a linear code for the sawtooth and cat maps abstract the sawtooth maps are a one parameter set of piecewise linear area preserving maps on the torus for positive integer values of the parameter k they are automorphisms of the torus known as the cat maps we present a symbolic dynamics for these maps in which the symbols are integers this code is related to a practical problem of the stabilisation of a system which is perturbed by impulses the code is linear in the sense that an orbit and its code are linearly related so it is not difficult to obtain a good approximation to one from the other in practice a stationary stochastic process for generating the code is given explicitly the theory uses green function methods which are also used to study ordered periodic orbits and cantori the problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed

transient measures in the standard map abstract for an area preserving map each chaotic orbit appears numerically to densely cover a region an irregular component of nonzero area surprisingly the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component most prominently there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles we show that this phenomena is transient and therefore numerical ergodicity on the irregular component eventually obtains though the times involved are extremely long 1010 iterates a markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class for example a rotational circle however the density spikes do occur in a markov tree model that includes the effects of islands around islands

boundary circles for area preserving maps abstract we report on numerical observations of various properties of boundary circles for area preserving maps we find that the continued fraction expansion of their rotation number generally has a special form we describe how their rotation number changes as parameters are varied we observe a statistically self similar structure in the neighbourhood of boundary circles which is probably relevant to the long time tails of the decay of correlations in the irregular components

quasiregularity and rigorous diffusion of strong hamiltonian chaos exact results are derived concerning quasiregularity and diffusion of strong chaos on resonances of the sawtooth map a chaotic ensemble of well defined quasiregularity type the sequence of resonances visited is generally a fractal set whose main characteristics the topological entropy and the hausdorff dimension are calculated exactly under some conditions using a symbolic dynamics the effect of quasiregularity on chaotic diffusion is characterized by an infinity of diffusion coefficients each associated with a fractal ensemble trapped in a periodic set of resonances in some cases these coefficients are calculated exactly and it is shown that rigorous diffusion takes place on the resonances

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

generalized tops and moment maps to loop algebras using a combination of hamiltonian reduction under continuous and discrete symmetry groups the lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable hamiltonian systems in loop algebras

classical functional bethe ansatz for sl n separation of variables for the magnetic chain it is desirable for one to be able to construct separation variables for systems with sl n r matrices or lax pairs this article relates and extends the two known methods of construction the sklyanin s functional bethe ansatz fba commun math phys 150 181 191 1992 was previously only known for sl 2 and sl 3 and associated r matrices the algebraic geometric method of adams harnad and hurtubise commun math phys 155 385 413 1993 has been shown to work for sl n but up until now only for linear poisson brackets in this article sklyanin s program is advanced by giving the fba for certain systems with sl n r matrices this is achieved by constructing rational functions a u and b u of the matrix elements of t u so that in the generic case the zeros xi of b u are the separation coordinates and the pi a xi provide their conjugate momenta it is shown that the separation variables thus defined are the same as those given as the simultaneous solutions of several equations in two

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

a devil s staircase into chaotic scattering abstract we discuss analytically the transition from a regular to a chaotic classical dynamics of a sample scattering model introduced earlier the transition arises as a homoclinic bifurcation of an atypical type in phase space as a discontinuity for the grammatical complexity and as an onset of a devil s staircase for the topological entropy

hyperbolic cantori have dimension zero it is proved that cantori for area preserving twist maps have dimension zero only finitely many orbits of gaps and gap widths go to zero exponentially in each orbit if they are hyperbolic

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

rigorous diffusion properties for the sawtooth map we get a rigorous bound for the diffusion constant of the hamiltonian dynamical system generated by a sawtooth map on a cylinder the momentum variable properly renormalized then behaves almost like a brownian motion in the limit of infinite coupling constants the strategy of the proof is a rigorous reformulation of the random phase approximation

poisson structures on moduli spaces of sheaves over poisson surfaces we introduce and study the notion of poisson surface we prove that the choice of a poisson structure on a surfaces canonically determines a poisson structure on the moduli space of stable sheaves ons this result generalizes previous results obtained by mukai 14 for abelian ork3 surfaces and by tyurin 16

wavelet frame based surface reconstruction from unorganized points applications of wavelet frames to image restoration problems e g image deblurring and inpainting have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images see e g 13 14 8 however wavelet frames have not yet been used for surface reconstruction problems recently in 6 connections between one of the wavelet frame based image restoration model 8 28 48 and variational models e g the rof model 45 were rigorously established such connections not only grant new insights to wavelet frame based image restorations it also case a geometric explanation to wavelet frame based approaches this leads us to a wavelet frame based model as well as a fast algorithm to reconstruct implicit surfaces from unorganized point sets in r 3 we will demonstrate the effectiveness of the proposed model using several commonly used examples

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

wavelet frame based scene reconstruction from range data abstract how to reconstruct the scene a visible surface from a set of scattered noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics as most real scenes can be modeled by piecewise smooth surfaces traditional surface fitting techniques e g smoothing spline generally can not preserve sharp discontinuities of surfaces based on sparse approximation of piecewise smooth functions in frame domain we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set which is robust to both additive noise and outliers furthermore the resulting minimization problem from our formulation can be efficiently solved by the split bregman method 1 2 the numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers

ergodic properties of the discontinuous sawtooth map we prove the ergodicity for the discontinuous sawtooth map adapting a technique previously used in billiard theory the core of the proof is the construction of a hopf chain passing through a countable dense set of discontinuity lines

diffusion in the standard map abstract diffusion in the standard map is studied numerically the stochasticity parameter k is near the critical value k c and the diffusion coefficient d is calculated it is found to satisfy to a good approximation the scaling relation d k k c with in good agreement with the value predicted by the scaling theory of the disappearance of the last bounding kam torus the critical region where this scaling relation holds is surprisingly large i e k 2 5 the mechanism of transport from the chaotic region to the remnants of the last kam torus is investigated evidence for the existence of a narrow stochastic channel mediating this transport is presented and its origin is discussed although the scaling of d agrees with the predictions of the scaling theory the transport mechanism is different from the one assumed in this theory

compactly supported tight affine spline frames in l 2 r d the theory of fiberization is applied to yield compactly supported tight affine frames wavelets in l 2 r d from box splines the wavelets obtained are smooth piecewise polynomials on a simple mesh furthermore they exhibit a wealth of symmetries and have a relatively small support the number of mother wavelets however increases with the increase of the required smoothness two bivariate constructions of potential practical value are highlighted in both the wavelets are derived from four direction mesh box splines that are refinable with respect to the dilation matrix 1 1 1 1

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

splitting of separatrices and symbolic dynamics for some degenerate standard like maps we consider standard like maps of class c1 with parabolic fixed points and such that the second derivative has a jump discontinuity at the fixed points an asymptotic formula for the splitting of the separatrices is obtained the structure of certain chaotic set in the stochastic layer around the separatrices is described in terms of symbolic dynamics it turns out in particular that swinging and rotating regimes of motion may alternate in time arbitrarily analogous results are stated for the so called separatrix mapping and for the more smooth standard like maps

chaotic scattering modelled by an inclined billiard abstract chaotic scattering as observed in satellite encounters and in other problems is modelled by a simple billiard a point particle bounces on two disks and in addition is subjected to a constant acceleration in an appropriate limit this problem reduces to a piecewise linear mapping we consider a one parameter family of orbits obtained by starting the particle at rest from a constant height we find that this family exhibits chaotic scattering it has an infinite number of discontinuities forming a self similar cantor set with the help of symbolic dynamics the structure of the family can be completely elucidated

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

vector bundles and lax equations on algebraic curves the hamiltonian theory of zero curvature equations with spectral parameter on an arbitrary compact riemann surface is constructed it is shown that the equations can be seen as commuting flows of an infinite dimensional field generalization of the hitchin system the field analog of the elliptic calogero moser system is proposed an explicit parameterization of hitchin system based on the tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained

periodic orbits of the sawtooth maps abstract the sawtooth maps are a one parameter family of chaotic area preserving maps of the 2 torus which become anosov systems for special values of the parameter we develop a technique for constructing explicitly all their periodic orbits as perturbations of anosov orbits combining methods of arithmetic and symbolic dynamics we find that the parameter dependence of periodic orbits is given by rational functions and compute numerically the population of periodic orbits as a function of the parameter

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

semiclassical quantization of sawtooth map with inclusion of next leading order and its failure the semiclassical quantization of sawtooth maps is reconsidered from the viewpoint of the higher order terms in the asymptotic series we carry out the semiclassical quantization in the leading and next leading orders however we fail in both cases the reasons are discussed in detail

darboux coordinates and liouville arnold integration in loop algebras darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part tilde mathfrak g of loop algebras these are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves defined within a given matrix representation a liouville generating function is obtained in completely separated form and shown through the liouvile arnold integration method to lead to the abel map linearization of all hamiltonian flows induced by the spectral invariants as illustrative examples the caseg sl 2 together with its real forms is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics with the liouville generating function expressed in hyperellipsoidal coordinates forg sl 3 the method is applied to the computation of quasi periodic solutions of the two component coupled nonlinear schrodinger equation a case which requires further symplectic constraints in order to deal with singularities in the spectral data at

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

hamiltonian transport on unstable periodic orbits transport in hamiltonian systems is approached on the basis of the unstable periodic orbits upos embedded in the chaotic region correlation functions and diffusion rates are defined for upo ensembles of two dimensional periodic maps global diffusion in these maps is interpreted in terms of the ensembles ut containing all the upos of period t as t the relative populations of closed orbits and accelerator modes in ut approach a gaussian distribution characterized by the diffusion coefficient d evidence for the validity of this mechanism is provided for the cat and sawtooth maps prime lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion these ensembles are generically small subsets of ut for some t it is shown however that the corresponding diffusion rates assume for t odd precisely the known value of d in the entire time interval t t for t even all orbits are closed and d is reproduced only in the interval t 12 t the diffusion rates associated with ensembles ut of the sawtooth maps appear to approximate well the value of d if t is of the order of the diffusion time the case of local transport rates associated with ensembles that are localized in phase space is illustrated

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

l minimization for wavelet frame based image restoration the theory of tight wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems the success of wavelet frame based models including balanced approach 18 7 and analysis based approach 11 31 50 is due to their capability of sparsely approximating piecewise smooth functions like images motivated by the balanced approach and analysis based approach we shall propose a wavelet frame based l0 minimization model where the l0 norm of the frame coefficients is penalized we adapt the penalty decomposition pd method of 40 to solve the proposed optimization problem some convergence analysis of the adapted pd method will also be provided numerical results showed that the proposed model solved by the pd method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images

resonances and transport in the sawtooth map we study transport in a completely chaotic hamiltonian system the hyperbolic sawtooth map analytical expressions are obtained for its cantori and resonances we show that resonances give a complete partition of phase space the flux leaking out of a resonance is given by its turnstiles whose form and areas are obtained analytically when the total flux out of a resonance becomes one third the area of an island the topology of the turnstiles changes at the same parameter value a horseshoe is formed corresponding to the orbits trapped within the resonance based on this a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained the partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles we calculate the survival probability within a resonance using the markov model the results are compared with results obtained numerically and from periodic orbit theory

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

markov tree model of transport in area preserving maps abstract transport in an area preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori a model retaining a discrete set of cantori approaching a boundary circle gives the markov chain description of hanson cary and meiss the inclusion of cantori surrounding island chains and islands about islands etc gives a markov tree model with a slower decay rate the survival probability distribution is shown to decay asymptotically as a power law the decay exponent agrees reasonably well with the computations of karney and of chirikov and shepelyanski

algebraic decay in self similar markov chains a continuous time markov chain is used to model motion in the neighborhood of a critical invariant circle for a hamiltonian map states in the infinite chain represent successive rational approximants to the frequency of the invariant circle for the case of a noble frequency the chain is self similar and the nonlinear integral equation for the first passage time distribution is solved exactly the asymptotic distribution is a power law times a function periodic in the logarithm of the time for parameters relevant to the critical noble circle the decay proceeds ast 4 05

markov shifts in the he non family abstract we give evidence for parameter regions in which the set of bounded orbits of the henon map is a hyperbolic set on which the map is equivalent to a subshift of finite type

extended chaos and disappearance of kam trajectories abstract in two dimensional area preserving maps kam trajectories serve as natural boundaries between stochastic regions as they disappear points may leak out from one region to the next this escape rate is defined and related to the action generating function analytic and numerical results are presented to describe the behavior of the escape rate near the disappearance of a kam trajectory

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases