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 admissibility of observation for perturbed c0 semigroups on banach spaces abstract this paper concerns linear infinite dimensional systems with unbounded observation operators we study the invariance of admissibility for observation operators under some unbounded perturbations of the generator we give also some relations between the extensions of such observation operators with respect to the original generator and the perturbed generator our results are applied to functional differential linear systems

quasi newton methods for implicit black box fsi coupling abstract in this paper we introduce a new multi vector update quasi newton mvqn method for implicit coupling of partitioned transient fsi solvers the new quasi newton method facilitates the use of black box field solvers and under certain circumstances can be demonstrated to provide newton like convergence behaviour for strongly coupled fsi benchmark problems we demonstrate the superior convergence behaviour and robust nature of the mvqn method compared to other well known quasi newton coupling schemes including the least squares reduced order modelling ibqn ls scheme the classical rank 1 update broyden s method and fixed point iterations with dynamic relaxation the quasi newton methods are analysed on a suite of strongly coupled fsi problems including but not limited to internal incompressible flow through a flexible tube where the solid density is an order of magnitude lower than the fluid density

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

stabilization of beams with nonlinear feedback this paper studies nonlinear feedback stabilization of euler bernoulli beams subject to various pointwise and concentrated actuators strong uniform and nonuniform stabilization are obtained with explicit decay estimates in appropriate spaces the results are obtained through the study of an abstract second order distributed system which encompasses the models under investigation

on invariance of p admissibility of control and observation operators to q type of perturbations of generator of c0 semigroup abstract in this paper it is proved in general setting that p admissibilities of control operators and observation operators are invariant to any q type of perturbations of generator of c 0 semigroups on banach space moreover some relations between the extensions of observation operators with respect to the original generator and the perturbed generator are also characterized so that the output can be expressed in the mild sense as an application the admissibility as well as the mild expressibility of output of a class of observation systems with time delay in state is deduced

admissibility of control operators for solution families to volterra integral equations we reduce the admissibility of finite dimensional control operators for an evolution system satisfying certain mild regularity conditions to that for a semigroup control system for which there are plenty of results available in the literature one such necessary and sufficient condition is the carleson measure criterion we prove this condition explicitly for evolution systems without the regularity assumption mentioned above we also look at some examples which show that our respective hypotheses are necessary

on laplace carleson embedding theorems this paper gives embedding theorems for a very general class of weighted bergman spaces the results include a number of classical carleson embedding theorems as special cases the little hankel operators on these bergman spaces are also considered next a study is made of carleson embeddings in the right half plane induced by taking the laplace transform of functions defined on the positive half line these embeddings have applications in control theory particular attention is given to the case of a sectorial measure or a measure supported on a strip and complete necessary and sufficient conditions for a bounded embedding are given in many cases c 2012 elsevier inc all rights reserved

interpolation by vector valued analytic functions with applications to controllability in this paper norm estimates are obtained for the problem of minimal norm tangential interpolation by vector valued analytic functions expressed in terms of the carleson constants of related scalar measures applications are given to the controllability properties of linear semigroup systems with a riesz basis of eigenvectors

carleson measures for hilbert spaces of analytic functions on the complex half plane the notion of a carleson measure was introduced by lennart carleson in his proof of the corona theorem for h d in this paper we will define it for certain type of reproducing kernel hilbert spaces of analytic functions of the complex half plane c which will include hardy bergman and dirichlet spaces we will obtain several necessary or sufficient conditions for a positive borel measure to be carleson by preforming tests on reproducing kernels weighted bergman kernels and studying the tree model obtained from a decomposition of the complex half plane the dirichlet space will be investigated in detail as a special case finally we will present a control theory application of carleson measures in determining admissibility of controls in well posed linear evolution equations

vertex operators and quasimodularity of chern numbers on the hilbert scheme abstract in carlsson and okounkov preprint 7 okounkov and the author defined a family of vertex operators on the equivariant cohomology groups of the hilbert scheme of points on a smooth quasi projective surface as a characteristic class of certain canonical bundles on hilb s hilb s we then proved a bosonization formula in terms of nakajima s heisenberg operators nakajima 1997 23 in this paper we apply this operator in the special case when s c 2 with a particular action of a torus and prove that the generating functions of equivariant chern numbers on hilb n over n are quasimodular forms in the generating variable this property determines the answer up to a finite dimensional vector space of functions of the generating variable q these generating functions can be thought of as the analogous correlation functions to nekrasov s partition function in rank 1 we present a different proof of the bosonization formula which is based on the proof of a more general formula in k theory given in an upcoming paper by nekrasov okounkov and the author in preparation 6 but specialized to the surface of interest by altering a certain bundle that appears in carlsson et al in preparation 6 and specializing the surface the proof actually reduces to a much simpler self contained application of the infinite wedge representation this picture is consistent with both the original introduction of this operator in nekrasov and okounkov 2006 26 and with haiman s character theory of the bridgeland king and reid isomorphism haiman 2003 11

new blow up rates for fast controls of schrodinger and heat equations abstract we consider the null controllability problem for the schrodinger and heat equations with boundary control we concentrate on short time or fast controls we improve recent estimates see l miller geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 202 226 l miller how violent are fast controls for schrodinger and plate vibrations arch ration mech anal 172 2004 429 456 l miller controllability cost of conservative systems resolvent condition and transmutation j funct anal 218 2005 425 444 l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 on the norm of the operator associating to any initial state the minimal norm control driving the system to zero our main results concern the schrodinger and heat equations in one space dimension they yield new estimates concerning window problems for series of exponentials as described in t i seidman the coefficient map for certain exponential sums nederl akad wetensch indag math 48 1986 463 478 and in t i seidman s a avdonin s a ivanov the window problem for series of complex exponentials j fourier anal appl 6 2000 233 254 these results are used following l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 to deal with the case of several space dimensions

admissible input elements for systems in hilbert space and a carleson measure criterion we study the control system dot x ax bu quad x in x quad u text scalar where a generates a semigroup on the hilbert space x but in general the control input element b notin x many boundary value control systems point control force situations etc can be studied in this context we define and analyze admissible input elements b and develop sufficient conditions for b to be admissible in terms of the carleson measure theorem of h p theory

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

a framework of rogers ramanujan identities and their arithmetic properties the two rogers ramanujan q series where 0 1 play many roles in mathematics and physics by the rogers ramanujan identities they are essentially modular functions their quotient the rogers ramanujan continued fraction has the special property that its singular values are algebraic integral units we find a framework which extends the rogers ramanujan identities to doubly infinite families of q series identities if a 1 2 and m n 1 then we have infinite product modular function where the p x x q are hall littlewood polynomials these q series are specialized characters of affine kac moody algebras generalizing the rogers ramanujan continued fraction we prove in the case of a that the relevant q series quotients are integral units

parametrizations of ideals in k x y and k x y z abstract we parametrize the affine space of artinian affine ideals of k x y which have a given initial ideal with respect to the degree reverse lexicographic term order the fact that the term order is degree compatible allows us to extend the parametrization to the projective case namely zero dimensional subschemes of p 2 with some extra assumption and to determine the betti strata of these grobner cells this allows us to prove a formula due to a iarrobino for the codimension of the betti strata of codimension two punctual schemes in p 2

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

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

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

the quasi newton least squares method a new and fast secant method analyzed for linear systems we present a new quasi newton method that can solve systems of equations of which no information is known explicitly and which requires no special structure of the system matrix like positive definiteness or sparseness the method builds an approximate jacobian based on input output combinations of a black box system uses a rank one update of this jacobian after each iteration and satisfies the secant equation while it has originally been developed for nonlinear equations we analyze its properties and performance when applied to linear systems analytically the method is shown to be convergent in n 1 iterations n being the number of unknowns irrespective of the nature of the system matrix the performance of this method is greatly superior to other quasi newton methods and comparable with gmres when tested on a number of standardized test cases

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

a korteweg de vries type of fifth order equations on a finite domain with point dissipation abstract the paper discusses initial value problem of a korteweg de vries type of fifth order equation w t w x x x w x x x x x j 1 n a j w j w x 0 w x 0 w 0 x posed on a periodic domain x 0 2 with periodic boundary conditions w i x 0 t w i x 2 t i 0 2 3 4 and an l 2 stabilizing feedback control law w x 2 t w x 0 t 1 w x x x 0 t where n is a fixed positive integer a j j 1 2 n are real constants and 1 it is shown that for w 0 x h 1 0 2 with the boundary conditions described above the problem is locally well posed for w c 0 t h 1 0 2 with a conserved volume of w w 0 2 w x t d x moreover the solution with small initial condition exists globally and approaches to w 0 x 2 as t

