a divergence free reconstruction of the nonconforming virtual element method for the stokes problem abstract in this paper we propose and investigate a divergence free reconstruction of the nonconforming virtual element for the stokes problem by constructing the computable raviart thomas like interpolation operator we guarantee the independence between the velocity error estimation u u h 1 h and the continuous pressure p as it happens for the divergence free flow solver moreover this modified scheme can also inherit the advantages of the classical nonconforming virtual element method such as very general meshes including non convex and degenerate elements a unified scheme for an arbitrary order approximation accuracy k etc then we provide the optimal l 2 error estimates for the velocity gradient and the pressure by taking advantage of the raviart thomas like interpolation operator and avoiding the use of a trace inequality finally three numerical experiments are presented to conform the theoretical analysis

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

the nonconforming virtual element method for the darcy stokes problem abstract we present the nonconforming virtual element method for the darcy stokes problem by using a gradient projection operator and a special polynomial subspace orthogonal to the gradient of some polynomial space we construct a h 1 nonconforming but h div conforming virtual element that allows us to compute the l 2 projection the optimal convergence is proved under the assumption of sufficient regularity and the uniform convergence is also obtained for the lowest order case besides we establish a discrete exact sequence of de rham complex related to the nonconforming virtual element finally we carry out some numerical tests to make a comparison of the convergence between different nonconforming virtual elements and confirm the optimal and uniform convergence of the nonconforming virtual element

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

discontinuous skeletal gradient discretisation methods on polytopal meshes in this work we develop arbitrary order discontinuous skeletal gradient discretisations dsgd on general polytopal meshes discontinuous skeletal refers to the fact that the globally coupled unknowns are broken polynomial on the mesh skeleton the key ingredient is a high order gradient reconstruction composed of two terms i a consistent contribution obtained mimicking an integration by parts formula inside each element and ii a stabilising term for which sufficient design conditions are provided an example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high order residuals on a raviart thomas nedelec subspace we prove that the novel dsgds satisfy coercivity consistency limit conformity and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems links with hybrid high order non conforming mimetic finite difference and non conforming virtual element methods are also studied numerical examples complete the exposition

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

partition congruences and the andrews garvan dyson crank in 1944 freeman dyson conjectured the existence of a crank function for partitions that would provide a combinatorial proof of ramanujan s congruence modulo 11 forty years later andrews and garvan successfully found such a function and proved the celebrated result that the crank simultaneously explains the three ramanujan congruences modulo 5 7 and 11 this note announces the proof of a conjecture of ono which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

bailey pairs and indefinite quadratic forms we construct classes of bailey pairs where the exponent of q in n is an indefinite quadratic form as an application we obtain families of q hypergeometric mock theta multisums

quasigeodesic pseudo anosov flows in hyperbolic 3 manifolds and connections with large scale geometry abstract in this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo anosov flow in a closed hyperbolic three manifold to be quasigeodesic quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover we prove that such flows are quasigeodesic if and only if there is an upper bound depending only on the flow on the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow the main ingredient is a proof that under the boundedness condition the fundamental group of the manifold acts as a uniform convergence group on a flow ideal boundary of the universal cover we also construct a flow ideal compactification of the universal cover and prove that it is equivariantly homeomorphic to the gromov compactification this implies the quasigeodesic behavior of the flow the flow ideal boundary and flow ideal compactification are constructed using only the structure of the flow

the nonconforming virtual element method for elasticity problems abstract we present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation this method is uniformly convergent for the nearly incompressible case the optimal convergence in the h 1 norm is proved under some regularity assumptions we mention that for the lowest order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by brenner and sung 1992 19 up to an approximation of the right hand side besides we also present the nonconforming virtual element method for the pure traction problem which is proved to be uniformly convergent with respect to the lame constant finally we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

a priori and a posterior error estimate of new weak galerkin finite element methods for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of a new weak galerkin wg finite element method for the second order elliptic interface problems instead of introducing the local weak derivatives on each element the bilinear form is computed through the inner product of traditional defined derivatives the proposed numerical scheme is symmetric and positive definite and can be applied to polygonal meshes in the a posteriori error estimate we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator have been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

a posteriori error analysis of a mixed virtual element method for a nonlinear brinkman model of porous media flow abstract in this paper we present an a posteriori error analysis of a mixed vem discretization for a nonlinear brinkman model of porous media flow which has been proposed by the authors in a previous work therein the system is formulated in terms of a pseudostress tensor and the velocity gradient whereas the velocity and the pressure of the fluid are computed via postprocessing formulas furthermore the well posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established we now propose reliable and efficient residual based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem the resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes for the analysis we make use of a global inf sup condition helmholtz decomposition local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions finally we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

