the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

a natural construction of borcherds fake baby monster lie algebra we use a z 2 orbifold of the vertex operator algebra associated to the niemeier lattice with root lattice a 8 3 and the no ghost theorem of string theory to construct a generalized kac moody algebra borcherds theory of automorphic products allows us to determine the simple roots and identify the algebra with the fake baby monster lie algebra

the theory of non linear elastic ship water interaction dynamics non linear mathematical models are developed to provide formulations of the equations of motion describing the dynamical interaction behaviour between an incompressible or compressible ideal fluid and a moving or fixed elastic or rigid structure the general theoretical approach is based on the fundamental equations of continuum mechanics the concept of hamilton s principle and suitably formulated variational principles the resultant mathematical model expressed in a fixed or a moving frame of reference allows the theoretical establishment of non linear problems associated with ship dynamics and offshore engineering through applications of the variational principles this is demonstrated by rigorously deriving the governing equations of motion for general non linear ship water interaction problems in particular the theory is applied to a rigid ship travelling in calm water or in waves a bottom fixed rigid rod or tower excited by an incident wave and a two dimensional elastic beam travelling in waves

properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

a unified approach for the formulation of interaction problems by the boundary element method a unified formulation is presented based on boundary element method in a form suitable for performing the interaction analyses by substructure method for solid solid and soil structure problems the proposed formulation permits the evaluation of all the elements of impedance and input motion matrices simultaneously at a single step in terms of system matrices of the boundary element method without solving any special problem such as unit displacement or load problem as required in conventional methods it eliminates further the complicated procedure and the need for using scattering analysis in the evaluation of input motion functions to explain the formulation it is first given for an inclusion interacting with an infinite surrounding medium under the influence of a seismic input where both the inclusion and surrounding medium are treated as viscoelastic it is shown that the formulation for a rigid inclusion may be obtained from that for flexible inclusion as a special case through a transformation then the formulation is extended to other types of interaction problems a multi inclusion problem and an interaction problem involving a foundation embedded in a viscoelastic half space it is found that the proposed formulation remains essentially the same for all kinds of interaction problems and it can be used not only in regular interaction analysis but also in the analysis involving diffraction of waves in a medium containing holes copyright 2005 john wiley sons ltd

generalized kac moody algebras automorphic forms and conway s group i abstract the moonshine properties imply that the twisted denominator identities coming from the action of the monster group on the monster algebra define modular forms in this paper we motivate the conjecture that the action of an extension of conway s simple group co 1 on the fake monster algebra gives rise to automorphic forms of singular weight on grassmannians we prove the conjecture for elements with square free level and nontrivial fixpoint lattice

modular forms and special cubic fourfolds abstract we study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ade singularities in this paper we show that the generating series of the degrees of such divisors is a level three modular form

domain decomposition methods for pseudo spectral approximations part i second order equations in one dimension multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered the equation is collocated at the chebyshev nodes inside each subinterval different patching conditions at the interfaces are analyzed results of stability and convergence are given

a comparison of localized finite element formulations for two dimensional wave diffraction and radiation problems two of the most promising localized finite element methods are compared the boundary series element method in which a series of eigenfunctions is used to represent the far field solution and the boundary integral element method in which an integral equation is satisfied at the boundary between localized finite element and outer regions the methods are applied to water of arbitrary depth the theory of the two methods is summarized and typical numerical results are discussed consideration is given to the well known hydrodynamical reciprocal relations and to the phenomenon of irregular frequencies the relative merits of the two methods are established

special values of hypergeometric functions and periods of cm elliptic curves let x x 0 6 1 w 6 be the quotient of the shimura curve x 0 6 1 by all the atkin lehner involutions by realizing modular forms on x in two ways one in terms of hypergeometric functions and the other in terms of borcherds forms and using schofer s formula for values of borcherds forms at cm points we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over overline q with complex multiplication

on newforms for kohnen plus spaces in this article we investigate the plus space of level n where 4 1n is a square free not necessarily odd integer this is a generalization of kohnen s work we define a hecke isomorphism wp k of mk 1 2 4m onto m k 1 2 8m for any odd positive integer m the methods of the proof of the newform theory are this isomorphism waldspurger s theorem and the dimension identity

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

shimura lifts of weakly holomorphic modular forms we show how to realize the shimura lift of arbitrary level and character using the vector valued theta lifts of borcherds using the regularization of borcherds lift we extend the shimura lift to take weakly holomorphic modular forms of half integral weight to meromorphic modular forms of even integral weight having poles at cm points

dynamic analysis of linear viscoelastic cylindrical and conical helicoidal rods using the mixed fem abstract the objective of this study is to investigate the influence of the rotary inertia on dynamic behavior of linear viscoelastic cylindrical and conical helixes by means of the laplace transform mixed finite element formulation and solution the element matrix is based on the timoshenko beam theory the influence of rotary inertias is considered in the dynamic analysis which is original in the literature rectangular sine and step type of impulsive loads are applied on helices having rectangular cross sections with various aspect ratios the kelvin and standard models are used for defining the linear viscoelastic material behavior and by means of the correspondence principle the elastic viscoelastic analogy the material parameters are replaced with their complex counterparts in the laplace domain the analysis is carried out in the laplace domain and the results are transformed back to time space numerically by modified durbin s algorithm first the solution algorithm is verified using the existing open sources in the literature and afterwards some benchmark examples such as conical viscoelastic rods are handled

the automorphism groups of kummer surfaces associated with the product of two elliptic curves we calculate the automorphism groups of several kummer surfaces associated with the product of two elliptic curves we give their generators explicitly

classification of integral lattices with large class number a detailed exposition of kneser s neighbour method for quadratic lattices over totally real number fields and of the sub procedures needed for its implementation is given using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices various results about genera of elementary lattices for small prime level are obtained for instance the class number of 1 2 dimensional 7 elementary even lattices of determinant 7 6 is 395 no extremal lattice in the sense of quebbemann exists the implementation incorporates as essential parts previous programs of w plesken and b souvignier

hydrodynamic forces on submerged oscillating cylinders at forward speed the hydrodynamic problem of submerged oscillating cylinders at forward speed is analysed by linearized potential theory the numerical method used combines a finite element approximation of the velocity potential in a region surrounding the cylinder with a boundary integral equation representation of the outer region this method avoids the calculation of the second order derivatives of the steady potential due to forward speed which appear in the body surface condition for the unsteady potential due to the oscillation of the cylinder numerical results from the present method for test cases of a circular cylinder in an unbounded fluid and below a free surface are in excellent agreement with the analytical solutions further results for elliptic cylinders are provided and the influence of forward speed on the hydrodynamic force on a submerged cylinder is investigated

numerical computation of a certain dirichlet series attached to siegel modular forms of degree two the rankin convolution type dirichlet series df g s of siegel modular forms f and g of degree two which was introduced by kohnen and the second author is computed numerically for various f and g in particular we prove that the series df g s which share the same functional equation and analytic behavior with the spinor l functions of eigenforms of the same weight are not linear combinations of those in order to conduct these experiments a numerical method to compute the petersson scalar products of jacobi forms is developed and discussed in detail

a fast iterative method to compute the flow around a submerged body the authors develop an efficient iterative method for computing steady linearized potential flow around a submerged body moving in a liquid of finite constant depth in this paper they restrict the presentation to the two dimensional problem but the method is readily generalizable to the three dimensional case i e the flow in a canal the problem is indefinite which makes the convergence of most iterative methods unstable to circumvent this difficulty the authors decompose the problem into two more easily solvable subproblems and form a schwarz type iteration to solve the original problem the first subproblem is definite and can therefore be solved by standard iterative methods the second subproblem is indefinite but has no body it is therefore easily and efficiently solvable by separation of variables the authors prove that the iteration converges for sufficiently small froude numbers in addition they present numerical results for a second order accurate discretization of the problem they demonstrate that the iterative method converges rapidly and that the convergences rate improves when the froude number decreases they also verify numerically that the convergence rate is essentially independent of the grid size 20 refs 6 figs 10 tabs

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

weil representations associated with finite quadratic modules to any finite quadratic module that is a finite abelian group together with a non degenerate quadratic form it is possible to associate a representation of mathrm mp 2 mathbb z the metaplectic cover of the modular group this representation is usually referred to as a weil representation and our main result is a general explicit formula for its matrix coefficients this result completes earlier work by scheithauer in the case when the representation factors through mathrm sl 2 mathbb z furthermore our formula is given in a such a way that it is easy to implement efficiently on a computer

the schwarz algorithm for spectral methods recently the schwarz alternating method has been successfully coupled to spatial discretizations of spectral type in order to solve boundary value problems in complex geometries with infinite order accuracy in this paper a simple version of the method is considered a proof of its convergence is given in the energy norm exploiting the properties of discrete harmonic polynomials and a discrete maximum principle for spectral methods more general situations can be handled theoretically in one space dimension

quaternionic loci in siegel s modular threefold let mathcal q d be the set of moduli points on siegel s modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order mathcal o in an indefinite quaternion algebra of discriminant d over mathbb q such that the rosati involution coincides with a positive involution of the form alpha mapsto mu 1 overline alpha mu on mathcal o for some mu in mathcal o with mu 2 d 0 in this paper we first give a formula for the number of irreducible components in mathcal q d strengthening an earlier result of rotger then for each irreducible component of genus 0 we determine its rational parameterization in terms of a hauptmodul of the associated shimura curve

finiteness of 2 reflective lattices of signature 2 n a modular form for an even lattice l of signature 2 n is said to be 2 reflective if its zero divisor is set theoretically contained in the heegner divisor defined by the 2 vectors in l we prove that there are only finitely many even lattices with n geq7 which admit 2 reflective modular forms in particular there is no such lattice in n geq26 except the even unimodular lattice of signature 2 26 this proves a conjecture of gritsenko and nikulin in the range n geq7

computations of siegel modular forms of genus two we explain the basic notions and theorems for doing computations in the theory of siegel modular forms of degree two on the full modular group and of even weight this synopsis concludes with a handy and computationally realistic algorithm for tabulating the fourier coefficients of such forms and the euler factors of their spinor zeta functions in the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm we point out two theoretically striking phenomena that are implied by the results of these computations

the weil representation and hecke operators for vector valued modular forms we define hecke operators for vector valued modular forms transforming with the weil representation associated to a discriminant form we describe the properties of the corresponding algebra of hecke operators and study the action on modular forms

state space boundary element finite element coupling for fluid structure interaction analysis state space coupling is a new approach for handling computational fluid structure interaction problems in the low to medium frequency range the method is based on standard boundary element bem and finite element fem discretizations of the acoustic and structural domains respectively the implicit frequency dependence of a traditional acoustic impedance matrix is made explicit through a power series expansion on circular frequency and considering time harmonic motion once expanded the acoustic energy of the fluid is directly coupled to that of the structure via hamilton s principle the coupled system is then recast in a canonical state space form which is also in the form of a standard eigenvalue problem solution of this system produces a series of complex eigenvalues and eigenvectors which represent a modal decomposition of the fluid loaded structure biorthogonality properties of the state space eigensystem allow for an uncoupled modal solution of the forced problem results for a submerged spherical shell and a finite plate in an infinite ridged baffle are given to illustrate the state space approach

the leech lattice and complex hyperbolic reflections we construct a natural sequence of finite covolume reflection groups acting on the complex hyperbolic spaces h 13 h 9 and h 5 and show that the 9 dimensional example coincides with the largest of the groups of mostow 11 our reflection groups arise as automorphism groups of certain lorentzian lattices over the eisenstein integers and we obtain our largest example by using the complex leech lattice in a manner inspired by conway 5 we also construct finite covolume reflection groups on the quaternionic hyperbolic spaces h 7 h 5 and h 3 again using the leech lattice and apply results of borcherds 4 to obtain automorphic forms for our groups

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

the fake monster superalgebra abstract we show that the physical states of a 10 dimensional superstring moving on a torus form a generalized kac moody superalgebra this gives the first explicit realizations of these algebras for a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots we call this algebra the fake monster superalgebra

automorphic products of singular weight for simple lattices we classify the simple even lattices of square free level and signature 2 n n ge 4 a lattice is called simple if the space of cusp forms of weight 1 n 2 for the dual weil representation of the lattice is trivial for a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form using this we determine all holomorphic borcherds products of singular weight arising from vector valued modular forms with non negative principal part for the simple lattices we construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps