thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

algebraic convergence of schottky groups a discrete faithful representation of the free group on g generators f g into isom h 3 is said to be a schottky group if h 3 d is homeomorphic to a handlebody h g where d is the domain of discontinuity for s action on the sphere at infinity for h 3 schottky space s g the space of all schottky groups is parameterized by the quotient of the teichmuller space f s g of the closed surface of genus g by mod 0 h g where mod 0 h g is the group of isotopy classes of homeomorphisms of s g which extend to homeomorphisms of h g which are homotopic to the identity masur exhibited a domain o h g of discontinuity for mod 0 h g s action on pl s g the space of projective measured laminations on s g so b h g o h g mod 0 h g may be appended to s g as a boundary

entropic approach to interior point solution of linear programs we present a maxentropic approach for obtaining interior suboptimal solutions to a linear programming problem not necessarily in karmarkar s canonical form each suboptimal solution lies in the relative interior set of feasible solutions we present an application to the problem of obtaining reconstructions of minimal weighted l 1 norm of a function from the knowledge of a few of its fourier coefficients

eigenvalues and entropy of a hitchin representation we show that the critical exponent of a representation rho in the hitchin component of mathrm psl d mathbb r is bounded above the least upper bound being attained only in the fuchsian locus this provides a rigid inequality for the area of a minimal surface on rho backslash x where x is the symmetric space of mathrm psl d mathbb r the proof relies in a construction useful to prove a regularity statement if the frenet equivariant curve of rho is smooth then rho is fuchsian

the margulis lemma and the thick and thin decomposition for convex real projective surfaces convex real projective surfaces are quotients of simply convex domains in the real projective planerp2under the properly discontinuous and free action of a projective automorphism group such surfaces carry hilbert metrics defined by logarithms of cross ratios we prove an analogous proposition to the margulis lemma in hyperbolic geometry holding for such a surface with a hilbert metric this allows us to decompose into thick and thin components we show a compactness result that given a certain collection of simple closed curves 1 non the subset of the deformation space of convex structures p corresponding to structures where the hilbert lengths of 1 nare bounded above is compact

an extension of the weil petersson metric to quasi fuchsian space we define a natural semi definite metric on quasi fuchsian space derived from geodesic current length functions and hausdorff dimension that extends the weil petersson metric on teichmuller space we use this to describe a metric on teichmuller space obtained by taking the second derivative of hausdorff dimension and show that this metric is bounded below by the weil petersson metric we relate the change in hausdorff dimension under bending along a measured lamination to the length in the weil petersson metric of the associated earthquake vector of the lamination

limit sets of teichm uller geodesics with minimal nonuniquely ergodic vertical foliation ii given a sequence of curves on a surface we provide con ditions which ensure that 1 the sequence is an innite quasi geodesic in the curve complex 2 the limit in the gromov boundary is repre sented by a nonuniquely ergodic ending lamination and 3 the sequence divides into a nite set of subsequences each of which projectively con verges to one of the ergodic measures on the ending lamination the conditions are suciently robust allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g 3 for example we also study the limit sets in the thurston boundary of teichmuller geodesic rays dened by quadratic dierentials whose vertical foliations are obtained from the constructions mentioned above we prove that such examples exist for which the limit is a cycle the 1 skeleton of the simplex of projective classes of measures visiting every vertex

bayesian methods and maximum entropy for ill posed inverse problems in this paper we study linear inverse problems where some generalized moments of an unknown positive measure are observed we introduce a new construction called the maximum entropy on the mean method mem which relies on a suitable sequence of finite dimensional discretized inverse problems its advantage is threefold it allows us to interpret all usual deterministic methods as bayesian methods it gives a very convenient way of taking into account prior information it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena the key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure

multilevel balancing domain decomposition by constraints deluxe algorithms with adaptive coarse spaces for flow in porous media multilevel balancing domain decomposition by constraints bddc deluxe algorithms are developed for the saddle point problems arising from mixed formulations of darcy flow in porous media in addit

symbolic dynamics for smale flows on introduit une classe de flots plus generaux que les flots de l axiome a et on montre que certains resultats pour ces derniers restent vrais

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

cyclic surfaces and hitchin components in rank 2 we prove that given a hitchin representation in a real split rank 2 group g 0 there exists a unique equivariant minimal surface in the corresponding symmetric space as a corollary we obtain a parametrization of the hitchin components by a hermit ian bundle over teichmuller space the proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of g 0 some partial extensions of the construction hold for cyclic bundles in higher rank

positive configurations of flags in a building and limits of positive representations parreau compactified the hitchin component of a closed surface s of negative euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of pi 1 s on an mathbb r euclidean building in this paper we use the positivity properties of hitchin representations introduced by fock and goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

blind deconvolution of discrete linear systems we study the blind deconvolution problem in the case where the input noise has a finite discrete support and the transfer linear system is not necessarily minimum phase we propose a new family of estimators built using algebraic considerations the estimates are consistent under very wide assumptions the input signal need not be independently distributed the cardinality of the finite support may be estimated simultaneously we consider in particular ar systems in this case we prove that the estimator of the parameters is perfect a s with a finite number of observations

a bddc method for mortar discretizations using a transformation of basis a bddc balancing domain decomposition by constraints method is developed for elliptic equations with discontinuous coefficients discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions the coarse component of the preconditioner is defined in terms of one mortar constraint for each edge face which is the intersection of the boundaries of a pair of subdomains a condition number bound of the form c max i 1 log h i h i 2 is established under certain assumptions on the geometrically nonconforming subdomain partition in the three dimensional case here h i and h i are the subdomain diameters and the mesh sizes respectively in the geometrically conforming case and the geometrically nonconforming cases in two dimensions no assumptions on the subdomain partition are required this bddc preconditioner is also shown to be closely related to the neumann dirichlet version of the feti dp algorithm the results are illustrated by numerical experiments which confirm the theoretical results

marked length spectral rigidity for flat metrics in this paper we prove that the space of at metrics nonpos itively curved euclidean cone metrics on a closed oriented surface is marked length spectrally rigid in other words two at metrics assigning the same lengths to all closed curves dier by an isometry isotopic to the identity

an adaptive bddc algorithm in variational form for mortar discretizations abstract a balancing domain decomposition by constraints bddc algorithm with adaptive primal constraints in variational form is introduced and analyzed for high order mortar discretization of two dimensional elliptic problems with high varying and random coefficients some vector valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm and the coarse space is formed by using a transformation operator on each interface compared with the adaptive bddc algorithms for conforming galerkin approximations our algorithm is simpler because there have not any continuity constraints at subdomain vertices in the mortar method involved in this paper the condition number of the preconditioned system is proved to be bounded above by a user defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain and independent of the mesh size and the contrast of the given coefficients numerical results show the robustness and efficiency of the algorithm for various model problems

bddc and feti dp without matrices or vectors abstract the connection between the bddc balancing domain decomposition by constraints algorithm and the feti dp finite element tearing and interconnecting dual primal algorithm is discussed in the language of function spaces operators and linear functionals

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

anosov flows surface groups and curves in projective space note that in 10 w goldman gives a complete description of these connected components in the case of finite covers of psl 2 r in the case of psl 2 r two homeomorphic components called teichmuller spaces play a central role these two components are well known to be homeomorphic to a ball of dimension 6g 6 n hitchin generalises this situation to psl n r indeed one of these components when n is odd and two when n is even has a very simple topology let us define an n fuchsian representation to be a representation which can be written as 0 where 0 is a cocompact representation with values in psl 2 r and is the irreducible representation of psl 2 r in psl n r we denote by reph 1 s psl n r a connected component that contains fuchsian representations and call it a hitchin component in fact when n is odd there is one hitchin component and when n is even two isomorphic ones n hitchin proves in 17

topological tameness of margulis spacetimes we show that margulis spacetimes without parabolic holonomy elements are topologically tame a margulis spacetime is the quotient of the 3 dimensional minkowski space by a free proper isometric action of the free group of rank geq 2 we will use our particular point of view that the margulis spacetime is a manifold with boundary with an bbb r sf p 3 structure in an essential way the basic tools are a bordification by a closed bbb r sf p 2 surface with free holonomy group and the work of goldman labourie and margulis on geodesics in the margulis spacetimes and 3 manifold topology

