optimal mock jacobi theta functions abstract we classify the optimal mock jacobi forms of weight one with rational coefficients the space they span is thirty four dimensional and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane we show that their fourier coefficients can be expressed explicitly in terms of singular moduli and obtain positivity conditions which distinguish the optimal mock jacobi forms that appear in umbral moonshine we find that many of ramanujan s mock theta functions can be expressed simply in terms of the optimal mock jacobi forms with rational coefficients

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

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

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

spherical polynomials and the periods of a certain modular form the space of cusp forms on sl2 z of weight 2k is spanned by certain modular forms with rational periods kohnen and zagier 5 have investigated those modular forms on sl2 z whose periods in the sense of eichler shimura are rational as an example they consider the function fk d z c e 2 1 k s bsc az bz c where d is a positive integer k an integer greater than one c a constant and the summation over all triples of integers a b c with b2 4ac d this function was introduced by zagier in 13 where it was shown to be a cusp form of weight 2k on sl2 z and was later shown by kohnen 4 to be the dth fourier coefficient of the holomorphic kernel function for the shimura shintani correspondence between modular forms of half integral and integral weight in 6 it is shown that the fk d span s2ks the space of cusp forms on sl2 z of weight 2k if and only if l f k the l series associated to the modular form f does not vanish at s k the center of the critical strip for every hecke eigenform fv it is at present only a conjecture supported by some numerical evidence that the fk d span s2k for all even k for k odd fk d we study the related functions fk d a z where now the summation is restricted to quadratic forms a b c belonging to the sl2 z equivalence class a we prove that certain finite sets of the fk d a span s2k the method of proof relies on the special structure of certain ideal classes and does not indicate how the conjecture that the fk d span s2k mlght be settled a similar result for modular forms on an arbitrary discrete cocompact subgroup of sl2 r has been obtained by svetlana katok 3 let be the set of all narrow equivalence classes of primitive binary quadratic torms ot posltlve olscrlmlnant throughout k wlll denote a posltlve lnteger and we set w 2k2 let a be an sl2 z equivalence class of quadratic forms we define the following classes associated to a aa c b a a b c e a a a b c a b c e a received by the editors may 1 1985 1980 mathematics subject classification 1985 revision primary llfll llf12 secondary llc08 r 1986 american mathematical society 0002 9947 86 1 00 25 per page

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

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

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

transition from pushed to pulled fronts in piecewise linear reaction diffusion systems abstract the front dynamics in reaction diffusion equations with a piecewise linear reaction term is studied a transition from pushed to pulled fronts when they propagate into the unstable state is found using a variational principle this transition occurs for a critical value of the discontinuity position in the reaction function in particular we study how the transition depends on the properties of the reaction term and on the delay time our results are in good agreement with the numerical solutions of the model

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

zagier type dualities and lifting maps for harmonic maass jacobi forms abstract the real analytic jacobi forms of zwegers phd thesis play an important role in the study of mock theta functions and related topics but have not been part of a rigorous theory yet in this paper we introduce harmonic maass jacobi forms which include the classical jacobi forms as well as zwegers functions as examples maass jacobi poincare series also provide prime examples we compute their fourier expansions which yield zagier type dualities and also yield a lift to skew holomorphic jacobi poincare series finally we link harmonic maass jacobi forms to different kinds of automorphic forms via a commutative diagram

implicit gradient elasticity a new formulation of gradient elasticity is derived and implemented pade approximations are used to introduce an implicit relation between non local strains and displacements as a result the finite element interpolation requires only 0 continuous rather than 1 continuous shape functions the underlying energy functional is presented and it is found that the present formulation is of the mixed type whereby the non local strains act as the primary unknowns and the displacements as the auxiliary unknowns the implications on the interpolation are studied finally the influence of the additional length scale parameter on the global response is assessed copyright 2006 john wiley sons ltd

speed shift in reaction diffusion equations with cutoff abstract front solutions in a reaction diffusion equations with cutoff are obtained analytically using piecewise linear approximations of the reaction term the piecewise emulation allows us to study analytically the effect of the cutoff for fronts propagating into metastable and unstable states a general relationship between the cutoff threshold and the parameters involved into the reaction term is derived our results are in excellent agreement with that existing in the literature and prove that the piecewise emulation is an useful procedure to deal with complex reaction diffusion equations under cutoff

two robust nonconforming hbox h 2 elements for linear strain gradient elasticity we propose two finite elements to approximate a boundary value problem arising from strain gradient elasticity which is a high order perturbation of the linearized elastic system our elements are hbox h 2 nonconforming while hbox h 1 conforming we show both elements converge in the energy norm uniformly with respect to the perturbation parameter

gradient elasticity in statics and dynamics an overview of formulations length scale identification procedures finite element implementations and new results in this paper we discuss various formats of gradient elasticity and their performance in static and dynamic applications gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher order spatial derivatives of strains stresses and or accelerations we focus on the versatile class of gradient elasticity theories whereby the higher order terms are the laplacian of the corresponding lower order terms one of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum we start with discussing the general mindlin theory that in its most general form has 903 constitutive elastic parameters but which were reduced by mindlin to three independent material length scales further simplifications are often possible in particular the aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures we also address how this can be extended to dynamics an overview of length scale identification and quantification procedures is given finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions details are provided for particular formats of gradient elasticity that can be implemented with simple linear finite element shape functions new numerical results show the removal of singularities in statics and dynamics as well as the size dependent mechanical response predicted by gradient elasticity

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

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

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

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

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

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

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

a control volume based finite element method for plane micropolar elasticity this paper describes the development of a numerical procedure for predicting deformations and stresses in a loaded two dimensional membrane exhibiting micropolar or cosserat constitutive behaviour the procedure employs a conventional finite element fe mesh together with a dual mesh of interconnected control volumes each of which must satisfy equilibrium a series of patch tests covering a variety of simple strain states are used to validate the procedure that is then employed to predict the stress concentration in a membrane containing a small hole the predictions provided by the procedure are compared with those given previously by fes

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

emergence of approximate translation invariance in finite intervals as a speed selection mechanism for propagating fronts we introduce a velocity selection criterion for fronts propagating into unstable and metastable states we restrict these fronts to large finite intervals in the comoving frame of reference and require that their centers be insensitive to the locations of the ends of the finite intervals thus exhibiting effectively an approximate translation invariance only one monotonic front has this behavior and its velocity is the one that is physically selected we present analytic results in the case of piecewise parabolic potentials and numerical results in other cases

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

a refined nonconforming quadrilateral element for couple stress strain gradient elasticity c0 1 patch test int j numer meth engng 2004 61 433 454 proposed by soh and chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress strain gradient elasticity the c0 1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations to pass the c0 1 patch test the element displacement functions used to calculate strains must satisfy c0 continuity or weak c0 continuity and quadratic completeness in this paper a 24 dof degrees of freedom quadrilateral element cq12 rdkq for the couple stress strain gradient elasticity is developed by combining the refined thin plate element rdkq and the nonconforming element cq12 the element rdkq which satisfies weak c1 continuity is used to calculate strain gradients whereas strains are computed by the element cq12 which is established based on an extended variational functional and satisfies weak c0 continuity and quadratic completeness numerical examples show that the element cq12 rdkq passes the c0 1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects copyright 2010 john wiley sons ltd

modular forms and period polynomials we study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group for the modular group this space is en dowed with a pairing corresponding to the petersson inner product on modular forms via a formula of haberland and with an action of hecke operators defined algebraically by zagier we generalize haberland s formula to not necessarily cuspidal modular forms for finite index subgroups and we show that it conceals two stronger formulas we extend the action of hecke operators to period polynomials of modular forms we show that the pairing on period polynomials appearing in haber land s formula is nondegenerate and we determine the adjoints of hecke operators with respect to it we give a few applications for 1 n an extension of the eichler shimura isomorphism to the entire space of modular forms the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms which are independent of the period rela tions and an explicit formula for fourier coefficients of hecke eigenforms in terms of their period polynomials generalizing the coefficients theorem of manin

an advanced boundary element method for solving 2d and 3d static problems in mindlin s strain gradient theory of elasticity an advanced boundary element method bem for solving two 2d and three dimensional 3d problems in materials with microstructural effects is presented the analysis is performed in the context of mindlin s form ii gradient elastic theory the fundamental solution of the equilibrium partial differential equation is explicitly derived the integral representation of the problem consisting of two boundary integral equations one for displacements and the other for its normal derivative is developed the global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2d and 3d problems respectively representative 2d and 3d numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2d problems copyright 2010 john wiley sons ltd

piecewise linear emulation of propagating fronts as a method for determining their speeds we use piecewise linear terms to emulate the polynomial nonlinear terms in a typical reaction diffusion equation replacing it thus with a set of simple linear inhomogeneous differential equations the resulting analytic solution constitutes an excellent approximation to the exact propagating front as is explicitly shown in the case of cubic and quintic nonlinearities yielding also the correct value for the physically selected speed of the observable front such a piecewise linear emulation can be used for any nonlinearity giving therefore a very reliable and accurate method for determining the selected speed of fronts invading unstable states especially pushed fronts

finite element formulations of strain gradient theory for microstructures and the c0 1 patch test based on finite element formulations for the strain gradient theory of microstructures a convergence criterion for the c0 1 patch test is introduced and a new approach to devise strain gradient finite elements that can pass the c0 1 patch test is proposed the displacement functions of several plane triangular elements which satisfy the c0 continuity and weak c1 continuity conditions are evaluated by the c0 1 patch test the difference between the proposed c0 1 patch test and the c0 constant stress and c1 constant curvature patch tests is elucidated an 18 dof plane strain gradient triangular element rct9 rt9 which passes the c0 1 patch test and has no spurious zero energy modes is proposed numerical examples are employed to examine the performance of the proposed element by carrying out the c0 1 patch test and eigenvalue test the proposed element is found to be without spurious zero energy modes and it possesses higher accuracy compared with other strain gradient elements copyright 2004 john wiley sons ltd

large strain two scale computational approach using c1 continuity finite element employing a second gradient theory a two scale computational algorithm employing second order computational homogenization under assumption of large strain is presented a micro macro transition has been proposed where the c1 continuity displacement based finite element is used for discretization at the macrolevel while the representative volume element at the microlevel is discretized by c0 continuity finite elements the performed finite element verifications show an advantage in comparison to available finite element formulations in order to improve numerical efficiency of the proposed computational procedures the algorithms are embedded into the finite element program abaqus the performance of the presented multiscale homogenization approach is demonstrated in numerical examples where the elastoplastic deformation responses are considered and compared with the small strain analysis

wavy front dynamics in a three component reaction diffusion system with one activator and two inhibitors an analytic description for traveling waves in a one dimensional reaction diffusion system with one activator and two inhibitors and with equal diffusion constants is developed using a piecewise linear approximation for the nonlinear activator reaction term the case of front waves is examined in more detail the monotonic and oscillating fronts being separately considered the corresponding wave profiles are constructed and the speed equation is obtained and discussed it is found that the fronts in the three component model propagate faster than the fronts in the two component system the front interaction is studied using numerical calculations the results show that at head on collisions two oscillating fronts produce a wavy domain which spreads in space with time