depend on the value of the proportion the dimensionality of the problem and whether or not the variance of the distributions is known

global action angle variables and the characterization of one degree of freedom rational hamiltonians we analyse the problem of constructing global complex action angle variables for a class of hamiltonians rational in q and quadratic in p we give a complete description of the monodromy of the action variables in terms of subgroups of unimodular transformations associated to the singular energy points we also consider a peculiar subclass of hamiltonian systems which are reducible to algebraically complete integrable systems and show that the reduction preserves the symplectic structure as a consequence the monodromy of the action variable is induced by the one of the reduced system finally we consider the problem of defining global angle variables we propose to define a set of angle variables which correspond to a group of poisson structures associated to

the full geometry of kowalewski s top and 1 2 abelian surfaces revue sur les varietes abeliennes les automorphismes d une courbe elliptique surfaces abeliennes t 2 de polarisation 1 2 jacobiens hyperelliptiques comme recouvrements doubles de t 2 les tores t 2 comme doubles recouvrements des jacobiens hyperelli courbe de kowalewski et surfaces abeliennes de polarisation 2 4

mixing two sets of noisy measurements changes the n dependence of resolution to a fourth root power law if noise is uncorrelated during repeated measurements of the same physical variable averaging these measurements improves the accuracy of estimating the variable when two values of a variable are measured separately the smallest separation of these two values that can be discriminated with a certain reliability resolution is inversely proportional to the square root of the number of measurements employed however if measurements for these two values are mixed together they need to be clustered before being averaged distinguishing mixed clusters with small separations can be thought of as a problem of deciding the number of components in a finite mixture model using the likelihood ratio the second moment estimator and the k means clustering methods we will show that a similarly defined resolution for the mixed scenario is approximately inversely proportional to the fourth root of the number of measurements the observed fourth root law is explained in terms of some more intuitive properties of the problem we also conclude that assuming that the fourth root law is universal the methods reported here are near optimal

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

total variation time flow with quantile regression for image restoration abstract we analyze the time flow of a modification of the standard rudin osher fatemi rof model for image restoration using the bounded variation semi norm the solution of the minimization problem is a restored image in the space of bounded variation bv recovered from a corrupted one and the functional may include a more general term s x u rather than the l 2 penalty term as used in the original rof model these new models were introduced in osher and scherzer 2004 31 in particular we will examine the non differentiable quantile regression penalty we will define an appropriate weak solution of the time flow and in particular prove the existence in l 0 b v and l stability

variable exponent bergman spaces abstract in this article we define variable exponent bergman spaces and show that polynomials are dense in the spaces we also show that the bergman projection and the berezin transform are bounded in these spaces

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

remarks on kowalevski s top we present a review of some results on kowalevski s top kt in classical and quantum mechanics the following items are considered i the generalized kt gkt on the lie algebras so 4 e 3 and so 3 1 ii kowalevski s gyrostat on these algebras iii action of the gkt iv quantum counterparts of the kt v semiclassical quantization of the gkt vi generalization of the kt by chaplygin and goryachev at l 0 and its lax representation unsolved questions are also discussed

a new integrable system on s2 with the second integral quartic in the momenta we construct a new integrable system on the sphere s2 with an additional integral of fourth order in the momenta using standard machinery of the reflection equation theory at the special values of parameters this system coincides with the kowalevski goryachev chaplygin system

testing the number of components in a normal mixture we demonstrate that under a theorem proposed by vuong the likelihood ratio statistic based on the kullback leibler information criterion of the null hypothesis that a random sample is drawn from a k 0 component normal mixture distribution against the alternative hypothesis that the sample is drawn from a k 1 component normal mixture distribution is asymptotically distributed as a weighted sum of independent chi squared random variables with one degree of freedom under general regularity conditions we report simulation studies of two cases where we are testing a single normal versus a two component normal mixture and a two component normal mixture versus a three component normal mixture an empirical adjustment to the likelihood ratio statistic is proposed that appears to improve the rate of convergence to the limiting distribution

an integrable hamiltonian motion on a sphere ii the separation of variables we consider the motions on the unit sphere in a potential which have been shown earlier to derive from a completely integrable hamiltonian the separation of variables is completed in terms of an independent variable u which differs from the canonical time t where m and are the two integrals of motion s is the action and is the integration constant constant along each trajectory finally a lax pair is deduced from a separable form of the system lax p d 1968 commun pure appl math 21 467

gas clouds rotating around a principal axis ii the separation of variables we present a new formulation of the spinning gas cloud model of ovsiannikov and dyson in the case of rotation around a fixed axis in terms of a 4 4 symmetric matrix which is a linear function of four coordinates xi we obtain a pair of separable variables l1 l2 related to the xi by a simple transformation and discuss their geometrical meaning

there s more than one way to skin schro dinger s cat abstract two methods for studying quantum discrete dynamical systes i e with a finite or countably infinite number of degrees of freedom are described and compared for i the discrete self trapping dst equation ii the ablowitz ladik al equation and iii a fermionic polaron fp model the first method called the quantum inverse scattering method qism emphasizes the relationship between the quantum theory and classical integrability the second method which we call the number state method nsm cam be extended to systems that are not classically integrable thus the two methods are not equivalent

lax representation for two particle dynamics splitting on two tori the lax representation in terms of matrices is constructed for separable multiply periodic systems splitting on two tori hyperelliptic kleinian functions and their reduction to elliptic functions are used

a special case of neumann s system and the kowalewski chaplygin goryachev top the authors present l operators both for the special case of neumann s system and the kowalewski chaplygin goryachev top these are quantum systems integrable by the quantum inverse scattering r matrix method l operators being 2 2 matrices satisfy an algebra generated by an r matrix of the xxx type the close connection between the two models is demonstrated they carry out a non obvious separation of variables and also give a dynamical group scheme for the eigenstate problem of neumann s system this separation differs from those in euler angles and allows them to find eigenenergies in an effective way

a completely integrable hamiltonian motion on the surface of a sphere it has been shown gaffet b 1996 j fluid mech 325 113 that the hamiltonian in three dimensional flat space defined by the simple equations of motion is reducible by transformations of variables to another hamiltonian governing the two dimensional motion of a particle on the surface of a sphere the equations of motion were shown to possess the painleve property and their integration was reduced to essentially one quadrature the earlier analysis essentially concerned the comparatively much simpler case where the second integral vanishes all such solutions were found to be described by elliptic functions we show in the present work that when the energy integral denoted m and the second integral denoted are related by the general solution is again represented by elliptic functions the separation of variables is completed and the solution presented in detail for the case m 3 chosen as an example

application of model selection criteria to some problems in multivariate analysis a review of model selection criteria is presented with a view toward showing their similarities it is suggested that some problems treated by sequences of hypothesis tests may be more expeditiously treated by the application of model selection criteria consideration is given to application of model selection criteria to some problems of multivariate analysis especially the clustering of variables factor analysis and more generally describing a complex of variables

hyperelliptic prym varieties and integrable systems we introduce two algebraic completely integrable analogues of the mumford systems which we call hyperelliptic prym systems because every hyperelliptic prym variety appears as a fiber of their momentum map as an application we show that the general fiber of the momentum map of the periodic volterra lattice is an affine part of a hyperelliptic prym variety obtained by removing n translates of the theta divisor and we conclude that this integrable system is algebraic completely integrable

additive deformations of the r matrix algebras we show how to construct new representations of the various r matrix algebras starting from known representations for linear r matrix algebras we investigate a dynamical r matrix which depends on the spectral parameter and half of the dynamical variables particle coordinates only the toda lattices and the henon heiles systems illustrate the scheme

the kowalewski top a new lax representation the 2 2 monodromy matrices for the kowalewski top on the lie algebras e 3 so 4 and so 3 1 are presented the corresponding quadratic r matrix structure is the dynamical deformation of the standard r matrix algebras some tops and toda lattices related to the kowalewski top are discussed

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

interpolation on variable morrey spaces defined on quasi metric measure spaces abstract in this paper we show the validity of a riesz thorin type interpolation theorem for linear operators acting from variable exponent lebesgue spaces into variable exponent morrey space in the framework of quasi metric measure spaces

multi hamiltonian structures for a class of degenerate completely integrable systems in this paper a class of degenerate i e associated to a degenerate poisson structure completely integrable systems is studied which generalizes the so called odd and even master systems introduced and studied by mumford and by vanhaecke it is shown that all these completely integrable systems called the generalized master systems admit a multi hamiltonian formulation and a systematic construction of this multi hamiltonian structure is described

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

on integrable deformations of the spherical top the motion on the sphere s2 in potential v x1x2x3 2 3 is considered the lax representation and the linearization procedure for this two dimensional integrable system are discussed

generalized polyspheroidal periodic functions and the quantum inverse scattering method bounded and periodic eigenfunctions of the operator d2 dv2 2 1 cos 2v sin 2v d dv 8 cos 2v 4 2 sin2 2v are studied these solutions are new special functions particular cases of which will be the spheroidal the coulomb spheroidal and the hyperspheroidal harmonics the algebraic technique due to the quantum inverse scattering method is applied to obtain the three term recursion relations for the coefficients of eigenfunctions expansion at the jacobi polynomials series and hence the suitable equation the continued fraction is equal to zero for eigenvalues the special functions considered generalize the truskova polyspheroidal periodic functions 0

from arzela ascoli to riesz kolmogorov abstract in this paper we study totally bounded sets in banach function spaces bfs from which we characterize compact sets via hausdorff criterion in some non standard function spaces which fall under the umbrella of bfs we obtain a riesz kolmogorov compactness theorem for the grand variable exponent lebesgue spaces

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

on time flows of minimizers of general convex functionals of linear growth with variable exponent in bv space and stability of pseudosolutions here we examine a method of selective smoothing in image restoration in bv space using a variable exponent functional of linear growth variable exponent growth was suggested in p blomgren et al 1997 6 and in y chen et al 2006 8 the notion of a pseudosolution of the minimization problem is discussed as well as the proof of existence and uniqueness of the solution of the time flow for the associated time dependent pde here we prove stability for pseudosolutions while also discussing the case for more general convex functionals of linear growth and variable exponent for use as an approximation for the time flow other than that which is used in y chen et al 2006 8

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

stationary problems for equation of the kdv type and dynamical r matrices we study a new quite general family of dynamical r matrices for an auxiliary loop algebra l su 2 related to restricted flows for equations of the kdv type this underlying r matrix structure allows us to reconstruct lax representations and to find variables of separation for a wide set of the integrable natural hamiltonian systems

kowalewski s top on the lie algebras o 4 e 3 and o 3 1 the authors extend the lax pair for kowalewski s top kt obtained by haine and horozov 1987 to the lie algebras o 4 and o 3 1 the results are expressed in terms of a solution of the neumann system they derive formulae for the action variables by analogy with the e 3 kt case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables