toeplitz and hankel operators on bergman spaces in this paper we consider toeplitz and hankel operators on the bergman spaces of the unit ball and the polydisk in c n whose symbols are bounded measurable functions we giv e necessary and sufficient conditions on the symbols for these operators to be compact we study the fredholm theory of topelitz operators for which the corresponding hankel operator is compact for these toeplitz operators the essential spectrum is computed and shown to be connected we also consider symbols that extend to continuous functions on the maximal ideal space of h for these symbols we describe when the toeplitz or hankel operators are compact

trace class criteria for toeplitz and composition operators on small bergman spaces abstract we characterize the schatten class toeplitz operators induced by a positive borel measure on the unit disc and the reproducing kernel of the bergman space a 2 where is a radial weight satisfying the doubling property r 1 s d s c 1 r 2 1 s d s by using this we describe the schatten class composition operators we also discuss basic properties of composition operators acting from a p to a v q

the bergman space the bloch space and commutators of multiplication operators soit d le disque unite ouvert dans le plan complexe c soit h d l ensemble des fonctions analytiques bornees sur d on caracterise les fonctions feh d telles que t f t f tft f est un operateur compact ou t f est l operateur multiplication t f l a 2 l a 2 defini par t f g fg

limit operators compactness and essential spectra on bounded symmetric domains abstract this paper is a follow up to a recent article about the essential spectrum of toeplitz operators acting on the bergman space over the unit ball as mentioned in the said article some of the arguments can be carried over to the case of bounded symmetric domains and some cannot the aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains in particular we show that a toeplitz operator on the bergman space a p is fredholm if and only if all of its limit operators are invertible even more generally we show that this is in fact true for all band dominated operators an algebra that contains the toeplitz algebra moreover we characterize compactness and explain how the berezin transform comes into play in particular we show that a bounded linear operator is compact if and only if it is band dominated and its berezin transform vanishes at the boundary for p 2 band dominated can be replaced by contained in the toeplitz algebra

functorial relationships between qh g b and qh g p we show a canonical injective morphism from the quantum cohomology ring qh g p to the associated graded algebra of qh g b which is with respect to a nice filtration on qh g b introduced by leung and the author this tells us the vanishing of a lot of genus zero three pointed gromov witten invariants of flag varieties g p

on algebras generated by toeplitz operators and their representations abstract we study banach and c algebras generated by toeplitz operators acting on weighted bergman spaces a 2 b 2 over the complex unit ball b 2 c 2 our key point is an orthogonal decomposition of a 2 b 2 into a countable sum of infinite dimensional spaces each one of which can be identified with a differently weighted bergman space a 2 d over the complex unit disk d moreover all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a toeplitz operator on a 2 d the symbols of the generating toeplitz operators are chosen to be suitable extensions to b 2 of families s of bounded functions on d symbol classes s that generate important classical commutative and non commutative toeplitz algebras in l a 2 d are of particular interest in this paper we discuss various examples in the case of s c d and s c d l 0 1 we characterize all irreducible representations of the resulting toeplitz operator c algebras their calkin algebras are described and index formulas are provided

multiplicative rule of schubert classes let g be a compact connected lie group and let h be the centralizer of a one parameter subgroup in g combining the ideas of bott samelson resolutions of schubert varieties and the enumerative formula on a twisted product of 2 spheres obtained in du2 we obtain an explicit formula for multiplying schubert classes in the flag manifold g diagup h

on the conjecture o of ggi for g p abstract in this paper we show that general homogeneous manifolds g p satisfy conjecture o of galkin golyshev and iritani which underlies gamma conjectures i and ii of them our main tools are the quantum chevalley formula for g p and a theory on nonnegative matrices including perron frobenius theorem

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

a mirror symmetric construction of qht g p q abstract let g be a simple simply connected complex algebraic group we give a lie theoretic construction of a conjectural mirror family associated to a general flag variety g p and show that it recovers the peterson variety presentation for the t equivariant quantum cohomology rings q h t g p q with quantum parameters inverted for sl n b we relate our construction to the mirror family defined by givental and its t equivariant analogue due to joe and kim

toeplitz operators and quantum mechanics abstract toeplitz operators on the segal bargmann spaces of gaussian measure square integrable entire functions on complex n space c n are studied the c algebra generated by the weyl form of the canonical commutation relations consists precisely of the uniform limits of almost periodic toeplitz operators the question of which toeplitz operators admit a symbol calculus modulo the compact operators is raised and sufficient conditions are given for such a calculus these conditions involve a notion of slow oscillation at infinity

lines in g p our goal is reached as follows fix a maximal torus t in g and a borel subgroup b t so that we can consider the corresponding root lattice q weight lattice set of simple roots 1 n and set of fundamental weights 1 n we then choose p b so that p corresponds to a subset s of or equivalently or also equivalently of the dynkin diagram of g recall that the picard group of g p can be identified with a sublattice of taken an ample l pic g p and setting

