eigenvalues for infinite matrices this paper is concerned with the problem of determining the location of eigenvalues for diagonally dominant infinite matrices upper and lower bounds for eigenvalues are established for tridiagonal matrices a numerical procedure for improving the bounds is given and the approximation of the eigenvectors is also discussed the techniques are illustrated for the solution of the well known mathieu s equation

nonlinear stability of θ methods for neutral differential equations in banach space abstract this paper is concerned with the stability properties of theoretical solution and numerical solution to nonlinear neutral differential equations with two delays in banach space some sufficient conditions for the stability and asymptotic stability of the systems are given the numerical stability of methods when applied to the systems is also considered a series of new stability results of methods are obtained

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

stability of linear multistep methods for nonlinear neutral delay differential equations in banach space this paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in banach space two approaches to numerically treating the neutral term are considered which allow us to prove several results on numerical stability of linear multistep methods these results provide some criteria for choosing the step size such that the numerical method is stable some examples of application and a numerical experiment which further confirms the main results are given

fourier mukai transform on abelian surfaces we study moduli spaces of stable sheaves on abelian surfaces whose mukai vectors are related by a cohomological fourier mukai transform we show that there is a fourier mukai transform inducing a birational map between them

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

nonlinear stability of runge kutta methods for neutral delay differential equations this paper deals with stability properties of runge kutta methods for the initial value problem in nonlinear neutral delay differential equationsy t f t y t y t t y t t t 0 y t f t t t 0 the new concepts of gs l stability gas l stability and weak gas l stability are introduced and it is shown that k l algebraically stable runge kutta methods with piecewise linear interpolation are gs l gas l and weakly gas l stable two numerical examples are given in the end of this paper which confirm our results

eigenvalues of the laplacian on an elliptic domain the importance of eigenvalue problems concerning the laplacian is well documented in classical and modern literature finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations asymptotic methods integral equations etc in this paper we present a comprehensive account of the general solutions to helmholtz s equations defined on simply connected regions using complex variable techniques we consider boundaries of the form zz f z z or its inverse z z g zz to illustrate the theory we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems where the coefficients of the infinite matrix are known polynomials of the eigenvalues we compute truncations of the infinite system for numerical values these values are compared to approximate values and some inequalities available in literature

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

virtual refinements of the vafa witten formula we conjecture a formula for the generating function of virtual chi y genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2 form specializing the conjecture to minimal surfaces of general type and to virtual euler characteristics we recover part of a formula of c vafa and e witten these virtual chi y genera can be written in terms of descendent donaldson invariants using t mochizuki s formula the latter can be expressed in terms of seiberg witten invariants and certain explicit integrals over hilbert schemes of points these integrals are governed by seven universal functions which are determined by their values on mathbb p 2 and mathbb p 1 times mathbb p 1 using localization we calculate these functions up to some order which allows us to check our conjecture in many cases in an appendix by h nakajima and the first named author the virtual euler characteristic specialization of our conjecture is extended to include mu classes thereby interpolating between vafa witten s formula and witten s conjecture for donaldson invariants

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

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

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

euler characteristics of su 2 instanton moduli spaces on rational elliptic surfaces recently minahan nemeschansky vafa and warner computed partition functions for n 4 topological yang mills theory on rational elliptic surfaces in particular they computed generating functions of euler characteristics of su 2 instanton moduli spaces in mathematics they are expected to coincide with those of gieseker compactifications in this paper we compute euler characteristics of these spaces and show that our results coincide with theirs we also check the modular property of z su 2 and z so 3 conjectured by vafa and witten

configurations in abelian categories iv invariants and changing stability conditions this is the last in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k are subsets of i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in a using artin stacks it showed well behaved moduli stacks obja m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod kq of representations of a quiver q the second studied algebras of constructible functions and stack functions on obja the third introduced stability conditions t on a and showed the moduli space objss of semistable objects in class is a constructible subset in obja so its characteristic function ss is a constructible function it formed algebras h pa h to h pa h to of constructible and stack functions on obja and proved many identities in them in this paper if t and t are stability conditions on a we write ss in terms of the ss and deduce the algebras h pa h to are independent of t we study invariantsiss or iss i counting semistable objects or configurations in a which satisfy additive and multiplicative identities we compute them completely when a mod kq or a coh p for p a smooth curve we also find invariants with special properties when a coh p for p a smooth surface with kp 1 nef or a calabi yau 3 fold

atiyah class and sheaf counting on local calabi yau fourfolds abstract we discuss donaldson thomas dt invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non compact calabi yau fourfold given by the total space of a rank 2 bundle on the surface we prove that in certain cases when the rank 2 bundle is chosen appropriately the universal truncated atiyah class of these codimension 2 sheaves reduces to one defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor threefold embedded in the ambient fourfold such reduction property of universal atiyah class enables us to relate our fourfold dt theory to a reduced dt theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in 15 16 we finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic k3

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

a review of infinite matrices and their applications abstract infinite matrices the forerunner and a main constituent of many branches of classical mathematics infinite quadratic forms integral equations differential equations etc and of the modern operator theory is revisited to demonstrate its deep influence on the development of many branches of mathematics classical and modern replete with applications this review does not claim to be exhaustive but attempts to present research by the authors in a variety of applications these include the theory of infinite and related finite matrices such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces matrices considered here have special structures like diagonal dominance tridiagonal sign distributions etc and are frequently nonsingular moreover diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations the main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems using techniques like conformal mapping iterations truncations etc to derive estimates based solutions particular attention is paid to computable precise error estimates and explicit lower and upper bounds topics include bessel s mathieu equations viscous fluid flow simply and doubly connected regions digital dynamics eigenvalues of the laplacian etc also presented are results in generalized inverses and semi infinite linear programming

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

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

integral operators and integral cohomology classes of hilbert schemes the methods of integral operators on the cohomology of hilbert schemes of points on surfaces are developed they are used to establish integral bases for the cohomology groups of hilbert schemes of points on a class of surfaces and conjecturally for all simply connected surfaces

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

choice of θ and mean square exponential stability in the stochastic theta method of stochastic differential equations this paper examines the relationship of choice of and mean square exponential stability in the stochastic theta method stm of stochastic differential equations sdes and mainly includes the following three results i under the linear growth condition for the drift term when 0 1 2 the stm may preserve the mean square exponential stability of the exact solution but the counterexample shows that the stm cannot reproduce this stability without this linear growth condition ii when 1 2 1 without the linear growth condition for the drift term the stm may reproduce the mean square exponential stability of the exact solution but the bound of the lyapunov exponent cannot be preserved iii when 1 this stm is called as the backward euler maruyama bem method the stm can reproduce not only the mean square exponential stability but also the bound of the lyapunov exponent this paper also gives the sufficient and necessary conditions of the mean square exponential stability of the stm for the linear sde when 0 1 2 and 1 2 1 respectively and the simulations also illustrate these theoretical results

some cubic modular identities of ramanujan there is a beautiful cubic analogue of jacobi s fundamental theta function identity it is sum n m infty infty q n 2 nm m 2 sum n m infty infty n m q n nm m sum n m infty infty q n 1 3 n 1 3 m 1 3 m 1 3 here exp 2 i 3 in this note we provide an elementary proof of this identity and of a related identity due to ramanujan we also indicate how to discover and prove such identities symbolically

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

estimations on numerically stable step size for neutral delay differential systems with multiple delays we derive two estimations of numerically stable step size for systems of neutral delay differential equations with multiple delays the stable step size for numerical integration of nddes with multiple delays can be easily selected by means of the logarithmic norm and the spectral radius of certain matrices both explicit linear multistep methods and explicit runge kutta methods are considered

nonlinear stability of linear multistep methods for stiff delay differential equations in banach spaces this paper is concerned with the nonlinear stability properties of a class of linear multistep methods when applied to initial value problems of stiff delay differential equations in banach spaces for the test problem classes d a l b and d a l d b in banach spaces a series of new stability results of the linear multistep methods are obtained

visualization of special eigenmode shapes of a vibrating elliptical membrane analysis of eigenfunctions and eigenvalues of the laplacian is important in the understanding of distributed parameter vibration systems and quantum mechanics the elliptical domain is advantageous for such work because it allows the explicit representation of eigenfunctions as products of the mathieu functions in ann phys 9 1960 pp 24 75 kelley and rubinow used wave propagation geometrical optics and wkb methods to show that two special types of eigenmodes whispering gallery and bouncing ball modes exist on a general convex domain and they illustrated the case of an elliptical domain in this paper we develop a numerical package of the mathieu and modified mathieu functions to actually construct profiles of a sequence of eigenfunctions in order to visualize such special eigenmodes and others in the process we have also been able to observe a new type of transition state named focusing modes herein which seem to have a complementary behavior to the bouncing ball modes numerical eige

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

asymptotic stability of solution to nonlinear neutral and volterra functional differential equations in banach spaces abstract this paper is concerned with the asymptotic stability properties of the solution to nonlinear neutral and volterra functional differential equations some sufficient conditions for the asymptotic stability of the systems are given as an illustration of the applications of these investigations the contractivity and asymptotic stability results of the solution to volterra partial functional differential equations and delay integro differential equations of hale s form are obtained respectively these results form the basis for obtaining insight into the analogous properties of their numerical solutions

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

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

stability analysis of lmms for systems of neutral multidelay differential equations this paper deals with the asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay differential equations nmdes in particular it is shown that a stability of the linear multistep methods lmms for odes is equivalent to ngpk stability of the induced methods for nmdes under the suitable conditions

stability and asymptotic stability of θ methods for nonlinear stiff volterra functional differential equations in banach spaces this paper is concerned with the stability and asymptotic stability of q methods for the initial value problems of nonlinear stiff volterra functional differential equations in banach spaces a series of new stability and asymptotic stability results of q methods are obtained

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

nonlinear stability of explicit and diagonally implicit runge kutta methods for neutral delay differential equations in banach space abstract this paper is devoted to the investigations into the nonlinear stability properties of explicit and diagonally implicit runge kutta methods for solving neutral delay differential equations in banach space two approaches to numerically treating neutral term are considered which allow us to prove several results on numerical stability and conditional contractivity of explicit and diagonally implicit runge kutta methods in particular some new results on stiff odes and stiff ddes which can be regarded as the generalizations of previous results in the literature are obtained

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class