structural modal reanalysis methods using homotopy perturbation and projection techniques abstract this paper focuses on the modal reanalysis of structures subjected to multiple modifications of various origins which can greatly affect the mode shapes of these structures the proposed methods are based on the results of a modal analysis of the original structure and on several concepts namely homotopy perturbation projection techniques and taylor series expansion a reduced basis is generated using perturbed eigenvectors and the iterative gram schmidt normalisation algorithm a finite element application highlights the advantages and disadvantages of each of the proposed methods in terms of precision and computational time

complex earthquakes and teichmuller theory it is known that any two points in teichmuller space are joined by an earthquake path in this paper we show any earthquake path r t s extends to a proper holomorphic mapping of a simplyconnected domain d into teichmuller space where r d c these complex earthquakes relate weil petersson geometry projective structures pleated surfaces and quasifuchsian groups using complex earthquakes we prove grafting is a homeomorphism for all 1 dimensional teichmuller spaces and we construct bending coordinates on bers slices and their generalizations in the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold

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

spherical functions on symmetric cones in this note we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real jordan algebra we use this formula as an inspiration for a similar recursive formula involving the jack polynomials

supersymmetric harry dym type equations a supersymmetric version is proposed for the well known harry dym system a general class super lax operator which leads to consistent equations is considered

closed geodesics and periods of automorphic forms we study the detailed structure of the distribution of eichler shimura periods of an automorphic form on a compact hyperbolic surface we show that these periods do not cluster around the asymptotic period over a homology class discovered by zelditch

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

multi scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates abstract a novel second order two scale asymptotic method is presented for the eigenvalue problems of the second order elliptic operator in the general composite domain the eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation and by applying the asymptotic expansion technique the eigenfunctions of the system are expanded to the second order terms using the argument of the so called corrector equations the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain the feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated the finite element algorithm is developed and three numerical experiments are carried out which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures it is also indicated that the second order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions by the coordinate transformation the asymptotic analysis method can be generalized to more general composite domain with quasi periodic and non periodic configurations

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

the bi hamiltonian structure of fully supersymmetric korteweg de vries systems the bi hamiltonian structure of integrable supersymmetric extensions of the korteweg de vries kdv equation related to then 1 and then 2 superconformal algebras is found it turns out that some of these extensions admit inverse hamiltonian formulations in terms of presymplectic operators rather than in terms of poisson tensors for one extension related to then 2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical kdv they can be explained by a factorization of the corresponding lax operator all the bi hamiltonian formulations are derived in a systematic way from the lax operators

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

super camassa holm type systems associated to the kuper ramond schwarz superalgebra we describe a class of super integrable systems with a local bi super hamiltonian structure including super camassa holm type systems which can be realized as euler type systems on the dual of the kuper ramond schwarz superalgebra

multi scale modal analysis for axisymmetric and spherical symmetric structures with periodic configurations abstract a new modal analysis method with second order two scale sots asymptotic expansion is presented for axisymmetric and spherical symmetric structures the symmetric structures considered are periodically distributed with homogeneous and isotropic constituent materials by the asymptotic expansion of the eigenfunctions the homogenized modal equations the effective materials coefficients the first and second order correctors are obtained the derived homogenized constitutive relationships are the same as the ones which serve to homogenize the corresponding static problems the eigenvalues are also expanded to the second order terms and using the so called corrector equation the correctors of the eigenvalues are expressed in terms of the first and second order correctors of the eigenfunctions the anisotropic materials are obtained by homogenization with different properties in the circumferential direction especially for the two dimensional axisymmetric layered structure the one dimensional plane axisymmetric and spherical symmetric structures the homogenized eigenfunctions and eigenvalues as well as their corresponding correctors are all solved analytically the finite element algorithm is established three typical numerical experiments are carried out and the necessity of the second order correctors is discussed based on the numerical results it is validated that the sots asymptotic expansion homogenization method is effective to identify the eigenvalues of the axisymmetric and spherical symmetric structures with periodic configurations and the original eigenfunctions with periodic oscillation can be reproduced by adding the correctors to the homogenized eigenfunctions

