formulas for the coefficients of half integral weight harmonic maas forms recently bruinier and ono proved that the coefficients of certain weight 1 2 harmonic weak maas forms are given as traces of singular moduli for harmonic weak maas forms here we prove that similar results hold for the coefficients of harmonic weak maas forms of weight 3 2 k k even and weight 1 2 k k odd by extending the theta lift of bruinier funke and bruinier ono moreover we generalize these results to include twisted traces of singular moduli using earlier work of the author and ehlen on the twisted bruinier funke lift employing a general duality result between weight k and 2 k we obtain formulas for all half integral weights we also show that the non holomorphic part of the theta lift in weight 1 2 k k odd is connected to the vanishing of the special value of the l function of a certain derivative of the lifted function

modularity quaternion kahler spaces and mirror symmetry we provide an explicit twistorial construction of quaternion kahler manifolds ob tained by deformation of c map spaces and carrying an isometric action of the modular group sl 2 z the deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing ns5 brane instanton effects in string compactifications with n 2 supersymmetry in this context the modular invariant parametrization of twistor lines found in this work yields the complete non perturbative mirror map between type iia and type iib physical fields

traces of cm values of modular functions zagier proved that the traces of singular moduli i e the sums of the values of the classical j invariant over quadratic irrationalities are the fourier coefficients of a modular form of weight 3 2 with poles at the cusps using the theta correspondence we generalize this result to traces of cm values of weakly holomorphic modular functions on modular curves of arbitrary genus we also study the theta lift for the weight 0 eisenstein series for sl2 and realize a certain generating series of arithmetic intersection numbers as the derivative of zagier s eisenstein series of weight 3 2 this recovers a result of kudla rapoport and yang

multiple d3 instantons and mock modular forms i we study d3 instanton corrections to the hypermultiplet moduli space in type iib string theory compactified on a calabi yau threefold in a previous work consistency of d3 instantons with s duality was established at first order in the instanton expansion using the modular properties of the m5 brane elliptic genus we extend this analysis to the two instanton level where wall crossing phenomena start playing a role we focus on the contact potential an analogue of the kahler potential which must transform as a modular form under s duality we show that it can be expressed in terms of a suitable modification of the partition function of d4 d2 d0 bps black holes constructed out of the generating function of msw invariants the latter coincide with donaldson thomas invariants in a particular chamber modular invariance of the contact potential then requires that in case where the d3 brane wraps a reducible divisor the generating function of msw invariants must transform as a vector valued mock modular form with a specific modular completion built from the msw invariants of the constituents physically this gives a powerful constraint on the degeneracies of bps black holes mathematically our result gives a universal prediction for the modular properties of donaldson thomas invariants of pure two dimensional sheaves

on the quantization of a self dual integrable system in this note we apply canonical quantization to the self dual particle system describing the motion of poles to a higher rank solution of the kp hierarchy explicitly determining both the quantum hamiltonian and the wavefunction it is verified that the quantum hamiltonian is trivially bispectral that is that the wavefunction can be taken to be symmetric as predicted by a widely held hypothesis of mathematical physics

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient nonlinear multigrid methods such as the full approximation scheme fas and newton multigrid newton mg are well established as fast solvers for nonlinear pdes of elliptic and parabolic type in this paper we consider newton mg and fas iterations applied to second order differential operators with nonlinear diffusion coefficient under mild assumptions arising in practical applications an approximation shown to be sharp of the execution time of the algorithms is derived which demonstrates that newton mg can be expected to be a faster iteration than a standard fas iteration for a finite element discretisation results are provided for elliptic and parabolic problems demonstrating a faster execution time as well as greater stability of the newton mg iteration results are explained using current theory for the convergence of multigrid methods giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice

special cycles on unitary shimura varieties i unramified local theory the supersingular locus in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over is uniformized by a formal scheme mathcal n in the case when p is an inert prime we define special cycles mathcal z bold x in mathcal n associated to collections bold x of m special homomorphisms with fundamental matrix t hermm ok when m n and t is nonsingular we show that the cycle mathcal z bold x is either empty or is a union of components of the ekedahl oort stratification and we give a necessary and sufficient condition in terms of t for mathcal z bold x to be irreducible when mathcal z bold x is zero dimensional in which case it reduces to a single point we determine the length of the corresponding local ring by using a variant of the theory of quasi canonical liftings we show that this length coincides with the derivative of a representation density for hermitian forms

donaldson thomas invariants of 2 dimensional sheaves inside threefolds and modular forms abstract motivated by the s duality conjecture we study the donaldson thomas invariants of the 2 dimensional gieseker stable sheaves on a threefold these sheaves are supported on the fibers of a nonsingular threefold x fibered over a nonsingular curve in the case where x is a k3 fibration we express these invariants in terms of the euler characteristic of the hilbert scheme of points on the k3 fiber and the noether lefschetz numbers of the fibration we prove that a certain generating function of these invariants is a vector modular form of weight 3 2 as predicted in s duality

the geometric theta correspondence for hilbert modular surfaces we give a new proof and an extension of the celebrated theorem of hirzebruch and zagier 17 that the generating function for the intersection numbers of the hirzebruch zagier cycles in certain hilbert modular surfaces is a classical modular form of weight 2 in our approach we replace hirzebuch s smooth complex analytic compactification of the hilbert modular surface with the real borel serre compactification the various algebro geometric quantities that occur in 17 are replaced by topological quantities associated to 4 manifolds with boundary in particular the boundary contribution in 17 is replaced by sums of linking numbers of circles the boundaries of the cycles in 3 manifolds of type sol torus bundle over a circle which comprise the borel serre boundary

a mesh independence principle for operator equations and their discretizations the mesh independence principle asserts that when newton s method is applied to a nonlinear equation between some banach spaces as well as to some finite dimensional discretization of that equation then the behavior of the discretized process is asymptotically the same as that for the original iteration and as a consequence the number of steps required by the two processes to converge to within a given tolerance is essentially the same so far this result has been proved only for certain classes of boundary value problems in this paper a proof is presented for a general class of operator equations and discretizations it covers the earlier results and extends them well beyond the cases that have been considered before

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

superconnections theta series and period domains abstract we use superconnections to define and study some natural differential forms on the period domain d that parametrizes polarized hodge structures of given type on a rational quadratic vector space v these forms depend on a choice of vectors v 1 v r v and have a gaussian shape that peaks on the locus where v 1 v r become hodge classes we show that they can be rescaled so that one can form theta series by summing over a lattice l r v r these series define differential forms on arithmetic quotients d we compute their cohomology class explicitly in terms of the cohomology classes of hodge loci in d when the period domain is a hermitian symmetric domain of type iv we show that the components of our forms of appropriate degree recover the forms introduced by kudla and millson in particular our results provide another way to establish the main properties of these forms

s duality and refined bps indices whenever available refined bps indices provide considerably more information on the spectrum of bps states than their unrefined version extending earlier work on the modularity of generalized donaldson thomas invariants counting d4 d2 d0 brane bound states in type iia strings on a calabi yau threefold mathfrak y we construct the modular completion of generating functions of refined bps indices supported on a divisor class although for compact mathfrak y the refined indices are not protected switching on the refinement considerably simplifies the construction of the modular completion furthermore it leads to a non commutative analogue of the tba equations which suggests a quantization of the moduli space consistent with s duality in contrast for a local cy threefold given by the canonical bundle over a complex surface s refined dt invariants are well defined and equal to vafa witten invariants of s our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank in cases where all reducible components of the divisor class are collinear which occurs e g when b 2 mathfrak y 1 or in the local case we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order in the local case it agrees with an earlier proposal by minahan et al for unrefined invariants and extends it to the refined level using the afore mentioned non commutative structure finally we show that these general predictions reproduce known results for u 2 and u 3 vafa witten theory on mathrm p 2 and make them explicit for u 4

regularized theta liftings and periods of modular functions in this paper we use regularized theta liftings to construct weak maass forms of weight 1 2 as lifts of weak maass forms of weight 0 as a special case we give a new proof of some of recent results of duke toth and the third author on cycle integrals of the modular j invariant and extend these to any congruence subgroup moreover our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane in particular we give the central value of the non existent l function for j the key to the proofs is the construction of a kind of simultaneous green function for both the cm points and the geodesic cycles which is of independent interest

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

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

bispectral darboux transformations the generalized airy case abstract this paper considers darboux transformations of a bispectral operator which preserve its bispectrality a sufficient condition for this to occur is given and applied to the case of generalized airy operators of arbitrary order r 1 as a result the bispectrality of a large family of algebras of rank r is demonstrated an involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters generalizing wilson s rank one bispectral involution spectral geometry and the relationship to the sato grassmannian are discussed

vector valued higher depth quantum modular forms and higher mordell integrals abstract we introduce vector valued higher depth quantum modular forms and investigate examples coming from characters of representations of vertex algebras these are expressed as rank two false theta series generalizing unary false theta series studied from several standpoints we also discuss certain double integrals which may be viewed as the obstruction to modularity of the depth two quantum modular forms we then find explicit formulas for the double error integrals in the form reminiscent of the classical mordell integral

algebraic formulas for the coefficients of half integral weight harmonic weak maass forms abstract we prove that the coefficients of certain weight 1 2 harmonic maass forms are traces of singular moduli for weak maass forms to prove this theorem we construct a theta lift from spaces of weight 2 harmonic weak maass forms to spaces of weight 1 2 vector valued harmonic weak maass forms on mp 2 z a result which is of independent interest we then prove a general theorem which guarantees with bounded denominator when such maass singular moduli are algebraic as an example of these results we derive a formula for the partition function p n as a finite sum of algebraic numbers which lie in the usual discriminant 24 n 1 ring class field

action angle maps and scattering theory for some finite dimensional integrable systems i the pure soliton case we construct an action angle transformation for the calogero moser systems with repulsive potentials and for relativistic generalizations thereof this map is shown to be closely related to the wave transformations for a large classl of hamiltonians and is shown to have remarkable duality properties all dynamics inl lead to the same scattering transformation which is obtained explicitly and exhibits a soliton structure an auxiliary result concerns the spectral asymptotics of matrices of the formm exp td ast it pertains to diagonal matricesd whose diagonal elements have pairwise different real parts and to matricesm for which certain principal minors are non zero

convergence of the multilevel full approximation scheme including thev cycle the multilevel full approximation scheme fas ml is a well known solver for nonlinear boundary value problems in this paper we prove local quantitative convergence statements for a class of fas ml algorithms in a general hilbertspace setting this setting clearly exhibits the structure of fas ml we prove local convergence of a nested iteration for a rather concrete class of fas ml algorithms in whichv cycles and only one jacobilike pre and post smoothing on each level are used

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

fourier coefficients of harmonic weak maass forms and the partition function in a recent paper bruinier and ono proved that certain harmonic weak maass forms have the property that the fourier coefficients of their holomorphic parts are algebraic traces of weak maass forms evaluated on heegner points as a special case they obtained a remarkable finite algebraic formula for the hardy ramanujan partition function p n which counts the number of partitions of a positive integer n we establish an asymptotic formula with a power saving error term for the fourier coefficients in the bruinier ono formula as a consequence we obtain a new asymptotic formula for p n one interesting feature of this formula is that the main term contains essentially 3 cdot h 24n 1 fewer terms than the truncated main term in rademacher s exact formula for p n where h 24n 1 is the class number of the imaginary quadratic field bbb q sqrt 24n 1

convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems the multigrid full approximation scheme fas mg is a well known solver for nonlinear boundary value problems in this paper we restrict ourselves to a class of second order elliptic mildly nonlinear problems and we give local conditions e g a local lipschitz condition on the derivative of the continuous operator under which the fas mg with suitably chosen parameters locally converges we prove quantitative convergence statements and deduce explicit bounds for important quantities such as the radius of a ball of guaranteed convergence the number of smoothings needed the number of coarse grid corrections needed and the number of fas mg iterations needed in a nested iteration these bounds show well known features of the fas mg scheme

vertical d4 d2 d0 bound states on k3 fibrations and modularity an explicit formula is derived for the generating function of vertical d4 d2 d0 bound states on smooth k3 fibered calabi yau threefolds generalizing previous results of gholampour and sheshmani it is also shown that this formula satisfies strong modularity properties as predicted by string theory this leads to a new construction of vector valued modular forms which exhibit some of the features of a generalized hecke transform

vafa witten theory and iterated integrals of modular forms vafa witten vw theory is a topologically twisted version of mathcal n 4 supersymmetric yang mills theory s duality suggests that the partition function of vw theory with gauge group su n transforms as a modular form under duality transformations interestingly vafa and witten demonstrated the presence of a modular anomaly when the theory has gauge group su 2 and is considered on the complex projective plane mathbb p 2 this modular anomaly could be expressed as an integral of a modular form and also be traded for a holomorphic anomaly we demonstrate that the modular anomaly for gauge group su 3 involves an iterated integral of modular forms moreover the modular anomaly for su 3 can be traded for a holomorphic anomaly which is shown to factor into a product of the partition functions for lower rank gauge groups the su 3 partition function is mathematically an example of a mock modular form of depth two

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

cycles with local coefficients for orthogonal groups and vector valued siegel modular forms the purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and fourier coefficients of siegel modular forms to the case where the cycles have local coefficients now the correspondence will involve vector valued siegel modular forms

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

bispectral kp solutions and linearization of calogero moser particle systems rational and soliton solutions of the kp hierarchy in the subgrassmanniangr1 are studied within the context of finite dimensional dual grassmannians in the rational case properties of the tau function which are equivalent to bispectrality of the associated wave function are identified in particular it is shown that there exists a bound on the degree of all time variables in if and only if is a rank one bispectral wave function the action of the bispectral involution in the generic rational case is determined explicitly in terms of dual grassmannian parameters using the correspondence between rational solutions and particle systems it is demonstrated that is a linearizing map of the calogero moser particle system and is essentially the map introduced by airault mckean and moser in 1977 2

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

wall crossing holomorphic anomaly and mock modularity of multiple m5 branes using wall crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two m5 branes wrapping a rigid divisor inside a calabi yau manifold the anomaly originates from restoring modularity of an indefinite theta function capturing the wall crossing of bps invariants associated to d4 d2 d0 brane systems we show the compatibility of this equation with anomaly equations previously observed in the context of mathcal n 4 topological yang mills theory on mathbbm p 2 and e strings obtained from wrapping m5 branes on a del pezzo surface the non holomorphic part is related to the contribution originating from bound states of singly wrapped m5 branes on the divisor we show in examples that the information provided by the anomaly is enough to compute the bps degeneracies for certain charges we further speculate on a natural extension of the anomaly to higher d4 brane charge

mock modular forms and geometric theta functions for indefinite quadratic forms theta functions for indefinite quadratic forms are an important tool to construct modular forms and mock modular forms in this note we recall the representation theoretic background in the construction of theta series with emphasis on the theory developed by the second named author with millson we then employ this machinery to define a theta integral for any signature for which we provide a natural splitting into a holomorphic part with geometric meaning and its non holomorphic modular completion in particular specializing to hyperbolic space we recover zweger s mock theta function from a geometric perspective

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

black holes and higher depth mock modular forms by enforcing invariance under s duality in type iib string theory compactified on a calabi yau threefold we derive modular properties of the gene rating function of bps degeneracies of d4 d2 d0 black holes in type iia string theory compactified on the same space mathematically these bps degeneracies are the generalized donaldson thomas invariants counting coherent sheaves with support on a divisor cal d at the large volume attractor point for cal d irreducible this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping m5 brane on cal d and is therefore known to be modular instead when cal d is the sum of n irreducible divisors cal d i we show that the generating function acquires a modular anomaly we characterize this anomaly for arbitrary n by providing an explicit expression for a non holomorphic modular completion in terms of generalized error functions as a result the generating function turns out to be a mixed mock modular form of depth n 1

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

analysis of a multilevel iterative method for nonlinear finite element equations the multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations we describe how this technique can be combined with a globally convergent approximate newton method to solve nonlinear partial differential equations we show that asymptotically only one newton iteration per level is required thus the complexity for linear and nonlinear problems is essentially equal

global laurent expansions on riemann surfaces we discuss global laurent expansions for meromorphich forms on a compact riemann surface of genusg 2 our approach is motivated by krichever and novikov s work on string theory

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

d instantons and twistors some exact results we present some results on instanton corrections to the hypermultiplet moduli space in calabi yau compactifications of type ii string theories previously using twistor methods only a class of d instantons d2 instantons wrapping a cycles was incorporated exactly and the rest was treated only linearly we go beyond the linear approximation and give a set of holomorphic functions which through a known procedure capture the effect of d instantons at all orders moreover we show that for a sector where all instanton charges have vanishing symplectic invariant scalar product the hypermultiplet metric can be computed explicitly

arithmetic siegel weil formula on x0 n abstract in this paper we proved an arithmetic siegel weil formula and the modularity of some arithmetic theta function on the modular curve x 0 n when n is square free in the process we also constructed some generalized delta function for 0 n and proved some explicit kronecker limit formula for eisenstein series on x 0 n