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

admissibility and controllability of diagonal volterra equations with scalar inputs this article studies volterra evolution equations from the point of view of control theory in the case that the generator of the underlying semigroup has a riesz basis of eigenvectors conditions for admissibility of the system s control operator are given in terms of the carleson embedding properties of certain discrete measures moreover exact and null controllability are expressed in terms of a new interpolation question for analytic functions providing a generalization of results known to hold for the standard cauchy problem the results are illustrated by examples involving heat conduction with memory

semigroup generation and stabilization by a r bounded feedback perturbations abstract let a be a generator of a strongly continuous semigroup of operators and assume that c and h are operators such that a ch generates a strongly continuous semigroup sh t on x let 0 be a real number in the resolvent set of a and let 1 1 then there are some fairly unrestrictive conditions under which a 0 a ch 0 a also generates a strongly continuous semigroup sk t on x which has the same exponential growth rate as sh t given an input operator b we can use this to identify a class of feedback perturbations k such that a bk generates a strongly continuous semigroup we can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system for example we show that if the control on a cantilever beam in the state space h02 0 1 l2 0 1 is a moment force on the free end then we cannot stabilize the beam with an a 1 2 bounded feedback but we can find an a e 1 4 bounded feedback for any e 0 which does stabilize the beam

a general necessary condition for exact observability suppose a generates an exponentially stable strongly continuous semigroup on the hilbert space x y is another hilbert space and c d a rightarrow y is an admissible observation operator for this semigroup this means that to any initial state in x we can associate an output function in l 2 0 infty y this paper gives a necessary condition for the exact observability of the system defined by a and c this condition called bf e is related to the hautus lemma from finite dimensional systems theory it is an estimate in terms of the operators a and c alone in particular it makes no reference to the semigroup this paper shows that bf e implies approximate observability and if a is bounded it implies exact observability this paper conjectures that bf e is in fact equivalent to exact observability the paper also shows that for diagonal semigroups bf e takes on a very simple form and applies the results to sequences of complex exponential functions

bounds on functions biorthogonal to sets of complex exponentials control of damped elastic systems abstract in this article we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials exp k t where the k belong to a sector of the positive real axis the bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties the dependence of the bounds upon the growth and separation parameters is carefully examined the main theorem then leads to some results regarding the solvability of moment problems of the form 0 t f t exp k t dt c k which arise in control problems associated with damped or dissipative systems as an application we establish an exact null controllability result for boundary control of a rectangular structurally damped plate

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

asymptotic behavior of coupled euler bernoulli beams with pointwise dissipation abstract examples of coupled euler bernoulli beams with pointwise dissipation are considered exponential decay of energy is shown independently of the support of the pointwise dissipation

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

weak admissibility does not imply admissibility for analytic semigroups two conjectures on admissible control operators by george weiss are disproved in this paper one conjecture says that an operator b defined on an infinite dimensional hilbert space u is an admissible control operator if for every element u in u the vector bu defines an admissible control operator the other conjecture says that b is an admissible control operator if a certain resolvent estimate is satisfied the examples given in this paper show that even for analytic semigroups the conjectures do not hold in the last section we construct a semigroup example showing that the first estimate in the hille yosida theorem is not sufficient to conclude boundedness of the semigroup

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

brief paper identification of variable spacial coefficients for a beam equation from boundary measurements we consider a system described by the euler bernoulli beam equation with one end clamped and with torque input at the other end the output function are the displacement and the angle velocity at the non clamped end of the beam we study the identification of the spatially variable coefficients in the beam equation from input output data we show that both the density and the flexural rigidity of the beam which are assumed to be of class c 4 can be uniquely determined if the input and output functions are known for all positive times

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

the operator carleson measure criterion for admissibility of control operators for diagonal semigroups on 1 2 abstract suppose a control system is described by a diagonal semigroup on the state space l 2 and by an unbounded control operator b defined on the input space l 2 we formulate a condition called the operator carleson measure criterion and show that this condition is necessary for the admissibility of b if the semigroup is analytic or invertible then the condition is sufficient as well our results extend the ones of ho russell and weiss concerning the case where the input space is one dimensional

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

boundary control of a linear thermoelastic beam abstract we show in this paper that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam the optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted as in the euler bernoulli beam theory

observability of compactly perturbed systems abstract we propose a general and constructive method for the study of observability of weakly coupled or more generally compactly perturbed linear distributed systems this approach allows us to avoid the usual indirect compactness uniqueness arguments in many cases our method is based on two observations first in many cases there exists a riesz basis formed by ordinary and generalized eigenfunctions of the underlying operator of the evolutionary equation second in the usual estimates some constants depend on the initial data in a rather particular way this allows us to write the solution as a trigonometric sum with vector coefficients and then to apply a generalization of an estimation method introduced by haraux for the study of scalar nonharmonic series applying this approach we improve and generalize some former theorems of lions haraux and jaffard

p admissible control elements for diagonal semigroups on l r spaces for linear systems described by dot x t a x t bu t where a is a diagonal operator on the state space l r for some 1 leq r infty we develop necessary and sufficient conditions for b to be p admissible this extends results by ho russell and weiss to the case r neq 2

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

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

robust multisecant quasi newton variants for parallel fluid structure simulations and other multiphysics applications multisecant quasi newton methods have been shown to be particularly suited to solve nonlinear fixed point equations that arise from partitioned multiphysics simulations where the exact jacobian is inaccessible in all these methods the underdetermined multisecant equation for the approximate inverse jacobian is enhanced by a norm minimization condition the standard choice is the minimization of the frobenius norm of the approximate inverse jacobian in this setting it is well known that transient fluid structure simulations typically require the use of secant information also from previous time steps to achieve a small enough number of iterations per implicit time step the number of these time steps highly depends on the application the physical parameters the used solvers and the mesh resolution using too few leads to a relatively high number of iterations while using too many leads not only to a computational overhead but also to an increase in the number of iterations as well determining th

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 cost of fast controls for some families of dispersive or parabolic equations in one space dimension in this paper we consider the cost of fast controls for a large class of linear equations of parabolic or dispersive type in one space dimension in small time by extending the work of tenenbaum and tucsnak j differential equations 243 2007 pp 70 100 we are able to give precise upper bounds on the time dependance of the cost of fast controls when the time of control t tends to 0 we also give a lower bound of the cost of fast controls for the same class of equations which proves the optimality of the power of t involved in the cost of the control these general results are then applied to treat notably the case of linear kdv equations and fractional heat or schrodinger equations

zero class admissibility of observation operators abstract an admissible observation operator is zero class admissible if the norm of the output map tends to zero as the time tends to zero sufficient and necessary conditions for zero class admissibility of observation operators are developed and a modified weiss condition is studied it is shown that the modified weiss condition is in general necessary but not sufficient for zero class admissibility for several important classes of c 0 semigroups it is proved that the modified weiss condition is indeed equivalent to zero class admissibility the methods are illustrated by certain pde examples

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

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

robustness and continuity of the spectrum for uncertain distribution parameter systems we analyze the spectrum of a class of abstract partial differential equations with boundary feedback control if we interpret the feedback as a perturbation then we obtain a robustness radius for spectrum placement we obtain continuity of the spectrum using this radius and rouche s theorem we exploit extensively recent developments in the representation theory for abstract linear systems especially regular systems we also analyze ill posed systems by interchanging the roles of input and output we apply these results to a wave equation and an ill posed beam equation

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

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

explicit lower bounds for the cost of fast controls for some 1 d parabolic or dispersive equations and a new lower bound concerning the uniform controllability of the 1 d transport diffusion equation in this paper we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional laplace operator we notably deduce the following striking result in the case of the heat equation controlled on the boundary the miller s conjecture formulated in geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 pp 202 226 is not verified moreover we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one dimensional convection diffusion equation with negative speed controlled on the left boundary proving that the conjecture formulated in j m coron and s guerrero singular optimal control a linear 1 d parabolic hyperbolic example asymptot anal 44 2005 pp 237 257 concerning this problem is also not verified at least for negative speeds the proof is based on complex analysis and more precisely on a representation formula for entire functions of exponential type and is quite related to the moment method of fattorini and russell

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

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

the asymptotic formulas for coefficients and algebraicity of jacobi forms expressed by infinite product abstract we determine asymptotic formulas for the fourier coefficients of jacobi forms expressed by infinite products with jacobi theta functions and the dedekind eta function these are generalizations of results about the growth of the fourier coefficients of jacobi forms given by an inverse of jacobi theta function to derive the asymptotic behavior of the betti numbers of the hilbert scheme of points on an algebraic surface by bringmann manschot and about the asymptotic behavior of the y genera of hilbert schemes of points on k3 surfaces by manschot rolon we also get the algebraicity of the generating functions given by gottsche for the hilbert schemes associated to general algebraic surfaces

characterizing projective spaces on deformations of hilbert schemes of k3 surfaces we seek to characterize homology classes of lagrangian projective spaces embedded in irreducible holomorphic symplectic manifolds up to the action of the monodromy group this paper addresses the case of manifolds deformation equivalent to the hilbert scheme of length 3 subschemes of a k3 surface the class of the projective space in the cohomology ring has prescribed intersection properties which translate into diophantine equations possible homology classes correspond to integral points on an explicit elliptic curve our proof entails showing that the only such point is two torsion 2011 wiley periodicals inc

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

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

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

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

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

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

applications of laplace carleson embeddings to admissibility and controllability it is shown how results on carleson embeddings induced by the laplace transform can be used to derive new and more general results concerning the weighted infinite time admissibility of control and observation operators for linear semigroup systems with q riesz bases of eigenvectors as an example the heat equation is considered next a new carleson embedding result is proved which gives further results on weighted admissibility for analytic semigroups finally controllability by smoother inputs is characterized by means of a new result about weighted interpolation

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