sixth order mock theta functions two new mock theta functions of the sixth order are defined the main theorem in this paper theorem 1 1 provides four transformation formulas relating the new mock theta functions with ramanujan s mock theta functions of the sixth order two further representations of the new mock theta functions are established lastly a hitherto unproved entry from ramanujan s lost notebook related to sixth order mock theta functions is proved

a hybrid high order method for the incompressible navier stokes equations based on temam s device abstract in this work we propose a novel hybrid high order method for the incompressible navier stokes equations based on a formulation of the convective term including temam s device for stability the proposed method has several advantageous features it supports arbitrary approximation orders on general meshes including polyhedral elements and non matching interfaces it is inf sup stable it is locally conservative it supports both the weak and strong enforcement of velocity boundary conditions it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element particular care is devoted to the design of the convective trilinear form which mimicks at the discrete level the non dissipation property of the continuous one the possibility to add a convective stabilisation term is also contemplated and a formulation covering various classical options is discussed the proposed method is theoretically analysed and an energy error estimate in h k 1 with h denoting the meshsize is proved under the usual data smallness assumption a thorough numerical validation on two and three dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations including in particular the well known two and three dimensional lid driven cavity problems

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

q identities for maass waveforms reinterpretation des theoremes de andrews dyson hickerson en termes de q identite pour une certaine forme d onde de maass

supg stabilization for the nonconforming virtual element method for advection diffusion reaction equations abstract we present the design convergence analysis and numerical investigations of the nonconforming virtual element method with streamline upwind petrov galerkin vem supg stabilization for the numerical resolution of convection diffusion reaction problems in the convective dominated regime according to the virtual discretization approach the bilinear form is split as the sum of a consistency and a stability term the consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form including the supg stabilization term polynomial projections can be computed exactly from the degrees of freedom the stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients the nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface optimal convergence properties of the method are proved in a suitable norm which includes contribution from the advective stabilization terms experimental results confirm the theoretical convergence rates

a virtual element method for stochastic stokes equations abstract in this paper a virtual element method for the stochastic stokes equations driven by an additive white noise is proposed and analyzed the velocity is approximated by the lowest order virtual element which is originally designed for the poisson equation and the projection is also taken as the one originally for the poisson equation while the pressure is approximated by the traditional discontinuous piecewise constant element for stable approximations we adopt a stabilization associating with the pressure jumps we show the inf sup condition and derive the stability we moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the green function numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained

a posteriori error estimates for a virtual element method for the steklov eigenvalue problem abstract the paper deals with the a posteriori error analysis of a virtual element method for the steklov eigenvalue problem the virtual element method has the advantage of using general polygonal meshes which allows implementing efficiently mesh refinement strategies we introduce a residual type a posteriori error estimator and prove its reliability and global efficiency local efficiency estimates also hold although in some elements they involve boundary terms that are not known to be locally negligible we use the corresponding error estimator to drive an adaptive scheme finally we report the results of a couple of numerical tests that allow us to assess the performance of this approach

the bailey chain and mock theta functions standard applications of the bailey chain preserve mixed mock modularity but not mock modularity after illustrating this with some examples we show how to use a change of base in bailey pairs due to bressoud ismail and stanton to explicitly construct families of q hypergeometric multisums which are mock theta functions we also prove identities involving some of these multisums and certain classical mock theta functions

bundles and finite foliations by a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume we recall that if is a kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space otherwise is geometrically in nite if happens to be a surface group then we say it is quasi fuchsian if the limit set for the group action is a jordan curve c and preserves the components of s n c the starting point for this work is the following theorem which is a combination of theorems due to marden thurston and bonahon

a low order locking free virtual element for linear elasticity problems abstract we propose and analyze a low order virtual element for linear elasticity problems it can be seen as an evolution of the bernardi raugel element to general polygonal meshes like the bernardi raugel element the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge the proposed method is well posed and optimal error estimates are obtained in the l 2 and h 1 norms moreover the generic constants in these estimates are shown to be uniform with respect to the lame coefficient numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions

generalized lambert series identities we present two infinite families of generalized lambert series identities and deduce several known identities from them they include an identity due to m jackson a corollary of ramanujan s 1 psi 1 summation formula and a recent identity of g e andrews r p lewis and z g liu

lifting cusp forms to maass forms with an application to partitions for 2 k abstract see text we define lifts of cuspidal poincare series in s k gamma 0 n to weight 2 k harmonic weak maass forms this construction answers a question of dyson by providing the general framework explaining ramanujan s mock theta functions as an application we show that the number of partitions of a positive integer n is the trace of singular moduli of a maass form arising from the lift of a weight 4 cusp form corresponding to a calabi yau threefold