local rigidity of discrete groups acting on complex hyperbolic space the superrigidity theorem of margulis see zimmer 17 classifies finite dimensional representations of lattices in semi simple lie groups of real rank strictly larger than 1 it is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank i semi simple lie groups that this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or in other words the existence of non trivial deformations in johnson millson 6 and kourouniotis 8 such deformations were constructed for certain representations of lattices in so n 1 based on a construction of thurston called bending the deformation space of the representation of a lattice f so n 1 obtained by restricting an inclusion so n 1 so n l 1 to f is of particular interest if n 2 the space of infinitesimal deformations is hi f r 1 where f acts on r 1 by the restriction of the standard action of so n 1 the space of infinitesimal deformations is non zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable in 6 it is also shown that some are not in this paper we study the complex analogue of the above example we let f be a cocompact torsion free lattice in su n 1 and consider the deformation space of the representation of f obtained by restricting an inclusion su n 1 su n i 1 if n l the space of infinitesimal deformations is h f r h f c 1 in the first summand f acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming f in u n 1 by a curve of homomorphisms into the center of u n 1 observe that the above inclusion factors as su n 1 u n 1 su n i 1 in the second summand f acts by the restriction of the standard action of su n 1 this summand is non zero for the standard arithmetic examples

adaptive selection of face coarse degrees of freedom in the bddc and the feti dp iterative substructuring methods we propose a class of method for the adaptive selection of the coarse space of the bddc and feti dp iterative substructuring methods the methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures it is assumed that the starting coarse degrees of freedom are already sufficientto prevent relative rigid body motions in any selected pair of adjacent substructures a heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold it is shown numerically on 2d elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost

iteration of mapping classes and limits of hyperbolic 3 manifolds let mod s be an element of the mapping class group of a surface s we classify algebraic and geometric limits of sequences q i x y i 1 of quasi fuchsian hyperbolic 3 manifolds ranging in a bers slice when has infinite order with finite order restrictions there is an essential subsurface d s so that the geometric limits have homeomorphism type s d 0 typically has pseudo anosov restrictions and d has components with negative euler characteristic these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit we show there is an s 1 depending on and bounded in terms of s so that q si x y i 1 converges algebraically and geometrically and we give explicit quasi isometric models for the limits

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

the modular action on psl2 r characters in genus 2 we explore the dynamics of the action of the mapping class group in genus 2 on the psl2 r character variety we prove that this action is ergodic on the connected components of euler class 1 as it was conjectured by goldman in the connected component of euler class 0 there are two invariant open subsets on one of them the action is ergodic in this process we give a partial answer to a question posed by bowditch

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

a bddc algorithm for mortar discretization of elasticity problems a balancing domain decomposition by constraints bddc algorithm is developed for compressible elasticity problems in three dimensions with mortar discretization on geometrically nonconforming subdomain partitions material parameters of the elasticity problems may have jump across the subdomain interface coarse basis functions in the bddc algorithm are constructed from primal constraints on faces which are similar to the average matching condition and the moment matching condition considered in a klawonn and o b widlund comm pure appl math 59 2006 pp 1523 1572 and h h kim a feti dp formulation of three dimensional elasticity problems with mortar discretization technical report new york university 2005 a condition number bound is proved to be c 1 log h h 3 for geometrically nonconforming partitions and to be c 1 log h h 2 for geometrically conforming partitions the bound is not affected by the jump of the material parameters across the subdomain interface numerical results are included

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

strong marked isospectrality of affine lorentzian groups the margulis invariant is a function on h1 2 1 where is a group of lorentzian transformations acting on 2 1 that contains no elliptic elements the spectrum of is the image of all id under the map if the underlying linear group of is fixed drumm and goldman proved that the spectrum defines the translational part completely in this note we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank up to conjugation in the group of affine transformations of 2 1 as long as it is non radiant and that its linear part is discrete and non elementary in particular isospectrality holds when the linear part is a schottky group

a bddc algorithm with adaptive primal constraints for staggered discontinuous galerkin approximation of elliptic problems with highly oscillating coefficients a bddc balancing domain decomposition by constraints algorithm for a staggered discontinuous galerkin approximation is considered after applying domain decomposition method a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner to construct a preconditioner that is robust to the coefficient variations a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance by the construction of the staggered discontinuous galerkin methods the degrees of freedom on subdomain interfaces are shared by only two subdomains and hence the construction of primal unknowns are simplified the resulting bddc algorithm is shown to have the condition number bounded by the predetermined tolerance a modified algorithm for parameter dependent problems is also introduced where the primal unknowns are only computed in an offline stage numerical results are included to show the performance of the proposed method and to verify the theoretical estimate

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

homotopy hyperbolic 3 manifolds are hyperbolic this paper introduces a rigorous computer assisted procedure for analyzing hyperbolic 3 manifolds this procedure is used to complete the proof of several long standing rigidity conjectures in 3 manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3 manifold

the grafting map of teichmuller space one of the underlying principles in the study of kleinian groups is that aspects of the complex projective geometry of quotients of by these groups reflect properties of the three dimensional hyperbolic geometry of the quotients ofh by these groups yet even though it has been over thirty five years since lipman bers exhibited a holomorphic embedding of the teichmuller space of riemann surfaces in terms of the projective geometry of a teichmuller space of quasi fuchsian manifolds no corresponding embedding in terms of the three dimensional hyperbolic geometry has been presented one of the goals of this paper is to give such an embedding this embedding is straightforward and has been expected for some time ta97 mc98 to each member of a bers slice of the space qf of quasi fuchsian 3manifolds we associate the bending measured lamination of the convex hull facing the fixed conformal end the geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting an operation on projective structures on surfaces that traces its roots back at least to klein kl33 p 230 50 with a modern history developed by many authors ma69 he75 fa83 st83 go87 gkm00 ta97 mc98 the main technical tool in our proof that bending measures give coordinates for bers slices and the second major goal of this paper is the completion of the proof of the grafting conjecture this conjecture states that for a fixed measured lamination the self map of teichmuller space induced by grafting a surface along is a homeomorphism of teichmuller space our contribution to this argument is a proof of the injectivity of the grafting map while the principal application of this result that we give is to geometric coordinates on the bers slice of qf one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of kleinian groups see 5 2 thus we feel that this result is of interest in its own right we now state our results and methods more precisely throughout s will denote a fixed differentiable surface which is closed orientable and of genus g 2 let tg be the corresponding teichmuller space of marked conformal structures on s and let pg denote the deformation space of complex projective structures on s see 2 for definitions

statistical regularities of self intersection counts for geodesics on negatively curved surfaces let s be a compact surface with constant negative curvature 1 from among all closed geodesics on of length t choose one at random and letnt be the number of its self intersections we prove that for a certain constant 0 the random variable nt t t has a limit distribution as t we conjecture that for surfaces of variable negative curvature the order of magnitude of typical variations is t rather than t we also prove analogous results for generic geodesics that is geodesics whose initial tangent vectors are chosen randomly according to normalized liouville measure date november 10 2011 1991 mathematics subject classification primary 57m05 secondary 53c22 37d40

simple length rigidity for hitchin representations abstract we show that a hitchin representation is determined by the spectral radii of the images of simple non separating closed curves as a consequence we classify isometries of the intersection function on hitchin components of dimension 3 and on the self dual hitchin components in all dimensions as an important tool in the proof we establish a transversality result for positive quadruples of flags

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

the klein hilbert and poincare metrics of a domain abstract let d be a bounded strictly convex domain in euclidean n space equipped with its hilbert metric h x y it is shown that as the points x and y of d approach distinct points on the boundary of d for any a in d the sum h x a h a y is asymptotic to h x y

adaptive selection of primal constraints for isogeometric bddc deluxe preconditioners isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer aided design cad software and the discretization of variational problems of continuum mechanics in contrast with the finite element case the basis functions of isogeometric analysis are often not nodal as a consequence there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains building on earlier work on the deluxe version of the bddc balancing domain decomposition by constraints family of domain decomposition algorithms several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global coarse component of these preconditioners numerical experiments provide evidence that this work can be successful yielding scalable and quasi optimal adaptive bddc algorithms for isogeometric discretizations

affine action and margulis invariant abstract in this paper we show that two zariski dense subgroups consisting of hyperbolic elements in so n 1 n r 2n 1 with the same marked margulis invariant are conjugate we also consider in affine deformations an analogue of quasifuchsian deformation of fuchsian groups

on entropy regularity and rigidity for convex representations of hyperbolic manifolds given a convex representation rho gamma rightarrow mathrm pgl d mathbb r of a convex cocompact group gamma of mathrm isom mathbb h k we find upper bounds for the quantity alpha h rho where h rho is the entropy of rho and alpha is the holder exponent of the equivariant map partial infty gamma rightarrow mathbb p mathbb r d we also give rigidity statements when the upper bound is attained this provides an analog of thurston s metric for convex cocompact groups of mathrm isom mathbb h k we then prove that if rho pi 1 sigma rightarrow mathrm psl d mathbb r is in the hitchin component then alpha h rho le 2 d 1 where alpha is the holder exponent of labourie s equivariant flag curve with equality if and only if rho is fuchsian

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

representation theoretic rigidity in psl 2 r let g be a connected simple lie group with trivial center let f be an abstract group and let 51 and 2 be inclusions of f in g assume throughout that each of the images 5j f is a lattice subgroup meaning that j f is discrete and that the g invariant measure on g s j f has total finite mass we say that tl and are equivalent if there is some automorphism 0 of g so that 2 qo51 if g is not isomorphic to psl 2 r then the mostow rigidity theorem see 18 19 16 and 24 says that 1 and t2 axe necessarily equivalent alternatively this says that any isomorphism between lattice subgroups of g extends to an automorphism of the whole group this remarkable result fails for psl 2 r see section 2 nonetheless taking g psl 2 r we have

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

duality relationships for entropy like minimization problems this paper considers the minimization of a convex integral functional over the positive cone of an l p space subject to a finite number of linear equality constraints such problems arise in spectral estimation where the bjective function is often entropy like and in constrained approximation the lagrangian dual problem is finite dimensional and unconstrained under a quasi interior constraint qualification the primal and dual values are equal with dual attainment examples show the primal value may not be attained conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum these conditions are satisfied in many examples

an implementation of a discretization method for semi infinite programming for linear semi infinite programming problems a discretization method is presented a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements a numerically stable simplex algorithm is used to solve the lp subproblems numerical examples from bivariate chebyshev approximation are presented

anosov representations domains of discontinuity and applications the notion of anosov representations has been introduced by labourie in his study of the hitchin component for sl n r subsequently anosov representations have been studied mainly for surface groups in particular in the context of higher teichmuller spaces and for lattices in so 1 n in this article we extend the notion of anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties in particular given an anosov representation g we explicitly construct open subsets of compact g spaces on which acts properly discontinuously and with compact quotient as a consequence we show that higher teichmuller spaces parametrize locally homogeneous geometric structures on compact manifolds we also obtain applications regarding non standard compact clifford klein forms and compactifications of locally symmetric spaces of infinite volume

a three level bddc algorithm for mortar discretizations in this paper a three level balancing domain decomposition by constraints bddc algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems the mortar discretization is considered on geometrically nonconforming subdomain partitions in two level bddc algorithms the coarse problem needs to be solved exactly however its size will increase with the increase of the number of the subdomains to overcome this limitation the three level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained this is an extension of previous work the three level bddc algorithms for standard finite element discretization estimates of the condition numbers are provided for the three level bddc method and numerical experiments are also discussed

length spectra and degeneration of flat metrics in this paper we consider flat metrics semi translation structures on surfaces of finite type there are two main results the first is a complete description of when a set of simple closed curves is spectrally rigid that is when the length vector determines a metric among the class of flat metrics secondly we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics the geometric interpretation is that flat metrics degenerate to mixed structures on the surface part flat metric and part measured foliation

geometric flow and rigidity on symmetric spaces of noncompact type in this paper we show that under a suitable condition every nonsingular geometric flow on a manifold which is modeled on the furstenberg boundary of x where x is a symmetric space of non compact type induces a torus action and in particular if the manifold is a rational homology sphere then the flow has a closed orbit

conformality for a robust class of non conformal attractors in this paper we investigate the hausdorff dimension of limit sets of anosov representations in this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them analogue to a differentiability property as an application of this convergence we prove that the hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent in the appendix in collaboration with m bridgeman we extend a classical result on the hessian of the hausdorff dimension on purely imaginary directions

application of the pade method to solving the noisy trigonometric moment problem some initial results the noisy trigonometric moment problem for a finite linear combination of box functions is considered and a research program possibly leading to a superresolving method is outlined and some initial steps are performed the method is based on the remark that the poles of the pade approximant to the z transform of the noiseless moments show asymptotically a regular pattern in the complex plane the pattern can be described by a set of arcs connecting points on the unit circle and a pole density function defined on the arcs when a moderate noise affects the moments more arcs are needed to describe the pole pattern but the noiseless pattern is slightly deformed still allowing its identification when this identification is possible a very effective noise filter and moment extrapolator should be easily constructed in this paper only some preliminary steps of the above research program are performed specifically the case of one box function is considered a method for computing the pole patterns

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd