explicit determination of the picard group of moduli spaces of semistable g bundles on curves let g be a connected simply connected simple affine algebraic group and cg be a smooth irreducible projective curve of any genus g 1 over c denote by mcg g the moduli space of semistable principal g bundles on cg let pic mcg g be the picard group of mcg g and let x be the infinite grassmannian of the affine kac moody group associated to g it is known that pic x z and is generated by a homogenous line bundle l 0 also as proved by kumarnarasimhan kn there exists a canonical injective group homomorphism

preconditioned conjugate gradient methods for three dimensional linear elasticity finite element modelling of three dimensional elasticity problems give rise to large sparse matrices various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques incomplete factorizations based on levels of fill drop tolerance and a two level hierarchical basis are developed various techniques for ensuring that the incomplete factors have positive pivots are presented computational tests are carried out for problems generated using unstructured tetrahedral meshes quadratic basis functions are used the performance of the iterative methods is compared to a standard direct sparse matrix solver problems with up to 70 000 degrees of freedom and small 1 element aspect ratio are considered

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

involutions on moduli spaces and refinements of the verlinde formula the moduli space m of semi stable rank 2 bundles with trivial deter minant over a complex curvecarries involutions naturally associated to 2 torsion points on the jacobian of the curve for every lift of a 2 torsion point to a 4 torsion point we define a lift of the involution to the determinant line bundle l we ob tain an explicit presentation of the group generated by these lifts in terms of the order 4 weil pairing this is related to the triple intersections of the components of the fixed point sets in m which we also determine completely using the order 4 weil pairing the lifted involutions act on the spaces of holomorphic sections of powers of l whose dimensions are given by the verlinde formula we compute the characters of these vector spaces as representations of the group generated by our lifts and we obtain an explicit isomorphism as group representations with the combinatorial topological tqft vector spaces of bhmv as an application we describe a brick decomposition with explicit dimension formulas of the verlinde vector spaces we also obtain similar results in the twisted i e degree one case

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

deformations of principal bundles on the projective line the study of bundles on ip 1 apparently has a long history see 22 chap i sect 2 4 grothendieck proved that any principal bundle on ip with a complex reductive lie groups as structure group admits a reduction of structure group to a maximal torus unique up to weyl group action 9 harder gave a simple proof of this result which works for ip 1 over arbitrary fields 1 11 in this paper we study the deformations of principal bundles over ipl let g be a split reductive group over the field k by the result of grothendieck harder and zariski locally trivial g bundle on ip is associated to the g bundle k z 0 i p 1 by a 1 ps 2 g g let us denote this g bundle by e let e s 1 be a g bundle with an isomorphism eso e s o x lp e we then call e a deformation of ex parametrized by s s o we say that the gbundle e tends or degenerates to the g bundle e and write e e if there is a deformation e s x l p 1 of e such that in every neighbourhood of the base point socs e o e there is an s such that e e we prove theorem 7 4 that if 2 are dominant 1 ps then e e z if and only if 2 i e 2 p is a positive integral combination of simple coroots or equivalently 2 oi e2g for every fundamental weight co i see sect 2 5 note that the set of dominant such that 2 is the same as the set of dominant weights occuring in the indecomposable or irreducible if char k 0 representation of the dual group g see sect 2 6 with highest weight 2 cf 16 sect 21 3 the deformation theory of g bundles on ip seems to be much the same as the representation theory of the dual group g cf 9 p 123 it would be interesting to find a more intrinsic connection between them the g bundles e and e are said to be algebraically equivalent if there is a g bundle e s x lp 1 with s connected such that e e s and e e s for some s s es we prove theorem 7 7 that the algebraic equivalence classes of zariski locally trivial g bundles are classified by the fundamental group of g i e the quotient of the lattice of 1 ps of g by the lattice of coroots this result

a bott borel weil theory for direct limits of algebraic groups we develop a bott borel weil theory for direct limits of algebraic groups some of our results apply to locally reductive ind groups g in general i e to arbitrary direct limits of connected reductive linear algebraic groups our most explicit results concern root reductive ind groups g the locally reductive ind groups whose lie algebras admit root decomposition given a parabolic subgroup p of g and a rational irreducible p module we consider the irreducible g sheaves inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i and their duals inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i these sheaves are locally free in general of infinite rank we prove a general analog of the bott borel weil theorem for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i namely that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i is nonzero for at most one index q q 0 and that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i is isomorphic to the dual of a rational irreducible g module v for q 0 0 we show that in contrast to the finite dimensional case v need not admit an irreducible p submodule there however one has a larger parabolic subgroup w p p constructed from p and a weyl group element w of length q 0 such that v is generated by an irreducible w p submodule consequently certain g modules v can appear only for q 0 0 never for q 0 0 for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i we show that there is no analog of bott s vanishing theorem more precisely that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i can have arbitrarily many nonzero cohomology groups finally we give an explicit criterion for the projectivity of the ind variety g p showing that g p is in general not projective

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

numerical solution of implicit neutral functional differential equations this paper is concerned with the numerical solution of implicit neutral functional differential equations based on the continuous runge kutta method for ordinary differential equations and the collocation method for functional equations two general one step methods are formulated and their uniform order of approximation is discussed numerical stability of a class of runge kutta collocation methods is analyzed in the case where a neutral equation can be expressed in an explicit as well as an implicit form the advantage of constructing a numerical scheme directly from the implicit form is explained theoretically and demonstrated by a numerical example

the numerical solution of volterra functional differential equations of neutral type in this paper methods of adams moulton type for neutral functional differential equations are investigated a general convergence theorem is presented and the order of convergence of these methods is examined some numerical examples are given

natural runge kutta and projection methods recently the author defined the class of natural runge kutta methods and observed that it includes all the collocation methods the present paper is devoted to a complete characterization of this class and it is shown that it coincides with the class of the projection methods in some polynomial spaces

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

the cone of type a level 1 conformal blocks divisors abstract we prove that the type a level 1 conformal blocks divisors on m 0 n span a finitely generated full dimensional subcone of the nef cone each such divisor induces a morphism from m 0 n and we identify its image as a git quotient parameterizing configurations of points supported on a flat limit of veronese curves we show how scaling git linearizations gives geometric meaning to certain identities among conformal blocks divisor classes this also gives modular interpretations in the form of git constructions to the images of the hyperelliptic and cyclic trigonal loci in m g under an extended torelli map

dissipativity of θ methods for a class of nonlinear neutral delay differential equations abstract this paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations at first a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given then the dissipativity results are obtained for one leg and linear methods when they are applied to these problems

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

solutions of neutral delay differential equations using a generalized lambert w function abstract the lambert w function is defined by w a e w a a 0 one of the many applications of the lambert w function is in solving delay differential equations ddes in 2003 asl and ulsoy provided a solution of some ddes in terms of the lambert w functions asl et al 2003 1 however the solutions are limited to differential equations with delay in the state variable scott et al 2006 2 introduced a generalized lambert function which was further studied by mez and baricz 2017 3 in our work we show that this generalization of the lambert w function provides an analytical solution to neutral delay differential equations nddes nddes are ddes with time delay not only in the state variables but also in the derivative terms this analytical solution is advantageous such that it is similar to the general solutions of linear odes also one can identify how the parameters affect the solution of the equation since our proposed solution is written in terms of these parameters we then propose a new numerical method to solve linear nddes using the generalized lambert w function we test our method to examples with known solutions we also provide a real world application by solving an ndde model of the population growth of an e coli culture using our proposed approach

stability analysis of one step methods for neutral delay differential equations in this paper stability properties of one step methods for neutral functional differential equations are investigate stability regions are characterized for runge kutta methods with respect to the linear test equation begin gathered y left t right ay left t right by left t tau right cy left t tau right t geqq 0 hfill y left t right g left t right tau leqq t leqq 0 hfill end gathered 0 where a b andc are complex parameters in particular it is shown that everya stable collocation method for ordinary differential equations can be extended to a method for neutrals delay differential equations with analogous stability properties the so called np stable method we also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations

stability analysis of general linear methods for nonlinear neutral delay differential equations abstract the present paper is concerned with the numerical solution to initial value problems of neutral delay differential equations nddes the sufficient conditions for the stability and asymptotic stability of the general linear methods for a class r of nonlinear systems of nddes are derived numerical test is given that confirm the theoretical results in the end

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

on convergence of difference schemes for delay parabolic equations the convergence of difference schemes for the approximate solutions of the initial boundary value problem for the delay parabolic differential equation dv t dt av t b t v email protected t 0 v t g t email protected email protected 0 in an arbitrary banach space e with the unbounded linear operators a and b t in e with dense domains d a d b t is investigated theorems on convergence estimates for the solutions of the first and the second order of accuracy difference schemes in fractional spaces e a are established in practice the convergence estimates in holder norms for the solutions of difference schemes of the first and the second order of approximation in t of the approximate solutions of multi dimensional delay parabolic equations are obtained the theoretical statements for the solution of these difference schemes are supported by the numerical example

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

numerical stability of nonlinear delay differential equations of neutral type this paper is devoted to the stability analysis of both the true solutions and the numerical approximations for nonlinear systems of neutral delay differential equations nddes of the form y t f t y t g t y t t y t t this work extends the results recently obtained by the authors bellen et al bit 39 1999 1 24 for the linear case this is accomplished by considering a suitable reformulation of the given system which transforms it into a nonlinear differential system coupled with an algebraic functional recursion numerical processes preserving the qualitative properties of the solutions are also investigated

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

global dissipativity for a stable methods this paper concerns the discretization of the initial value problem u t f u under the three structural conditions i f bbb c n longrightarrow bbb c n re e langle f u u rangle leq a b u 2 a geq 0 b 0 for all u in bbb c n ii f bbb c n longrightarrow bbb c n re e langle f u u rangle 0 iii f w longrightarrow h re e langle f w w rangle h leq a b w h 2 a geq 0 b 0 for all w in w for complex hilbert spaces w subseteq h dahlquist s g stability theory is used to show that linear multistep and one leg methods yield dissipative discretizations for all f satisfying i if and only if the method rho sigma is a stable extensions of g stability theory are made to find necessary and sufficient conditions on rho sigma for similar properties to hold in cases ii and iii in every case conditions are found for the strict contractivity of solutions for large initial data and bounds for the asymptotic rate of decay are calculated in cases i and iii

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

dissipativity of linear θ methods for integro differential equations this paper is concerned with the analytic and numerical dissipativity of nonlinear integro differential equations ides a dissipativity criteria for ides is given it is shown that any a stable linear 0 method for the systems is dissipative numerical examples are given to confirm the theoretical results

borel weil bott theory on the moduli stack of g bundles over a curve let g be a semi simple group and m the moduli stack of g bundles over a smooth complex projective curve using representation theoretic methods i prove the pure dimensionality of sheaf cohomology for certain evaluation vector bundles over m twisted by powers of the fundamental line bundle this result is used to prove a borel weil bott theorem conjectured by g segal for certain generalized flag varieties of loop groups along the way the homotopy type of the group of algebraic maps from an affine curve to g and the homotopy type the hodge theory and the picard group of m are described one auxiliary result in appendix a is the alexander cohomology theorem conjectured in gro2 a self contained account of the uniformization theorem of ls for the stack m is given including a proof of a key result of drinfeld and simpson in characteristic 0 a basic survey of the simplicial theory of stacks is outlined in appendix b

dissipativity of volterra functional differential equations this paper is concerned with the dissipativity of volterra functional differential equations in a hilbert space a sufficient condition for dissipativity of one class of such equations is obtained this result is applied to delay differential equations and integro differential equations to obtain dissipativity results that are more general and deeper than related results in the previous literature

homology of affine springer fibers in the unramified case assuming a certain purity conjecture we derive a formula for the complex cohomology groups of the affine springer fiber corresponding to any unramified regular semisimple element we use this calculation to present a complex analog of the fundamental lemma for function fields we show that the kappa orbital integral that arises in the fundamental lemma is equal to the lefschetz trace of the frobenius acting on the etale cohomology of a related variety

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

one step methods of any order for neutral functional differential equations a new class of one step methods for the numerical solution of neutral functional differential equations is considered these methods can be taken to have arbitrarily high order this should be compared with other one step methods such as those of tavernini for which the obvious modification to neutral functional differential equations leads only to methods of order one

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

an abelianization of su 2 wzw model the purpose of this paper is to carry out the abelianization program pro posed by atiyah 1 and hitchin 9 for the geometric quantization of su 2 wess zumino witten model let c be a riemann surface of genus g let mg be the moduli space of semi stable rank 2 holomorphic vector bundles on c with trivial determinant for a positive integer k let t mg ck be the space of holomorphic sections of the fc th tensor product of the determinant line bundle c on mg an element of t mg ck is called a rank 2 theta function of level k the main result of our abelianization is to give an explicit representation of a base of t mg ck as well as its transformation formula in terms of classical riemann theta functions with automorphic form coefficients defined on the prym variety p associated with a two fold branched covering surface c of c r mg ck can be identified with the conformai block of level k of the su 2 wzw model 5 15 the abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way firstly we construct a projectively flat connection on the vector bundle over the teichm ller space with fibre t mg ck secondly making use of our ex plicit representation of rank 2 theta functions we construct a hermitian product on the vector bundle preserved by the connection also our explicit represen tation enables us to prove that y mg ck has the predicted dimension from the quantum clebsh gordan conditions a natural connection on the said vector bundle for the su iv wzw model was first constructed by hitchin 11 it will turn out that the connection constructed in this paper coincides with the hitchin connection laszlo 16 showed that the hitchin connection coincides with the con nection constructed by tsuchiya ueno and yamada 21 through the above identification on the other hand kirillov 13 14 constructed a hermitian product on the conformai block compatible with the tsuchiya ueno yamada connection using the representation theory of affine lie algebras together with the theory of hermitian modular tensor categories cf 22 laszlo s result

sur l homologie des fibres de springer affines tronquees en suivant les travaux de goresky kottwitz et macpherson 12 on calcule l homologie des fibres de springer affines tronquees dans le cas non ramifie mais sous une hypothese de purete dans le cas equivalue on demontre cette hypothese de purete la troncature depend d un parametre qui est un diviseur sur une variete torique l adique dans les cas non ramifies et equivalues pour chaque fibre de springer affine tronquee on definit un faisceau coherent gradue sur cette variete torique dont l espace des sections globales s identifie a l homologie l adique de la fibre consideree pour certaines familles de groupes endoscopiques ces faisceaux apparaissent dans des suites exactes on en deduit alors le lemme fondamental pondere d arthur 5 conjecture 5 1 dans les cas equivalues et non ramifies

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

degeneration of the strange duality map for symplectic bundles global sections of the line bundles on a moduli space of vector bundles or more generally principal g bundles are called generalized theta functions the dimension of the vector spaces of generalized theta functions is given by the celebrated verlinde formula if you compute several verlinde numbers you will find that some of them coincide unexpectedly behind the coincidence there is often geometric meaning let su r be the moduli space of rank r bundles with trivial determinant on a smooth projective curve c of genus g and let l be the ample generator of picsu r let u n be the moduli space of rank n bundles of slope g 1 on c and let u n be the locus of e u n such that h c e 6 0 then by the verlinde formula you see that the dimensions of the vector spaces h0 su r l n and h0 u n o r are equal there is a strange duality map cf b1 h0 su r l n h0 u n o r

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

ngp α stability of general linear methods for nddes abstract this paper deals with the asymptotic behavior of general linear methods for a class of neutral delay differential equations nddes new equivalent stability relations between the ode methods and the induced ndde methods are obtained moreover the obtained results are also extended to the case of nddes with distributed delays

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

estimation of numerically stable step size for neutral delay differential equations via spectral radius we derive the estimates of numerically stable step size for systems of neutral delay differential equations nddes which only need to be calculated the spectral radius of the corresponding matrices the stable step size for numerical integration of nddes can be easily selected by means of the estimates the stability regions of both linear multistep methods and explicit runge kutta methods are presented

dissipativity and asymptotic stability of nonlinear neutral delay integro differential equations this paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro differential equations ndides we first give a generalization of the halanay inequality which plays an important role in the study of dissipativity and stability of differential equations then we apply the generalization of the halanay inequality to ndides and the dissipativity and the asymptotic stability results of the theoretical solution of ndides are obtained from a numerical point of view it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions therefore the results presented in this paper provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

multiple acceptable solutions in structural model improvement this paper examines the question as to whether it is possible to obtain the correct discrete linear analytical model of an actual structure by the use of analysis and test data it is shown that no such correct model exists an illustration of an application is shown where multiple acceptable and very different analytical models are all representative of the structure the implications regarding methods of improving analytical models and detecting damage are discussed

conformal blocks verlinde formula and diagram automorphisms abstract the verlinde formula computes the dimension of the space of conformal blocks associated to simple lie algebras and stable pointed curves if a simply laced simple lie algebra admits a nontrivial diagram automorphism then this automorphism acts on the space of conformal blocks naturally we prove an analogue of the verlinde formula for the trace of the diagram automorphism on the space of conformal blocks along the way we get an analogue of the kac walton formula for the trace of the diagram automorphism we also get a twining type formula between the conformal blocks for the pair s l 2 n 1 s p 2 n

adams methods for neutral functional differential equations in this paper adams type methods for the special case of neutral functional differential equations are examined it is shown thatk step methods maintain orderk 1 for sufficiently small step size in a sufficiently smooth situation however when these methods are applied to an equation with a non smooth solution the order of convergence is only one some computational considerations are given and numerical experiments are presented

stability analysis of numerical methods for systems of functional differential and functional equations abstract this paper is concerned with the numerical solution of functional differential and functional equations which include functional differential equations of neutral type as special cases the adaptation of linear multistep methods one leg methods and runge kutta methods is considered the emphasis is on the linear stability of numerical methods it is proved that a stable methods can inherit the asymptotic stability of underlying linear systems some general results of stability on explicit and implicit methods are also given

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

strange duality of verlinde spaces for g_2 and f_4 we prove that the pull back of the canonical theta divisor for e 8 bundles at level one induces a strange duality between verlinde spaces for g 2 and f 4 at level one on smooth curves of genus g we also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in text pic overline text m g n mathbb q

stability of the radau ia and lobatto iiic methods for neutral delay differential system abstract numerical stability is considered for several runge kutta methods to systems of neutral delay differential equations the linear stability analysis is adopted to the system adapted with the equistage interpolation process as well as the continuous extension the runge kutta methods are shown to have the numerical stability similar to the analytical asymptotic stability with arbitrary stepsize when certain assumptions hold for the logarithmic matrix norm on the coefficient matrices of the ndde system

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

spline approximations for neutral functional differential equations based on an abstract approximation theorem for text c 0 semigroups trotter kato theorem we present an algorithm where linear autonomous functional differential equations of neutral type are approximated by sequences of ordinary differential equations of increasing dimensions numerical examples using cubic splines and cubic hermite splines illustrate the theoretical results

stochastic optimization using the stochastic preconditioned conjugate gradient method the spcg solver has been integrated into nike3d and used to conduct a monte carlo simulation and a stochastic shape optimization of a cantilever beam