model checking computation tree logic over finite lattices multi valued model checking is an extension of classical model checking used to verify the properties of systems with uncertain information content it is used in systems where the semantic domain of both the models of the systems and the specification is a de morgan algebra or more abstract structure such as semirings a common feature of de morgan algebras and semirings is that they satisfy the distributive law which is crucial for all of the multi valued model checking algorithms developed so far in this paper we study computation tree logic with membership values in a finite lattice which are called multi valued computation tree logic mctl we introduce three semantics for mctl over the multi valued kripke structure path semantics fixpoint semantics and algebraic semantics and prove that they coincide if and only if the underlying lattice is distributive furthermore we provide model checking algorithms with respect to the three semantics our algorithms show that mctl model checking problems with respect to the fixpoint and algebraic semantics can be solved in polynomial time in the size of the state space of the multi valued kripke structure the size of the lattice and the length of the formula while mctl model checking problem with respect to the path semantics is in pspace we also provide a lower bound for the mctl model checking problem with respect to the path semantics

global stability and chaos in a population model with piecewise constant arguments sufficient conditions are obtained for the global stability of the positive equilibrium of dx dt rx t 1 ax t b j 0 c jx t j it is shown that for certain special cases solutions of the equation can have chaotic behaviour through period doubling bifurcations

higher genus wall crossing in the gauged linear sigma model we introduce a technique for proving all genus wall crossing formulas in the gauged linear sigma model as the stability parameter varies without assuming factorization properties of the virtual class implementing this technique to the gauged linear sigma model associated to a complete intersection in weighted projective space we obtain a uniform proof of the wall crossing formula in both the geometric and the landau ginzburg phase

normal crossings singularities for symplectic topology abstract we introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent in a suitable sense to the desired geometric notions our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology this partially answers gromov s question on the feasibility of defining singular symplectic sub varieties and lays foundation for rich developments in the future in subsequent papers we establish a smoothability criterion for symplectic normal crossings varieties in the process providing the multifold symplectic sum envisioned by gromov and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors

comparison theorems and oscillation criteria for difference equations abstract first we establish the equivalence of the oscillation of the delay difference equation x n p n x n k 0 and the second order difference equation 2 y n 1 2 k 1 k k k 1 p n k k k 1 k 1 y n 0 where pn is a sequence of nonnegative real numbers and k is a positive integer next we obtain some sharp conditions for oscillations and nonoscillations of the first equation

the persistence and global attractivity in general nonautonomous discrete single species kolmogorov model with delays in this paper general nonautonomous discrete time single species kolmogorov model with delays is studied some sufficient conditions on the boundedness persistence permanence and global attractivity of solutions are established as application the discrete time nonautonomous delayed allee effect model is discussed the sufficient conditions on the permanence and global attractivity are obtained

sheaves on artin stacks we develop a theory of quasi coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of laumon and moret bailly we study basic cohomological properties of such sheaves and prove stack theoretic versions of grothendieck s fundamental theorem for proper morphisms grothendieck s existence theorem zariski s connectedness theorem as well as finiteness theorems for proper pushforwards of coherent and constructible sheaves we also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to laumon and moret bailly 1 1 in the book lm b the lisse etale topos of an algebraic stack was introduced and a theory of quasi coherent and constructible sheaves in this topology was developed unfortunately it was since observed by gabber and behrend independently that the lisse etale topos is not functorial as asserted in loc cit and hence the development of the theory of sheaves in this book is not satisfactory as is in addition since the publication of the book lm b several new results have been obtained such as finiteness of coherent and etale cohomology fa ol and various other consequences of chow s lemma ol the purpose of this paper is to explain how one can modify the arguments of lm b to obtain good theories of quasi coherent and constructible sheaves on algebraic stacks and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above 1 2 the paper is organized as follows in section 2 we recall some aspects of the theory of cohomological descent sga4 v which will be used in what follows in section 3 we review the basic definitions of the lisse etale site cartesian sheaves over a sheaf of algebras and verify some basic properties of such sheaves in section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology in section 5 we generalize these results to comparisons between ext groups computed in the lisse etale topos and ext groups computed using the etale topology on a hypercovering in section 6 we specialize the discussion of sections 3 5 to quasi coherent sheaves we show that if x is an algebraic stack and oxlis et denotes the structure sheaf of the lisse etale topos then the triangulated category d qcoh x of bounded below complexes of oxlis et modules with quasi coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes for example we show in this section that if f x y is a quasi compact morphism of algebraic stacks and m is a quasi coherent sheaf on x date november 2 2005 1

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

existence of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p n w c n x q n w where a n b n and c n are positive sequences of period w and p and q are positive integers by using the coincidence degree theorem as well as some priori estimates we will establish a sufficient condition for the existence of positive periodic solution x n

weighted automata and multi valued logics over arbitrary bounded lattices we show that l weighted automata l rational series and l valued monadic second order logic have the same expressive power for any bounded lattice l and for finite and infinite words we also prove that aperiodicity star freeness and l valued first order and ltl definability coincide this extends classical results of kleene buchi elgot trakhtenbrot and others to arbitrary bounded lattices without any distributivity assumption that is fundamental in the theory of weighted automata over semirings in fact we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices

motivic and quantum invariance under stratified mukai flops for stratified mukai flops of type a n k d 2k 1 and e 6 i it is shown the fiber product induces isomorphisms on chow motives in contrast to standard mukai flops the cup product is generally not preserved for a n 2 d 5 and e 6 i flops quantum corrections are found through degeneration deformation to ordinary flops

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

canonical rational equivalence of intersections of divisors one way to define an operation in intersection theory is to define a map on the group of algebraic cycles together with a map on the group of rational equivalences which commutes with the boundary operation assuming the maps commute with smooth pullback the extension of the operation to the setting of algebraic stacks is automatic the goal of section 2 of this paper is to present the operation of intersecting with a principal cartier divisor in this light we then show how this operation lets us obtain a rational equivalence which is fundamental to intersection theory a one dimensional family of cycles on an algebraic variety always admits a unique limiting cycle but a family of cycles over the punctured affine plane may yield different limiting cycles if one approaches the origin from different directions an important step in the historical development of intersection theory was realizing how to prove that any two such limiting cycles are rationally equivalent the results of section 2 yield as a corollary a new explicit formula for this rational equivalence another important rational equivalence in intersection theory is the one that is used to demonstrate commutativity of gysin maps associated to regularly embedded subschemes in section 3 we exhibit a two dimensional family of cycles such that the cycles we obtain from specializing in two different ways are precisely the ones we need to show to be rationally equivalent to obtain the commutativity result our explicit rational equivalence respects smooth pullback and hence the generalization to stacks is automatic this simplifies intersection theory on deligne mumford stacks as in 8 where construction of such a rational equivalence fills the most difficult section of that important paper since our rational equivalence arises by considering families of cycles on a larger total space we are able to deduce section 4 that the rational equivalence is invariant under a certain naturally arising group action the key observation is that we can manipulate the situation so that the group action extends to the total space this equivariance result is used but appears with mistaken proof in 2 where an important new tool of modern intersection theory the theory of virtual fundamental classes is developed acknowledgement the author would like to thank s bloch w fulton t graber and r pandharipande for helpful advice and the organizers and staff of the mittagleffler institute for hospitality during the 1996 97 program in algebraic geometry

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

landau ginzburg calabi yau correspondence for the complete intersections x3 3 and x2 2 2 2 abstract we define a generalization of fan jarvis ruan witten theory a hybrid model associated to a collection of quasihomogeneous polynomials of the same weights and degree which is expected to match the gromov witten theory of the calabi yau complete intersection cut out by the polynomials in genus zero we prove that the correspondence holds for any such complete intersection of dimension three in ordinary rather than weighted projective space these results generalize those of chiodo ruan for the quintic threefold and as in that setting givental s quantization can be used to yield a conjectural relation between the full higher genus theories

oscillation and global attractivity in hematopoiesis model with periodic coefficients in this paper we shall consider the nonlinear delay differential equation p t email protected t p m email protected 1 p n email protected email protected t p t where k is a positive integer b t and c t are positive periodic functions of period w in the nondelay case we shall show that has a unique positive periodic solution email protected t and we will study the global attractivity of email protected t in the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of about email protected t and establish some sufficient conditions for the global attractivity of email protected t our results in this paper extend as well as improve the results in the autonomous case

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

a new method toward the landau ginzburg calabi yau correspondence via quasi maps the landau ginzburg calabi yau correspondence claims that the gromov witten invariant of the quintic calabi yau 3 fold should be related to the fan jarvis ruan witten invariant of the associated landau ginzburg model via wall crossings in this paper we consider the stack of quasi maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for gromov witten invariants and the moduli stack for fan jarvis ruan witten invariants

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

the complexity of computing the behaviour of lattice automata on infinite trees abstract several logic based decision problems have been shown to be reducible to the emptiness problem of automata in a similar way non standard reasoning problems can be reduced to the computation of the behaviour of weighted automata in this paper we consider a variant of weighted buchi automata working on unlabelled infinite trees where the weights belong to a lattice we analyse the complexity of computing the behaviour of this kind of automata if the underlying lattice is not distributive we show that the decision version of this problem is in exptime and pspace hard in general assuming that the lattice operations are polynomial time computable if the lattice is what we call linear space computable encoded then the upper bound can be reduced to pspace but the lower bound also decreases to np hard and co np hard we conjecture that the upper bounds provided are in fact tight

oscillatory and asymptotic behavior of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p email protected c n x q email protected where a n b n and c n are positive sequences of period w and p and q are positive integers we will establish some sufficient conditions for the oscillation of all positive solutions about its positive periodic solution x n and prove that every nonoscillatory solution converges to x n monotonically as n

gromov witten pairs correspondence for the quintic 3 fold we use the gromov witten pairs descendent correspondence for toric 3 folds and degeneration arguments to establish the gw p correspondence for several compact calabi yau 3 folds including all cy complete intersections in products of projective spaces a crucial aspect of the proof is the study of the gw p correspondence for descendents in relative geometries projective bundles over surfaces relative to a section play a special role the gw p correspondence for calabi yau complete intersections provides a structure result for the gromov witten invariants in a fixed curve class after change of variables the gromov witten series is a rational function in the variable q eiu invariant under q q 1

oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics abstract in this paper we shall consider the nonlinear delay differential equation where m and n are positive integers and v t and t are positive periodic functions of period in the nondelay case we shall show that has a unique positive periodic solution x t and provides sufficient conditions for the global attractivity of x t in the delay case we shall present sufficient conditions for the oscillation of all positive solutions of about x t and establish sufficient conditions for the global attractivity of x t

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

a remark on gromov witten invariants of quintic threefold abstract the purpose of the article is to give a proof of a conjecture of maulik and pandharipande for genus 2 and 3 as a result it gives a way to determine gromov witten invariants of the quintic threefold for genus 2 and 3

absolute and relative gromov witten invariants of very ample hypersurfaces for any smooth complex projective variety x and any smooth very ample hypersurface y subset x we develop the technique of genus zero relative gromov witten invariants of y in x in algebro geometric terms we prove an equality of cycles in the chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the gromov witten invariants of x and y given the gromov witten invariants of x we show that these relations are sufficient to compute all relative invariants as well as all genus zero gromov witten invariants of y whose homology and cohomology classes are induced by x

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

stable logarithmic maps to deligne faltings pairs i we introduce a new compactication of the space of relative stable maps this approach uses logarithmic geoemetry in the sense of kato fontaineillusie without taking expansions of the target the underlying structures of the stable logarithmic maps are stable in the usual sense

generating functions for hurwitz hodge integrals we describe explicit generating functions for a large class of hurwitz hodge integrals these are integrals of tautological classes on moduli spaces of admissible covers a stackily smooth compactification of the hurwitz schemes admissible covers and their tautological classes are interesting mathematical objects on their own but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves and the orbifold gromov witten theory of dm stacks

weighted finite automata over strong bimonoids we investigate weighted finite automata over strings and strong bimonoids such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold we define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions moreover we show that in this case the given weighted automata can be determinized

periodic solutions of a single species discrete population model with periodic harvest stock abstract we discuss a discrete population model describing single species growth with periodic harvest stock the theory of coincidence degree is applied to show that the model equation admits two periodic solutions under minor technical assumptions we show that one of these two periodic solutions is positive and attracts almost all positive solutions

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

the witten equation mirror symmetry and quantum singularity theory for any nondegenerate quasi homogeneous hypersurface singularity we describe a family of moduli spaces a virtual cycle and a corresponding cohomological eld theory associated to the singularity this theory is analogous to gromov witten theory and generalizes the theory of r spin curves which corresponds to the simple singularity ar 1 we also resolve two outstanding conjectures of witten the rst conjecture is that ade singularities are self dual and the second conjecture is that the total potential functions of ade singularities satisfy corresponding ade integrable hierarchies other cases of integrable hierarchies are also discussed

torus localization and wall crossing for cosection localized virtual cycles abstract the theory of virtual fundamental class defines important invariants such as the gromov witten and the donaldson thomas invariants it has been generalized to the cosection localized virtual cycle which has applications in seiberg witten fan jarvis ruan witten and other invariants in this paper we prove the formulas of virtual pullback torus localization and wall crossing for cosection localized virtual cycles