the solution methods for the largest eigenvalue singular value of nonnegative tensors and convergence analysis abstract in this paper we study two solution methods for finding the largest eigenvalue singular value of general square rectangular nonnegative tensors for a positive tensor one can find the largest eigenvalue singular value based on the properties of the positive tensor and the power type method while for a general nonnegative tensor we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power type method for finding the largest eigenvalue singular value of a positive tensor with an inexact strategy we prove the convergence of the method for the general nonnegative tensor under a certain assumption the computing complexity of the method is established motivated by the interior point method for the convex optimization we put forward a one step inner iteration power type method whose convergence is also established under certain assumption additionally by using embedding technique we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power type method for this problem finally numerical examples of our algorithms are reported which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising

local search for the probabilistic traveling salesman problem correction to the 2 p opt and 1 shift algorithms abstract the probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space where each customer requires a visit only with some known probability a solution to this problem is an a priori tour which visits all customers and the objective is to minimize the expected length of the a priori tour over all customer subsets assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour this problem belongs to the class of stochastic vehicle routing problems a class which has received increasing attention in recent years and which is of major importance in real world applications several heuristics have been proposed and tested for the probabilistic traveling salesman problem many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem in particular two local search algorithms 2 p opt and 1 shift were introduced by bertsimas in a previous report we have shown that the expressions for the cost evaluation of 2 p opt and 1 shift moves as proposed by bertsimas are not correct in this paper we derive the correct versions of these expressions and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

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

diversification strategies in local search for a nonbifurcated network loading problem abstract this paper examines a variant of the network loading problem a network design problem found in the telecommunications industry in this problem facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices the objective is to minimize the total installation costs in this work the nonbifurcated version of the problem is considered where the demand at any given vertex must be satisfied through a single path the proposed heuristics alternate between a construction phase and a local search phase each new construction phase except the first one is part of a diversification strategy aimed at providing a new starting point for the following local search phase different diversification strategies are tested and compared on large scale instances with up to 500 vertices

further results on the probabilistic traveling salesman problem abstract in 1985 jaillet introduced the probabilistic traveling salesman problem ptsp a variant of the classical tsp in which only a subset of the nodes may be present in any given instance of the problem the goal is to find an a priori tour of minimal expected length with the strategy of visiting the present nodes in a particular instance in the same order as they appear in the a priori tour in this paper we reexamine the ptsp using a variety of theoretical and computational approaches we sharpen the best known bounds for the ptsp derive several asymptotic relations and compare from various veiwpoints the ptsp with the re optimization strategy i e finding an optimal tour in every problem instance when a euclidean metric is used and the nodes are uniformly distributed in the unit square a heuristic for the ptsp is shown to be very close to the re optimization strategy we examine some ptsp heuristics with provable worst case performance and address the question of finding constant guarantee heuristics implementations of various heuristics some based on sorting and some on local optimality permit us to discuss the qualitative and quantitative properties of computational problems with up to 5000 nodes

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

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

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

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

newton noda iteration for finding the perron pair of a weakly irreducible nonnegative tensor we present a newton noda iteration nni for computing the perron pair of a weakly irreducible nonnegative mth order tensor mathscr a a by combining the idea of newton s method with the idea of the noda iteration the method requires the selection of a positive parameter theta k k in the kth iteration and produces a scalar sequence approximating the spectral radius of mathscr a a and a positive vector sequence approximating the perron vector we propose a halving procedure to determine the parameters theta k k starting with theta k 1 k 1 for each k such that the scalar sequence is monotonically decreasing convergence of this sequence to the spectral radius of mathscr a a and convergence of the vector sequence to the perron vector is guaranteed for any initial positive unit vector as long as the sequence theta k k so chosen is bounded below by a positive constant in this case we always have theta k 1 k 1 near convergence and the convergence is quadratic very often the halving procedure will return theta k 1 k 1 i e no halving is actually used for each k if the tensor is semisymmetric m ge 4 m 4 and theta k 1 k 1 then the computational work in the kth iteration of nni is roughly the same as that for one iteration of the ng qi zhou algorithm which is linearly convergent for the smaller class of weakly primitive tensors

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

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

a priori optimization consider a complete graph g v e in which each node is present with probability p we are interested in solving combinatorial optimization problems on subsets of nodes which are present with a certain probability we introduce the idea of a priori optimization as a strategy competitive to the strategy of reoptimization under which the combinatorial optimization problem is solved optimally for every instance we consider four problems the traveling salesman problem tsp the minimum spanning tree vehicle routing and traveling salesman facility location we discuss the applicability of a priori optimization strategies in several areas and show that if the nodes are randomly distributed in the plane the a priori and reoptimization strategies are very close in terms of performance we characterize the complexity of a priori optimization and address the question of approximating the optimal a priori solutions with polynomial time heuristics with provable worst case guarantees finally we use the tsp as an example to find practical solutions based on ideas of local optimality

local convergence analysis of inverse iteration algorithm for computing the h spectral radius of a nonnegative weakly irreducible tensor abstract in this paper we present an inverse iteration algorithm to find the h spectral radius and the associated positive eigenvector of a nonnegative weakly irreducible tensor which always preserve the positivity of approximate eigenvectors the local quadratical convergence of the proposed algorithm is established based on a basic result of the newton s method for solving nonlinear equations some numerical examples illustrate the efficiency of the proposed algorithm

extension of the 2 p opt and 1 shift algorithms to the heterogeneous probabilistic traveling salesman problem abstract the probabilistic traveling salesman problem is a well known problem that is quite challenging to solve it involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability there are several proposed algorithms for the homogeneous version of the problem where all customers have identical probability of being realized from the literature the most successful approaches involve local search procedures with the most famous being the 2 p opt and 1 shift procedures proposed by bertsimas d j bertsimas l howell further results on the probabilistic traveling salesman problem european journal of operational research 65 1 1993 68 95 recently however evidence has emerged that indicates the equations offered for these procedures are not correct and even when corrected the translation to the heterogeneous version of the problem is not simple in this paper we extend the analysis and correction to the heterogeneous case we derive new expressions for computing the cost of 2 p opt and 1 shift local search moves and we show that the neighborhood of a solution may be explored in o n2 time the same as for the homogeneous case instead of o n3 as first reported in the literature

different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is a topic of theoretical and practical importance in the study of stochastic network problems it provides researchers with a modeling framework for exploring the stochastic effects in routing problems this paper proposed three initial solution generators nn1 nn2 ran under a genetic algorithm ga framework for solving the ptsp a set of numerical experiments based on heterogeneous and homogeneous ptsp instances were conducted to test the effectiveness and efficiency of the proposed algorithms the results from the heterogeneous ptsp show that the average e t values obtained by the three generators under a ga framework are similar to those obtained by the previous best but with an average computation time saving of 50 2 as for the homogeneous ptsp instances nn1 is a relatively better generator among the three examined while ran consistently performs worse than the other two generators in terms of average e t values additionally as compared to previously reported studies no one single algorithm consistently outperformed the others across all homogeneous ptsp instances in terms of the best e t values the fact that no one initial solution generator consistently performs best in terms of the e t value obtained across all instances in heterogeneous cases and that the performance of each examined algorithm is dependent on the number of nodes n and probability p for homogeneous cases suggest the possibility of context dependent phenomenon finally to obtain valid results researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting ptsp experiments

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

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

diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem this paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem ptsp different combinations of three commonly used local search methods in the ptsp i e 1 shift 2 opt and 3 opt were used to investigate its effects a set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances the numerical results and the permutation test showed that the diversified local search strategy especially by combining 1 shift and 2 opt algorithms can most effectively solve the homogeneous and heterogeneous ptsp in most of the tested instances in comparison with the single local search strategy under scatter search framework

a priori solution of a travelling salesman problem in which a random subset of the customers are visited consider a problem of routing through a set of n points on any given instance of the problem only a subset consisting of k out of n points 0 k n has to be visited with the number k random with known probability distribution we wish to find a priori a tour through all n points on any given instance the k points present will then be visited in the same order as they appear in the a priori tour the problem of finding such a tour of minimum length in the expected value sense is defined as a probabilistic traveling salesman problem ptsp what distinguishes one ptsp from another is the probability distribution or more generally the probability law that specifies the number k and the identity of the points that need to be visited on any given instance of the problem after motivating the problem by applications we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions we then provide in a unified way

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

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

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

a priori optimization of the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is defined on a graph g v e where v is the vertex set and e is the edge set each vertex vi has a probability pi of being present with each edge vi vj is associated a distance or cost cij in a first stage an a priori hamiltonian tour on g is designed the list of present vertices is then revealed in a second stage the a priori tour is followed by skipping the absent vertices the ptsp consists of determining a first stage solution that minimizes the expected cost of the second stage tour the problem is formulated as an integer linear stochastic program and solved by means of a branch and cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function problems involving up to 50 vertices are solved to optimality

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

incorporating scatter search and threshold accepting in finding maximum likelihood estimates for the multinomial probit model this paper presents a procedure that incorporates scatter search and threshold accepting to find the maximum likelihood estimates for the multinomial probit mnp model scatter search widely used in optimization related studies is a type of evolutionary algorithm that uses a small set of solutions as the selection pool for mating and generating new solutions to search for a globally optimal solution threshold accepting is applied to the scatter search to improve computational efficiency while maintaining the same level of solution quality a set of numerical experiments based on synthetic data sets with known model specifications and error structures were conducted to test the effectiveness and efficiency of the proposed framework the results indicated that the proposed procedure enhanced performance in terms of likelihood function value and computational efficiency for mnp model estimation as compared to the original scatter search framework

an always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor in this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor we prove this method converges for any irreducible nonnegative tensor we also apply this method to study the positive definiteness of a multivariate form

characterization of the probabilistic traveling salesman problem we show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem the probabilistic traveling salesman must decide on an a priori order in which to visit n cities randomly distributed over a unit square before learning that some cities can be omitted we find the optimized average length of the pruned tour follows e l over bar pruned rootnp 0 872 0 105p f np where p is the probability of a city needing to be visited and f np 1 as np infinity the average length of the a priori tour before omitting any cities is found to follow e l a priori rootn pbeta p where beta p 1 1 25 0 82 ln p is measured for 0 05less than or equal topless than or equal to0 6 scaling arguments and indirect measurements suggest that beta p tends towards a constant for p 0 03 our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths exploiting the sampling error to provide the analog of thermal fluctuations in simulated thermal annealing the method has general application to the optimization of functions whose cost to evaluate rises with the precision required

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

heuristics based on spacefilling curves for combinatorial problems in euclidean space we describe a family of heuristics to solve combinatorial problems such as routing and partitioning these heuristics exploit geometry but ignore specific distance measures consequently they are simple and fast but nonetheless fairly accurate and so seem well suited to operational problems where time or computing resources are limited we survey promising new application areas and show how procedures may be customized to reflect the structure of particular applications

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