hankel operators and toeplitz operators on the bergman space abstract we consider in this paper the question of when the semi commutator t fg t f t g on the bergman space with bounded harmonic symbols is compact several conditions equivalent to compactness of t fg t f t g are given as a consequence we prove a conjecture of axler that for bounded analytic functions f and g on the unit disk t fg t g t f is compact iff either f or g is constant on each gleason part p m except d

toeplltz operators and the berezin transform on h2 abstract for t a bounded linear operator on the hardy space h2 its berezin transform is the function t on the unit disc defined by t tk k where k z 1 2 1 2 1 z is the normalized reproducing szego kernel when t t gf is a toeplitz operator it turns out that the nontangential boundary values of t gf coincide with gf almost everywhere this allows us to construct extensions and or generalizations of the classical symbol calculus for toeplitz operators to larger operator algebras briefly discussed is also the relation between our results and the recent results of berger coburn and zhu for toeplitz and hankel operators on the bergman space

parametric integer linear programming a synthesis of branch and bound with cutting planes abstract a branch and bound algorithm is designed to solve the general integer linear programming problem with parametric right hand sides the right hand sides have the form b d where b and d are comformable vectors d consists of nonnegative constants and varies from zero to one the method consists of first determining all possible right hand side integer constants and appending this set of integer constants to the initial tableau to form an expanded problem with a finite number of family members the implicit enumeration method gives a lower bound on the integer solutions the branch and bound method is used with fathoming tests which allow one family member possibly to fathom other family members a cutting plane option applies a finite number of cuts to each node before branching in addition the cutting plane method is invoked whenever some members are feasible at a node and others are infeasible the branching and cutting process is repeated until the entire family of problem has been solved

parametric uncapacitated facility location we consider the parametric uncapacitated facility location problem uflp where the demand of each customer is allowed to vary over a certain interval despite the value of parametric analysis no such efforts are found for the uflp that mainly seems to be due to the fact that parametric analysis usually requires the computation of a number of uflp s with heavy computational burden in this paper we will show that the special structure of the uflp provides a nice property for its parametric version based on which an efficient branch and bound procedure for the parametric analysis is developed

a shadow price in integer programming for management decision abstract the main purpose of this paper is to propose a new concept of shadow prices in integer programming with rich economic interpretation which is useful in management decision and to examine closely some related properties as well as computational methods this shadow price is based not on marginal analysis but on average analysis the existence and the uniqueness that have not been achieved until now from other studies on dual prices of integer programming are guaranteed for this shadow price a version of complementary slackness theorem in integer programming has been achieved for this shadow price an easy method of finding upper and lower bounds of the shadow price as well as a rigorous algorithm of obtaining its exact value is presented

function theory on cartan domains and the berezin toeplitz symbol calculus cn with dv z normalized lebesgue measure on 9 we let l l2 q dv be the usual space of lebesgue square integrable complex valued functions on 9 by h2 h2 q dv we denote the bergman subspace of l2 consisting of holomorphic functions the orthogonal projection operator from l2 onto h2 is denoted by p forf in l q the space of essentially bounded measurable functions on 9 we will consider the multiplication operator

a contraction algorithm for the multiparametric integer linear programming problem we designed and implemented an algorithm to solve the continuos right hand side multiparametric integer linear programming ilp problem that is to solve a family of ilp problems in which the problems are related by having identical objective and matrix coefficients our algorithm works by choosing an appropiate finite sequence of nonparametric mixed integer linear programming milp problems in order to obtain a complete multiparametrical analysis the algorithm may be implemented by using any software capable of solving milp problems

an algorithm to perform a complete right hand side parametrical analysis for a 0 1 integer linear programming problem abstract we designed and implemented an algorithm to solve the continuous right hand side parametric 0 1 integer linear programming ilp problem that is to solve a family of 0 1 ilp problems in which the problems are related by having identical objective and matrix coefficients our algorithm works by choosing an appropiate finite sequence of non parametric 0 1 mixed integer linear programming milp problems in order to obtain a complete parametrical analysis the algorithm may be implemented by using any software capable of solving milp problems

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

norm estimates for iterated commutators on the bergman spaces of the unit ball introduction we study the iterated commutators of the bergman projection and the multiplication by measurable functions called the symbols unlike the commutators the iterated commutators are no longer depending linearly on their symbols therefore most of the techniques used successfully in studying the first order commutators fail to work effectively for the nth order iterated commutators these operators are also very different from their analogs on l2 rm which has been studied by l peng t qian in pq using the s jason and j peeter s technique established in jp and r rochberg s semmes in rs where deeper tools of harmonic analysis in r are employed in this paper by using the bergman geometry properties of holomorphic functions and related analyses we characterize the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded compact or in the schatten p class some lp estimates are also included in our discussion let 1m be the open unit ball of cm and dv z be the volume measure on blm

compactness of hankel operators and analytic discs in the boundary of pseudoconvex domains abstract using several complex variables techniques we investigate the interplay between the geometry of the boundary and compactness of hankel operators let be a function smooth up to the boundary on a smooth bounded pseudoconvex domain c n we show that if is convex or the levi form of the boundary of is of rank at least n 2 then compactness of the hankel operator h implies that is holomorphic along analytic discs in the boundary furthermore when is convex in c 2 we show that the condition on is necessary and sufficient for compactness of h

gromov witten invariants for and pontryagin product for ω we give an explicit formula for t equivariant 3 pointed genus zero gromov witten invariants for g b we derive it by finding an explicit formula for the pontryagin product on the equivariant homology of the based loop group k

parametric objective integer programming using knapsack facets and gomory cutting planes abstract this paper presents a method for finding the optimal solutions to an integer program ip as the coefficients of the objective function are varied continuously over a wide range the method requires identifying an optimal solution to the ip at a number of point values of the objective coefficients these point value ps are solved using knapsack facets or gomory cutting plane if there are n different optimal solutions in the range of coefficient variation then optimal solutions must be identified at 2 n 1 different coefficient values by using a facet or cutting plane algorithm and employing post optimal analysis and re optimization the computational effort required may be much less than would be needed to solve 2 n 1 ips independently

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

quantum pieri rules for tautological subbundles abstract we give quantum pieri rules for quantum cohomology of grassmannians of classical types expressing the quantum product of chern classes of the tautological subbundles with general cohomology classes we derive them by showing the relevant genus zero three pointed gromov witten invariants coincide with certain classical intersection numbers

totally positive toeplitz matrices and quantum cohomology of partial flag varieties we show that the set of totally positive unipotent lower triangular toeplitz matrices in gln form a real semi algebraic cell of dimension n 1 furthermore we prove a natural cell decomposition for its closure the proof uses properties of the quantum cohomology rings of the partial flag varieties of gln c relying in particular on the positivity of the structure constants which are enumerative gromov witten invariants we also give a characterization of total positivity for toeplitz matrices in terms of the quantum schubert classes this work builds on some results of dale peterson s which we explain with proofs in the type a case

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

gamma classes and quantum cohomology of fano manifolds gamma conjectures we propose gamma conjectures for fano manifolds which can be thought of as a square root of the index theorem studying the exponential asymptotics of solutions to the quantum differential equation we associate a principal asymptotic class a f to a fano manifold f we say that f satisfies gamma conjecture i if a f equals the gamma class widehat gamma f when the quantum cohomology of f is semisimple we say that f satisfies gamma conjecture ii if the columns of the central connection matrix of the quantum cohomology are formed by widehat gamma f operatorname ch e i for an exceptional collection e i in the derived category of coherent sheaves mathcal d b mathrm coh f gamma conjecture ii refines a part of a conjecture by dubrovin we prove gamma conjectures for projective spaces and grassmannians

parameterisation algorithms for the integer linear programs in binary variables abstract in this paper we consider the integer programmiing problem minimize z x c x subject to ax b x binary roodman appended the objective function z x to the body of the constraints and presented a modified version of the balas additive algorithm by which each fathomed partial solution is attributed to the constraint which caused the fathoming exploiting this information he conducted a ranging analysis i e calculating bounds on the values of the parameters which leave the original optimal solution unchanged and b parameter change analysis i e determining new optimal solutions if any for revised values of the parameters outside the ranging bounds we extend roodman s results and construct parametric functions of the following form let be any parameter of c or b or a and replace by then holding every other parameter of the program fixed and varying in the set of real numbers we construct the parametric function z which matches non overlapping intervals of with optimal solutions this replaces by exact bounds in the linear programming sense the bounds underestimated by roodman s ranging analysis it also determines optimal solutions for any values of rather than for a revised set of values finally some results of computational experience are presented

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

hankel operators induced by radial bekolle bonami weights on bergman spaces we study big hankel operators h f nu a p omega rightarrow l q nu generated by radial bekolle bonami weights nu when 1 p le q infty here the radial weight omega is assumed to satisfy a two sided doubling condition and a p omega denotes the corresponding weighted bergman space a characterization for simultaneous boundedness of h f nu and h overline f nu is provided in terms of a general weighted mean oscillation compared to the case of standard weights that was recently obtained by pau et al indiana univ math j 65 5 1639 1673 2016 the respective spaces depend on the weights omega and nu in an essentially stronger sense this makes our analysis deviate from the blueprint of this more classical setting as a consequence of our main result we also study the case of anti analytic symbols

toeplitz operators on the segal bargmann space in this paper we give a complete characterization of those functions on 2n dimensional euclidean space for which the berezin toeplitz quantizations admit a symbol calculus modulo the compact operators the functions in question are characterized by a condition of small oscillation at infinity

quantum pieri rules for isotropic grassmannians we study the three point genus zero gromov witten invariants on the grassmannians which parametrize non maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew symmetric form we establish pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties which give a combinatorial formula for the product of any schubert class with certain special schubert classes we also give presentations of these rings with integer coefficients in terms of special schubert class generators and relations