moduli of sheaves supported on curves of genus four contained in a quadric surface abstract we study the moduli space of stable sheaves of euler characteristic one supported on curves of degree 3 3 contained in a smooth quadric surface we compute its hodge numbers by investigating the variation of the moduli spaces of semi stable coherent systems we find the classification of the semi stable sheaves by means of extensions or resolutions

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

general regular variation popa groups and quantifier weakening abstract we introduce general regular variation a theory of regular variation containing the existing karamata bojanic karamata de haan and beurling theories as special cases the unifying theme is the popa groups of our title viewed as locally compact abelian ordered topological groups together with their haar measure and fourier theory the power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening so important in this field

transformation de fourier et temps d occupation browniens in this paper we study oscillatory stochastic integrals of the form gamma lambda int limits 0 infty exp i lambda b s g s d s where is a non zero parameter andg a square integrable function we study integrability properties of and its behavior as a function of using stochastic calculus techniques martingale theory representation of ito for a random variable of the wiener space lemma of garsia rodemich rumsey we also obtain limit theorems in law related to the variables based upon an asymptotic version of a theorem of knight on orthogonal continuous martingales

zero one laws for the excursions and range of a levy process a 01 law for the excursions of a levy process x from 0 is established and is used to prove that the closed range of x is either a nowhere dense set or else contains an interval including x0

the douady space of a complex surface abstract we prove that a standard realization of the direct image complex via the so called douady barlet hilbert chow in the algebraic case morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi isomorphism of complexes this is a more precise form of a special case of the decomposition theorems of beilinson bernstein deligne gabber and m saito of which the proof we present is independent in addition it is elementary and transparent and does not use either perverse sheaves the methods of positive characteristic nor saito s theory of mixed hodge modules the proof hinges on the special case of the bi disk in the complex affine plane where we make explicit use of a construction of nakajima s and of the corresponding representation theoretic interpretation foreseen by vafa and witten some consequences of the decomposition theorem the gottsche formula holds for complex surfaces interpretation of the rational cohomologies of douady spaces as a kind of fock space new proofs of results of briancon and ellingsrud and stromme on punctual hilbert schemes and computation of the mixed hodge structure of the douady spaces in the kahler case we also derive a natural connection with equivariant k theory for which in the case of algebraic surfaces bezrukavnikov and ginzburg have proposed a different approach

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

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

j lossless conjugation and factorization for discrete time systems the j j lossless factorization plays an important role in h control theory just like the wiener hopf factorization does in lqg theory for continuous time systems in this paper the theory of j lossless factorization is extended to discrete time systems the notion j lossless conjugation which is a powerful tool for computing j lossless factorization in continuous time systems is also extended to discrete time systems these extensions are far from trivial the discrete time version j lossless conjugations and factorizations turns out to be much more complicated reflecting the complicated nature of the discrete time riccati equations

small gaps in the range of stable processes we examine the structure of the range of a stable process with drift near its initial position and derive an integral test to determine the rate of growth of the size of intervals in the complement of the range this integral test depends on the random number of excursions that the process makes away from the initial point

a lifting technique for linear periodic systems with applications to sampled data control abstract a lifting technique is developed for periodic linear systems and applied to the h and h 2 sampled data control problems

the local structure of the sample paths of asymmetric cauchy processes precise information about the local evolution of the sample path is obtained using asymptotic estimates for the potential kernel for the capacity of small intervals and unions of intervals and for the hitting distribution of such sets a new construction of local time is obtained and it is shown that except for the completely asymmetric case the range of the process on a line is a nowhere dense set of positive lebesgue measure

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

random fourier series and continuous additive functionals of levy processes on the torus let x be an exponentially killed levy process on t n the n dimensional torus that satisfies a sector condition this includes symmetric levy processes let f e denote the extended dirichlet space of x let h f e and let h y y tn denote the set of translates of h that is h y h y we consider the family of zero energy continuous additive functions n t hy y t t n x r as defined by fukushima for a very large class of random functions h we show that j p t n log n p t n 1 2 d is a necessary and sufficient condition for the family n t h y y t t n x r to have a continuous version almost surely here n p t n is the minimum number of balls of radius e in the metric that covers t n where the metric is the energy metric we argue that this condition is the natural extension of the necessary and sufficient condition for continuity of local times of levy processes of barlow and hawkes results on the bounded variation and p variation in t of n t hy for y fixed are also obtained for a large class of random functions h

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

a local time correspondence for stochastic partial differential equations it is frequently the case that a white noise driven parabolic and or hyperbolic stochastic partial differential equation spde can have random field solutions only in spatial dimension one here we show that in many cases where the spatial operator is the l generator of a levy process x a linear spde has a random field solution if and only if the symmetrization of x possesses local times this result gives a probabilistic reason for the lack of existence of random field solutions in dimensions strictly bigger than one in addition we prove that the solution to the spde is holder continuous in its spatial variable if and only if the said local time is holder continuous in its spatial variable we also produce examples where the random field solution exists but is almost surely unbounded in every open subset of space time our results are based on first establishing a quasi isometry between the linear l space of the weak solutions of a family of linear spdes on one hand and the dirichlet space generated by the symmetrization of x on the other hand we study mainly linear equations in order to present the local time correspondence at a modest technical level however some of our work has consequences for nonlinear spdes as well we demonstrate this assertion by studying a family of parabolic spdes that have additive nonlinearities for those equations we prove that if the linearized problem has a random field solution then so does the nonlinear spde moreover the solution to the linearized equation is holder continuous if and only if the solution to the nonlinear equation is and the solutions are bounded and unbounded together as well finally we prove that in the cases that the solutions are unbounded they almost surely blow up at exactly the same points date october 7 2007 2000 mathematics subject classification primary 60h15 60j55 secondary 35r60 35d05

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

constructions of local time for a markov process a sequence of functions defined on the space of excursions of a markov process from a fixed point is considered for each of the functions the sum over the excursions that begin by time t is normalized in an appropriate manner conditions are obtained for the convergence of the sequence of normalized sums to the local time evaluated at time t we obtain a unified structure for convergence theorems which includes some new constructions of local time as well as many constructions previously obtained by quite varied techniques

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

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

on the algebras of bps states we define an algebra on the space of bps states in theories with extended supersymmetry we show that the algebra of perturbative bps states in toroidal compactification of the heterotic string is closely related to a generalized kac moody algebra we use d brane theory to compare the formulation of rr charged bps algebras in type ii compactification with the requirements of string string duality and find that the rr charged bps states should be regarded as cohomology classes on moduli spaces of coherent sheaves the equivalence of the algebra of bps states in heterotic iia dual pairs elucidates certain results and conjectures of nakajima and gritsenko nikulin on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras in particular to any calabi yau 3 fold there are two canonically associated algebras exchanged by mirror symmetry

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

vertex algebras and the cohomology ring structure of hilbert schemes of points on surfaces using vertex algebra techniques we determine a set of generators for the cohomology ring of the hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers

the chow groups and the motive of the hilbert scheme of points on a surface abstract we compute the chow motive and the chow groups with rational coefficients of the hilbert scheme of points on a smooth algebraic surface

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

twisted cubics on cubic fourfolds we construct a new twenty dimensional family of projective eight dimen sional irreducible holomorphic symplectic manifolds the compactified moduli space m3 y of twisted cubics on a smooth cubic fourfold y that does not contain a plane is shown to be smooth and to admit a contractionm3 y z y to a projective eight dimensional symplec tic manifoldz y the construction is based on results on linear determinantal representations of singular cubic surfaces

on the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface let j be an abelian surface with a generic ample line bundle open image in new window for n 1 the moduli space mj 2 0 2n of open image in new window 1 semistable sheaves f of rank 2 with chern classes c1 f 0 c2 f 2n is a singular projective variety endowed with a holomorphic symplectic structure on the smooth locus in this paper we show that there does not exist a crepant resolution of mj 2 0 2n for n 2 this certainly implies that there is no symplectic desingularization of mj 2 0 2n for n 2

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

on the universal family of hilbert schemes of points on a surface abstract for a smooth quasi projective surface x and an integer n 3 we show that the universal family z n over the hilbert scheme hilb n x of n points has non q gorenstein rational singularities and that the samuel multiplicity at a closed point on z n can be computed in terms of the dimension of the socle we also show that n

ideals of the cohomology rings of hilbert schemes and their applications we study the ideals of the rational cohomology ring of the hilbert scheme x n of n points on a smooth projective surface x as an application for a large class of smooth quasi projective surfaces x we show that every cup product structure constant of h x n is independent of n moreover we obtain two sets of ring generators for the cohomology ring h x n similar results are established for the chen ruan orbifold cohomology ring of the symmetric product in particular we prove a ring isomorphism between h x n c and h orb x n sn c for a large class of smooth quasi projective surfaces with numerically trivial canonical class

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

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

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

the cup product of hilbert schemes for k3 surfaces to any graded frobenius algebra a we associate a sequence of graded frobenius algebras a n so that there is canonical isomorphism of rings h x 2 n h x n 2n for the hilbert scheme x n of generalised n tuples of any smooth projective surface x with numerically trivial canonical bundle

calabi yau volumes and reflexive polytopes we study various geometrical quantities for calabi yau varieties realized as cones over gorenstein fano varieties obtained as toric varieties from reflexive polytopes in various dimensions focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the sasaki einstein base of the corresponding calabi yau cone are calculated by doing so we conjecture new bounds for the sasaki einstein volume with respect to various topological quantities of the corresponding toric varieties we give interpretations about these volume bounds in the context of associated field theories via the ads cft correspondence

conjugation and h control of discrete time systems we establish conjugation notion in discrete time systems first introduced into the h control theory of continuous time systems by kimura 1989 in discrete time systems conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle with the aid of j lossless conjugation conjugation by a j lossless system it is shown that the parametrization of sub optimal solutions of h model matching problems is reduced to a lyapunov type equation the parametrization of all solutions is given in an extremely simple way it is further proved that the j lossless conjugation of the h model matching problem is a natural state space representation of classical interpolation in discrete time systems

on the continuity of local times of borel right markov processes the problem of finding a necessary and sufficient condition for the continuity of the local times for a general markov process is still open barlow and hawkes have completely treated the case of the levy processes and marcus and rosen have solved the case of the strongly symmetric markov processes we treat here the continuity of the local times of borel right processes our approach unifies that of barlow and hawkes and of marcus and rosen by using an associated gaussian process that appears as a limit in a clt involving the local time process

an intersection number for the punctual hilbert scheme of a surface we compute the intersection number between two cycles a and b of complementary dimensions in the hilbert scheme h parameterizingr subschemes of given finite length n of a smooth projective surface s the n 1 cycle a corresponds to the set of finite closed subschemes the support of which has cardinality 1 the n 1 cycle b consists of the closed subschemes the support of which is one given point of the surface since b is contained in a indirect methods are needed the intersection number is a b 1 n 1n answering a question by h nakajima

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

nested hilbert schemes on surfaces virtual fundamental class abstract we construct virtual fundamental classes on nested hilbert schemes of points and curves in complex nonsingular projective surfaces these classes recover the virtual classes of seiberg witten theory as well as the reduced stable theory and play a crucial role in the reduced donaldson thomas theory of local surface threefolds that we study in 15 we show that certain integrals against the virtual fundamental classes of punctual nested hilbert schemes are expressed as integrals over the products of the hilbert scheme of points we are able to find explicit formulas for some of these integrals by relating them to carlsson okounkov s vertex operator formulas

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

chern classes of tautological sheaves on hilbert schemes of points on surfaces we give an algorithmic description of the action of the chern classes of tautological bundles on the cohomology of hilbert schemes of points on a smooth surface within the framework of nakajima s oscillator algebra this leads to an identification of the cohomology ring of hilb n a2 with a ring of explicitly given differential operators on a fock space we end with the computation of the top segre classes of tautological bundles associated to line bundles on hilb n up to n 7 extending computations of severi lebarz tikhomirov and troshina and give a conjecture for the generating series

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

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

direct state space solution to the discrete time h sub infinity one block problem a general state space representation is used to allow a complete formulation of the h sub infinity optimization problem without any invertibility condition on the system matrix unlike existing solutions a straightforward approach is used to solve the one block h sub infinity optimization problem the parameterization of all solutions to the discrete time h sub infinity suboptimal one block problem is first given in transfer function form in terms of a set of functions in h sub infinity that satisfy a norm bound the parameterization of all solutions is also given as a linear fractional representation

on the character of convergence to brownian local time ii in this paper we consider the sequences of stochastic processes which converge weakly asn to brownian local time these processes are generated by a recurrent random walk with finite variance the main result is the following it is possible to redefine a random walk in such a way that for a wide class of processes the normalized differences between them and brownian local time converge in distribution to some stochastic process we also prove that such differences with probability one have the logarithmic upper bound it is so called strong invariance principles for local times

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