second order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates a new second order two scale sots asymptotic analysis method is presented for the heat conduction problems concerning composite materials with periodic configuration under the coordinate transformation the heat conduction problems are solved on the transformed regular domain with quasi periodic structure in the general curvilinear coordinate system by the asymptotic expansion the cell problems effective material coefficients and homogenized heat conduction problems are obtained successively the main characteristic of the approximate model is that each cell problem defined on the microscopic cell domain is associated with the macroscopic coordinate the error estimation of the asymptotic analysis method is established on some regularity hypothesis some common coordinate transformations are discussed and the reduced sots solutions are presented especially by considering the general one dimensional problem the explicit expressions of the sots solutions are derived and stronger error estimation is presented finally the corresponding finite element algorithms are presented and numerical results are analyzed the numerical errors presented agree well with the theoretical prediction which demonstrate the effectiveness of the second order asymptotic analysis method by the coordinate transformation the asymptotic analysis method can be extended to more general domain with periodic microscopic structures

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

an analytic construction of the deligne mumford compactification of the moduli space of curves in 1969 p deligne and d mumford compactified the moduli space of curves their compactification is a projective algebraic variety and as such it has an underlying analytic structure alternatively the quotient of the augmented teichmueller space by the action of the mapping class group gives a compactification of the moduli space we put an analytic structure on this compact quotient and prove that with respect to this structure it is canonically isomorphic as an analytic space to the deligne mumford compactification

the polyakov path integral over bordered surfaces the open string amplitudes we use the feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic bdhp string in a position representation the consistent functional integral representation of the open string propagator is derived and evaluated this result is used as a starting point for two kinds of constructions of the off shell multiloop open string amplitudes the general idea of the presented approach is to consider the off shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space

a super hierarchy of the vector nonlinear schrodinger equations and hamiltonian structures conservation laws a super hierarchy of the vector nonlinear schrodinger equations associated with the 3 3 matrix spectral problem is proposed by using the lenard recursion equations and the zero curvature equation the corresponding super bi hamiltonian structures are established with the help of the super trace identity furthermore we derive the infinite conservation flows of the super vector nonlinear schrodinger equation and the second equation in the hierarchy by utilizing spectral parameter expansions

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

holomorphic extensions of laplacians and their determinants the teichmuller space teich s of a surface s in genus g 1 is a totally real submanifold of the quasifuchsian space qf s we show that the determinant of the laplacian det on teich s has a unique holomorphic extension to qf s to realize this holomorphic extension as the determinant of differential operators on s we introduce a holomorphic family of elliptic second order differential operators on s whose parameter space is the space of pairs of beltrami differentials on s and which naturally extends the laplace operators of hyperbolic metrics on s we study the determinant of this family and show how this family realizes the holomorphic extension of det as its determinant

asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the dirichlet problem for second order elliptic equations in perforated domains in this paper we study the spectral properties of dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain the asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained and the multiscale finite element algorithms and numerical results are proposed

infinitesimal deformations of nodal stable curves abstract an analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers for sections of the square of the relative dualizing sheaf the residue map at a node gives rise to an exact sequence the residue kernel defines the vanishing residue subsheaf for suitable stable curve families the direct image sheaf on the base is locally free and the sequence of direct images is exact recent work of hubbard koch and a formal argument provide that the direct image sheaf is naturally identified with the moduli cotangent sheaf the result generalizes the role of holomorphic quadratic differentials as cotangents for smooth curve families formulas are developed for the pairing of an infinitesimal opening of a node and a section of the direct image sheaf applications include an analytic description of the conormal sheaf for the locus of noded stable curves and a formula comparing infinitesimal openings of a node the moduli action of the automorphism group of a stable curve is described an example of plumbing an abelian differential and the corresponding period variation is presented

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

fuchsian affine actions of surface groups let q be the irreducible representation of sl 2 r i nsl q r define a fuchsian subgroup of sl q r to be a subgroup conjugate to a discrete subgroup of q sl 2 r we prove in this paper that the fundamental group of a compact surface does not act properly on the affine space by affine tranformations if its linear part is fuchsian

a new class of euler equation on the dual of the n 1 extended neveu schwarz algebra let g be the n 1 extended neveu schwarz algebra and greg its regular dual in this paper we will study a super euler system with seven parameters s1 s2 c1 c5 associated with greg we will show that the super euler system is 1 local bi superbihamiltonian if s1 14c1 and s2 12c2 2 supersymmetric if s1 c1 and s2 c2 3 local bi superbihamiltonian and supersymmetric if s1 c1 0 and s2 c2 0 by choosing different parameters we could obtain several supersymmetric or bi superhamiltonian generalizations of some well known integrable systems including the ito equation the 2 component camassa holm equation the 2 component hunter saxton equation and especially the whitham broer kaup dispersive water wave system let g be the n 1 extended neveu schwarz algebra and greg its regular dual in this paper we will study a super euler system with seven parameters s1 s2 c1 c5 associated with greg we will show that the super euler system is 1 local bi superbihamiltonian if s1 14c1 and s2 12c2 2 supersymmetric if s1 c1 and s2 c2 3 local bi superbihamiltonian and supersymmetric if s1 c1 0 and s2 c2 0 by choosing different parameters we could obtain several supersymmetric or bi superhamiltonian generalizations of some well known integrable systems including the ito equation the 2 component camassa holm equation the 2 component hunter saxton equation and especially the whitham broer kaup dispersive water wave system

the polyakov path integral over bordered surfaces ii the closed string off shell amplitudes following the general scheme of the convariant path integral quantization of gauge systems two alternative formulations of the first quantized closed bosonic string in a position representation are presented in both approaches the covariant path integral representations of the propagator and of the higher order off shell amplitudes are constructed for a wide class of gauges the explicit formulae for off shell amplitudes are obtained this paper is the continuation of our previous work where the corresponding problems in the open string case were considered 20

infinitesimal bending and twisting in one dimensional dynamics an infinitesimal theory for bending and earthquaking in one dimensional dynamics is developed it is shown that any tangent vector to teichmuller space is the initial data for a bending and for an earthquaking or dinary differential equation the discussion involves an analysis of infinitesimal bendings and earthquakes the hilbert transform natural bounded linear oper ators from a banach space of measures on the mobius strip to tangent vectors to teichmuller space and the construction of a nonlinear right inverse for these operators the inverse is constructed by establishing an infinitesimal version of thurston s earthquake theorem

a non standard lax formulation of the harry dym hierarchy and its supersymmetric extension for the harry dym hierarchy a non standard lax formulation is deduced from that of the korteweg de vries kdv equation through a reciprocal transformation by supersymmetrizing this lax operator a new n 2 supersymmetric extension of the harry dym hierarchy is constructed and is further shown to be linked to one of the n 2 supersymmetric kdv equations through the superconformal transformation the bosonic limit of this new n 2 supersymmetric harry dym equation is related to a coupled system of kdv mkdv equations

eichler shimura homology intersection numbers and rational structures on spaces of modular forms in this paper we reinterpret the main results of 8 using the intersection theory of cycles with coefficients to this end we give a functorial interpretation of eichler schimura periods introduction in this paper we reformulate the main results of s katok 8 in terms of homology theory with local coefficients in order to do so we srst give an interpretation of the period of a cusp form f of weight 2m 2 m 1 over a closed geodesic y as the kronecker index of the cohomoiogy class sh f with local coefficients associated to f by shimura 13 and a 1 cycle with dual local coefficients associated to y we will call this period the eichler shimura period there are other interpretations of eichler shimura periods but these seem to work only for arithmetic fundamental groups whereas the defining integral of the period makes sense quite generally there is a simple idea from group homology theory which is the basis of our reformulation that of a decomposable cycle we now give the definition suppose a is a representation of a group r into gl v where v is a finite dimensional vector space then we may consider the eilenberg mac lane homology group hl r we observe that if ey is an element of r and v is an element of v satisfying a y v v a ey invariant then the 1 chain oy 3 v satisfies a ty x v a ty v v o consequently ty x v is a 1 cycle and gives rise to an element of hl r a which we also denote oy q3 v definitions we will call such cycles decomposable cycles and we will say a homology class is decomposable if it has a decomposable representative remark a general 1 cycle is of the form e 1 1 aj x vj where n n 5 tyj vj vj o j l j l for more details on homology and cohomology of groups see brown 3 we can now give our formulation of eichler shimura periods let v be the space of the standard 2 dimensional complex representation of sl2 r and v be its dual the corresponding real space we will denote by vr we let smv denote received by the editors january 31 1986 1980 mathematics subject classification 1985 revision primary llf06 20h10 55n25 secondary 22e40 the second author was partially supported by an nsf grant t1987 american mathematical society 0002 9947 87 1 00 25 per page

amenability poincare series and quasiconformal maps any coveringy x of a hyperbolic riemann surfacex of finite area determines an inclusion of teichmuller spaces teich x teich y we show this map is an isometry for the teichmuller metric if the covering isamenable and contracting otherwise in particular we establish 1 for classical poincare series kra s theta conjecture

limits of quasi fuchsian groups with small bending we study limits of quasi fuchsian groups for which the bending measures on the convex hull boundary tend to zero giving necessary and sufficient conditions for the limit group to exist and be fuchsian as an application we complete the proof of a conjecture made in 24 conjecture 6 5 that the closures of pleating varieties for quasi fuchsian groups meet fuchsian space exactly in kerckhoff s lines of minima of length functions doubling our examples gives rise to a large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach 2

exotic projective structures and quasi fuchsian space ii let p s be the space of projective structures on a closed surface s of genus g 1 and let q s be the subset of p s of projective structures with quasi fuchsian holonomy it is known that q s consists of infinitely many connected components in this article we show that the closure of any exotic component of q s is not a topological manifold with boundary and that any two components of q s have intersecting closures

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

second order two scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations abstract a new second order two scale sots analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations the axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions the sots asymptotic expansions for the space problem plane axisymmetric problem and spherical symmetric problem are presented and the main feature is that the anisotropic material is obtained by the homogenization the analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained and the error estimations of the expansions are established the second order asymptotic analysis finite element algorithm is presented and the numerical examples are solved including the hollow cylinder rotating disk and hollow sphere composed of periodic composite materials the computational results demonstrate the effectiveness and accuracy of the sots asymptotic analysis algorithm and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction it is also indicated that the stress distributions can be correctly computed only by adding the second order correctors

numerical methods for elliptic partial differential equations with rapidly oscillating coefficients this paper presents two methods for the numerical solution of the classical homogenization problem of elliptic operators with periodically oscillating coefficients the numerical solution of such problems is difficult because of the presence of rapidly oscillating coefficients the first method based on the classical one which consists of the homogenized solution the first and second order correctors whereas the second one is based on the bloch wave approach further for the calculation of the homogenized coefficients and some auxiliary functions involved in this method we applied both methods and compared their accuracies the bloch approximation consists in determining an oscillating integral numerically the bloch method provides a better approximation to the exact solution than the classical first order corrector term in the smooth coefficients case moreover we provided taylor approximations for the bloch approximation function and implemented it numerically in order to show the efficiency of these methods exhaustive numerical examples in both one and two dimensional cases are presented

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

closed geodesics periods and arithmetic of modular forms is the upper half plane and f is a discrete cocompact subgroup of sl 2 n acting by fractional linear transformations we demonstrate that in this case the periods over closed geodesics play a role somewhat similar to that of fourier coefficients of modular forms on sl 2 z and its congruence subgroups more specifically those periods uniquely determine a modular form theorem 2 this result is valid for cusp forms on any fuchsian group of the first kind with or without cusps and is closely related to the study of relative poincar6 series associated to closed geodesics for each integer k 2 and each closed geodesic 7o we define special cusp forms of weight 2k called relative poincar6 series ok t ol and prove that they generate the whole space s2k f of cusp forms theorem 1 in w 3 we give an expression for periods of a relative poincar6 series over closed geodesics in purely geometrical terms through the intersection of the corresponding geodesics theorem 3 an application of theorems 1 and 3 to arithmetic subgroups of sl 2 1r gives two natural rational structures o n s2k l n theorem 4 the relative poincar6 series have been studied for general f by petersson 18 19 and hejhal 5 g 2 wolpert 23 gives a basis of s4 f for g 2 for sl 2 tz the relative poincar6 series have been studied by zagier 25 kohnen 8 kohnen and zagier 9 and kramer 13 a related problem of constructing cusp forms of weight two associated to closed geodesics has been treated by kudla and millson in 143 in connection with the problem of

limits of geometrically tame kleinian groups let f be a finitely generated geometrically tame kleinian group with h3 f m when m has a finite volume the hyperbolic structure of m is unique up to homotopy by mostow s rigidity theorem but when m has infinite volume it is meaningful to consider the deformation space of the hyperbolic structure of m let ah m denote the space of faithful representations with discrete image from nl m to psl2 c modulo conjugacy with the topology of algebraic convergence until now only a little is known about ah m when f is of second kind there exists the space of quasi conformal deformation of f denoted by qho m with the quasi conformal or quasi isometric topology there is a canonical continuous injection t qho m ah m we abuse the notation qho m for t qho m the topological type of qho m has been determined by sullivan 21 one method for investigating ah m is to consider limits of sequences in qho m boundary groups in bers 3 and double limits in thurston 24 are such examples for considering limits it is important to determine when a sequence in ah m converges it was shown by thurston 24 and morgan shalen 16 that when m is acylindrical every sequence in ah m has a converging subsequence we gave in ohshika 19 a sufficient condition of convergence for geometrically finite groups in w 2 of this paper we prove a theorem which implies the existence of converging subsequence in all of the above cases the case of boundary groups double limits acylindrical manifolds and the theorem in 19 the statement is as follows

on bosonic limits of two recent supersymmetric extensions of the harry dym hierarchy two generalized harry dym equations recently found by brunelli das and popowicz in the bosonic limit of new supersymmetric extensions of the harry dym hierarchy j math phys 44 4756 2003 are transformed into previously known integrable systems one into a pair of decoupled kdv equations the other one into a pair of coupled mkdv equations from a bi hamiltonian hierarchy of kupershmidt

geodesic flow on super bott virasoro group and harry dym family we show various harry dym type equations and super harry dym equations introduced by brunelli das and popowicz 2003 supersymmetric extensions of the harry dym hierarchy j math phys 44 4756 4767 2003 follow from the geodesic flows on the bott virasoro group and its supersymmetric generalization superconformal group in fact their bi hamiltonian structures can be derived from the lie poisson structures on the super bott virasoro orbit we also show that that m2kdv or calogero degasperis equation is also connected to the bott virasoro group

supersymmetric reciprocal transformation and its applications the supersymmetric analog of the reciprocal transformation is introduced this is used to establish a transformation between one of the supersymmetric harry dym equations and the supersymmetric modified korteweg de vries equation the reciprocal transformation as a backlund type transformation between these two equations is adopted to construct a recursion operator for the supersymmetric harry dym equation by proper factorization of the recursion operator a bi hamiltonian structure is found for the supersymmetric harry dym equation furthermore a supersymmetric kawamoto equation is proposed and is associated with the supersymmetric sawada kotera equation the recursion operator and odd bi hamiltonian structure of the supersymmetric kawamoto equation are also constructed

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

supersymmetric extensions of the harry dym hierarchy we study the supersymmetric extensions of the harry dym hierarchy of equations we obtain the susy b extension the doubly susy b extension as well as the n 1 and the n 2 supersymmetric extensions for this system the n 2 supersymmetric extension is particularly interesting since it leads to new classical integrable systems in the bosonic limit we prove the integrability of these systems through the bi hamiltonian formulation of integrable models and through the lax description we also discuss the supersymmetric extension of the hunter zheng equation which belongs to the harry dym hierarchy of equations

a method for nonlinear modal analysis and synthesis application to harmonically forced and self excited mechanical systems abstract the recently developed generalized fourier galerkin method is complemented by a numerical continuation with respect to the kinetic energy which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes in order to enhance the practicability regarding the investigation of complex large scale systems it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations a novel reduced order model rom is developed for those regimes where internal resonances are absent the approach allows for an accurate approximation of the multi harmonic content of the resonant mode and accounts for the contributions of the off resonant modes in their linearized forms the rom facilitates the efficient analysis of self excited limit cycle oscillations frequency response functions and the direct tracing of forced resonances the rom is equipped with a large parameter space including parameters associated with linear damping and near resonant harmonic forcing terms an important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology this is achieved by selected numerical examples including finite element models of structures with strongly nonlinear non conservative contact constraints

second order two scale finite element algorithm for dynamic thermo mechanical coupling problem in symmetric structure the new second order two scale sots finite element algorithm is developed for the dynamic thermo mechanical coupling problems in axisymmetric and spherical symmetric structures made of composite materials the axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction the spherical symmetric structure is only periodic in radial direction the dynamic thermo mechanical coupling model is presented and the equivalent compact form is derived then the cell problems effective material coefficients and the homogenized thermo mechanical coupling problem are obtained successively by the second order asymptotic expansion of the temperature increment and displacement the homogenized material obtained is manifested with the anisotropic property in the circumferential direction the explicit expressions of the homogenized coefficients in the plane axisymmetric and spherical symmetric cases are given and both the derivation of the analytical solutions of the cell functions and the quasi static thermoelasticity problems are discussed based on the sots method the corresponding finite element procedure is presented and the unconditionally stable implicit algorithm is established some numerical examples are solved and the mutual interaction between the temperature and displacement field is studied under the condition of structural vibration the computational results demonstrate that the second order asymptotic analysis finite element algorithm is feasible and effective in simulating and predicting the dynamic thermo mechanical behaviors of the composite materials with small periodic configurations in axisymmetric and spherical symmetric structures this may provide a vital computational tool for analyzing composite material internal temperature distribution and structural deformation induced by the dynamic thermo mechanical coupling response under strong aerothermodynamic environment second order two scale sots expansions for dynamic thermo mechanical coupling problems in symmetric structure are obtained sots finite element algorithm is proposed and unconditional stable implicit scheme is established anisotropic material is obtained by homogenization with enhanced strength and minor thermal expansion in circumferential direction the mutual interaction and simultaneous vibration of the temperature and displacement field are simulated a new computational tool is presented for analyzing thermo mechanical response of composites under strong aerothermodynamic environment

supersymmetric extensions of the nonlinear schrodinger equation symmetries and coverings a construction is proposed for a supersymmetric generalization of the cubic schrodinger equation resulting in two supersymmetric systems one of which contains a free parameter both systems are proven to admit an infinite set of higher order local and nonlocal symmetries and a seemingly infinite set of conservation laws the lowest order terms of which are given explicitly moreover the theory of coverings equivalent to the prolongation method of wahlquist and estabrook is applied to both systems both are seen to admit an infinite dimensional covering algebra the structure of which is determined explicitly resulting in a related supersymmetric system of differential equations as well as an auto backlund transformation for each equation this indicates the complete integrability of both systems

a new n 2 supersymmetric korteweg de vries equation there are two known integrable n 2 space supersymmetric extensions of the kdv equation both can be written as hamiltonian systems with a common poisson structure which corresponds to the n 2 supersymmetric form of the second kdv structure by analyzing the conservation laws of the one parameter family of equations that share this hamiltonian structure one finds that a third system is singled out by the existence of nontrivial conservation laws it is conjectured to be integrable

bosonic reduction of susy generalized harry dym equation in this paper we construct the two component supersymmetric generalized harry dym equation which is integrable and study various properties of this model in the bosonic limit in particular in the bosonic limit we obtain a new integrable system which under a hodograph transformation reduces to a coupled three component system we show how the hamiltonian structure transforms under a hodograph transformation and study the properties of the model under a further reduction to a two component system we find a third hamiltonian structure for this system which has been shown earlier to be a bi hamiltonian system making this a genuinely tri hamiltonian system the connection of this system to the modified dispersive water wave equation is clarified we also study various properties in the dispersionless limit of our model

multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains in this paper we will discuss the mixed boundary value problems for the second order elliptic equation with rapidly oscillating coefficients in perforated domains and will present the higher order multiscale asymptotic expansion of the solution for the problem which will play an important role in the numerical computation the convergence theorems and their rigorous proofs will be given finally a multiscale finite element method and some numerical results will be presented

homogenization of periodic structures via bloch decomposition in this paper the classical problem of homogenization of elliptic operators in arbitrary domains with periodically oscillating coefficients is considered using bloch wave decomposition a new proof of convergence is furnished it sheds new light and offers an alternate way to view the classical results in a natural way this method leads us to work in the fourier space and thus in a framework dual to the one used by l tartar problemes d homogeneisation dans les equations aux derivees partielles cours peccot au college de france 1977 in his method of homogenization further this technique offers a nontraditional way of calculating the homogenized coefficients which is easy to implement in the computer

deformed harry dym and hunter zheng equations we study the deformed harry dym and hunter zheng equations with two arbitrary deformation parameters these reduce to various other known models in appropriate limits we show that both these systems are bi hamiltonian with the same hamiltonian structures they are integrable and belong to the same hierarchy corresponding to positive and negative flows we present the lax pair description for both the systems and construct the conserved charges of negative order from the lax operator for the deformed harry dym equation we construct the nonstandard lax representation for two special classes of values of the deformation parameters in general we argue that a nonstandard description will involve a pseudodifferential operator of infinite order