tilting and refined donaldson thomas invariants abstract we study tilting for the heart a of the canonical t structure of the finite dimensional derived category of the ginzburg algebra for a quiver with potential q w we give conditions on that the stable objects for a central charge on a define a sequence of simple tilts from a to a 1 on that conditions the refined donaldson thomas invariant associated to q w is independent of the chosen central charge

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

on generalized bisection of n simplices a generalized procedure of bisection of n simplices is introduced where the bisection point can be an almost arbitrary point at one of the longest edges it is shown that nested sequences of simplices generated by successive generalized bisection converge to a singleton and an exact bound of the convergence speed in terms of diameter reduction is given for regular simplices which mark the worst case the edge lengths of each worst and best simplex generated by successive bisection are given up to depth n for n 2 and 3 the sequence of worst case diameters is provided until it is halved

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

moduli stacks and invariants of semistable objects on k3 surfaces abstract for a k3 surface x and its bounded derived category of coherent sheaves d x we have the notion of stability conditions on d x in the sense of t bridgeland in this paper we show that the moduli stack of semistable objects in d x with a fixed numerical class and a phase is represented by an artin stack of finite type over c then following d joyce s work we introduce the invariants counting semistable objects in d x and show that the invariants are independent of a choice of a stability condition

finite volume method for the calculation of compressible chemically reacting flows several efficient pseudo time techniques have been developed for calculating steady state chemically reacting flows the techniques include the implicit treatment of the chemical source term point implicit multiple grid accelerator and a constant cfl condition it turns out that these methods can be viewed as ways of rescaling the equations in time such that all chemical and convective phenomena evolve at comparable pseudo time scales consequently the number of iterations needed to solve reacting problems is approximately the same as for non reacting problems the techniques are demonstrated for a simple dissociation model and a nontrivial h2 air combustion model

longest edge n section algorithms in this paper we survey all known including own recent results properties of the longest edge n section algorithms these algorithms in classical and recently designed conforming form are nowadays used in many applications including finite element simulations computer graphics etc as a reliable tool for controllable mesh generation in addition we present a list of open problems arising in and around this topic

cusps of the kahler moduli space and stability conditions on k3 surfaces ma int j math 20 6 727 750 2009 established a bijection between fourier mukai partners of a k3 surface and cusps of the kahler moduli space the kahler moduli space can be described as a quotient of bridgeland s stability manifold we study the relation between stability conditions near to a cusp and the associated fourier mukai partner y in the following ways 1 we compare the heart of to the heart of coherent sheaves on y 2 we construct y as moduli space of stable objects an appendix is devoted to the group of auto equivalences of mathcal d b x which respect the component stab dagger x of the stability manifold

stability conditions and extremal contractions we study extremal contractions from smooth projective varieties via a moduli theoretic approach in the two dimensional case we show that any extremal contraction appears as a moduli space of bridgeland stable objects in the derived category of coherent sheaves in the three dimensional case we show that a a similar result holds with respect to conjectural bridgeland stability conditions

birational geometry of singular moduli spaces of o grady type abstract following bayer and macri we study the birational geometry of singular moduli spaces m of sheaves on a k3 surface x which admit symplectic resolutions more precisely we use the bayer macri map from the space of bridgeland stability conditions stab x to the cone of movable divisors on m to relate wall crossing in stab x to birational transformations of m we give a complete classification of walls in stab x and show that every minimal birational model of m in the sense of the log minimal model program appears as a moduli space of bridgeland semistable objects on x an essential ingredient of our proof is an isometry between the orthogonal complement of a mukai vector inside the algebraic mukai lattice of x and the neron severi lattice of m which generalises results of yoshioka as well as perego and rapagnetta moreover this allows us to conclude that the symplectic resolution of m is deformation equivalent to the 10 dimensional irreducible holomorphic symplectic manifold found by o grady

categorical entropy for fourier mukai transforms on generic abelian surfaces abstract in this note we shall compute the categorical entropy of an autoequivalence on a generic abelian surface

limit stable objects on calabi yau 3 folds the notion of limit stability is introduced in 31 to give a limiting degeneration of bridgeland stability in this article we give the survey of the results in 31 30 we introduce the limit stability counting invariants of limit stable objects and see that the wall crossing phenomena yields the rationality conjecture of pandharipandethomas generating functions

complex surfaces with equivalent derived categories we examine the extent to which a smooth minimal complex projective surface x is determined by its derived category of coherent sheaves dx to do this we find for each such surface x the set of surfaces y for which there exists a fourier mukai transform d y to d x

fano varieties of cubic fourfolds containing a plane we prove that the fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a k3 surface on the other hand we show that the fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a k3 surface the moduli spaces of complexes and of sheaves are related by wall crossing in the derived category of twisted sheaves on the corresponding k3 surface

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

two sided estimation of diameters reduction rate for the longest edge n section of triangles with n 4 in this work we study the diameters reduction rate for the iterative application of the longest edge le n section of triangles for n p 4 the maximum diameter d n k of all triangles generated at the kth iteration of the le n section is closely connected with the properties of the triangular mesh generated by this refinement scheme the upper and the lower bounds for d 2 k were proved by kearfott in 9 and for d

spectral methods for modeling supersonic chemically reacting flowfields on emploie une methode de runge kutta a deux stades pour integrer les equations en temps et une methode spectrale de chebyshev pour l integration des equations dans l espace comparaison avec une methode de differences finies implicites

an efficient multigrid algorithm for compressible reactive flows this paper presents a parallel multigrid method for computing inviscid and viscous high speed steady state reactive flows the governing equations for reactive flow are solved using an explicit multigrid algorithm while treating the chemical source terms in a point implicit manner the cusp convective upwind and split pressure scheme is used to provide necessary artificial dissipation without contaminating the solution this explicit method yields excellent parallel speedups thus enabling the calculation of reactive flows with detailed chemical kinetics including large numbers of species and reactions results indicate good multigrid speedups and adequate resolution of the reaction zone in both inviscid axisymmetric and viscous two dimensional hydrogen oxygen and hydrogen air test cases

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

mmp via wall crossing for moduli spaces of stable sheaves on an enriques surface abstract we use wall crossing in the bridgeland stability manifold to systematically study the birational geometry of the moduli space m v of semistable objects of class v for a generic stability condition on an arbitrary enriques surface x in particular we show that for any other generic stability condition the two moduli spaces m v and m v are birational as a consequence we show that for primitive v of odd rank m v is birational to a hilbert scheme of points similarly in even rank we show that m v is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when l v 1 as an added bonus of our work we prove that the donaldson mukai map v v pic m v is an isomorphism for these classes finally we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on x that are uniruled and thus not k trivial

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

automorphisms of positive entropy on some hyperkahler manifolds via derived automorphisms of k3 surfaces abstract we construct examples of hyperkahler manifolds of picard number two with automorphisms of positive entropy via derived automorphisms of k3 surfaces of picard number one our hyperkahler manifolds are constructed as moduli spaces of bridgeland stable objects in derived categories of k3 surfaces then automorphisms of positive entropy are induced by derived automorphisms of positive entropy on k3 surfaces

stability conditions on threefolds with nef tangent bundles abstract in this paper we prove the bogomolov gieseker type inequality conjecture for threefolds with nef tangent bundles as a corollary there exist bridgeland stability conditions on these threefolds

a short proof of the deformation property of bridgeland stability conditions the key result in the theory of bridgeland stability conditions is the property that they form a complex manifold this comes from the fact that given any small deformation of the central charge there is a unique way to correspondingly deform the stability condition we give a short direct proof of an effective version of this deformation property

on weyl groups and artin groups associated to orbifold projective lines abstract we associate a generalized root system in the sense of kyoji saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra we generalize results by saito and takebayashi and yamada for elliptic weyl groups and elliptic artin groups to the weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2 calabi yau completion of the bound quiver algebra

smoothness and poisson structures of bridgeland moduli spaces on poisson surfaces let x be a projective smooth holomorphic poisson surface in other words whose anti canonical bundle is effective we show that moduli spaces of certain bridgeland stable objects on x are smooth moreover we construct poisson structures on these moduli spaces

orthogonally spherical objects and spherical fibrations abstract we introduce a relative version of the spherical objects of seidel and thomas 32 define an object e in the derived category d z x to be spherical over z if the corresponding functor from d z to d x gives rise to autoequivalences of d z and d x in a certain natural way most known examples come from subschemes of x fibred over z this categorifies to the notion of an object of d z x orthogonal over z we prove that such an object is spherical over z if and only if it possesses certain cohomological properties similar to those in the original definition of a spherical object we then interpret this geometrically in the case when our objects are actual flat fibrations in x over z

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

derived equivalences via hrs tilting abstract let a be an abelian category and b be the happel reiten smalo tilt of a with respect to a torsion pair we give necessary and sufficient conditions for the existence of a derived equivalence between a and b which is compatible with the inclusion of b into the derived category of a as applications we obtain new derived equivalences related to splitting torsion pairs ttf triples and two term silting subcategories respectively we prove that for the realization functor of any bounded t structure its denseness implies its fully faithfulness

derived automorphism groups of k3 surfaces of picard rank 1 we give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a k3 surface of picard rank 1 we do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible

on stability conditions for the quintic threefold we study the clifford type inequality for a particular type of curves c 2 2 5 which are contained in smooth quintic threefolds this allows us to prove some stronger bogomolov gieseker type inequalities for chern characters of stable sheaves and tilt stable objects on smooth quintic threefolds employing the previous framework by bayer bertram macri stellari and toda we construct an open subset of stability conditions on every smooth quintic threefold in mathbf p 4 mathbb c

on global and local mesh refinements by a generalized conforming bisection algorithm we examine a generalized conforming bisection gcb algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes it is based on the notion of a mesh density function which prescribes where and how much to refine the mesh some regularity properties of generated sequences of refined triangulations are proved several numerical tests demonstrate the efficiency of the proposed bisection algorithm it is also shown how to modify the gcb algorithm in order to generate anisotropic meshes with high aspect ratios

derived equivalences of k3 surfaces and orientation every fourier mukai equivalence between the derived categories of two k3 surfaces induces a hodge isometry of their cohomologies viewed as hodge structures of weight two endowed with the mukai pairing we prove that this hodge isometry preserves the natural orientation of the four positive directions this leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical torelli theorem for k3 surfaces determining all hodge isometries that are induced by automorphisms

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

a new proof of the degeneracy property of the longest edge n section refinement scheme for triangular meshes in this note by using complex variable functions we present a new simpler proof of the degeneracy property of the longest edge n section of triangles for n p 4 this means that the longest edge n section of triangles for n p 4 produces a sequence of triangles with minimum interior angle converging to zero

stability and fourier mukai transforms on elliptic fibrations abstract we systematically develop bridgeland s 7 and bridgeland maciocia s 10 techniques for studying elliptic fibrations and identify criteria that ensure 2 term complexes are mapped to torsion free sheaves under a fourier mukai transform as an application we construct an open immersion from a moduli of stable complexes to a moduli of gieseker stable sheaves on elliptic threefolds as another application we give various 1 1 correspondences between fibrewise semistable torsion free sheaves and codimension 1 sheaves on weierstrass surfaces

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

the space of stability conditions on abelian threefolds and on some calabi yau threefolds we describe a connected component of the space of stability conditions on abelian threefolds and on calabi yau threefolds obtained as the crepant resolution of a finite quotient of an abelian threefold our proof includes the following essential steps 1 we simultaneously strengthen a conjecture by the first two authors and toda and prove that it follows from a more natural and seemingly weaker statement this conjecture is a bogomolov gieseker type inequality involving the third chern character of tilt stable two term complexes on smooth projective threefolds we extend it from complexes of tilt slope zero to arbitrary tilt slope 2 we show that this stronger conjecture implies the so called support property of bridgeland stability conditions and the existence of an explicit open subset of the space of stability conditions 3 we prove our conjecture for abelian threefolds thereby reproving and generalizing a result by maciocia and piyaratne important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall crossing closely related to the support property

bridgeland stability conditions on wild kronecker quivers abstract we prove that for l 3 the space of bridgeland stability conditions on the kronecker quiver with l parallel arrows is biholomorphic to c h as a complex manifold

bridgeland stability conditions on the acyclic triangular quiver abstract using results in our previous paper non semistable exceptional objects in hereditary categories we focus here on studying the topology of the space of bridgeland stability conditions on d b rep k q where q and k is an algebraically closed field in particular we prove that this space is contractible in the previous paper it was shown that it is connected

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

efficient calculation of chemically reacting flow a semi implicit finite volume formulation is used to jrvstudy flow with chemical reactions in this formulation the source terms resulting from the chemical reaction are treated implicitly and the resulting system of partial differential equations is sovled using two time stepping schemes the first is based on the runge kutta method and the second on an adams predictor corrector method results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated furthermore analysis and computation indicate that the rungekutta method is more efficient than the adams methods for these systems of differential equations finally an adaptive time stepping scheme is developed to study problems involving shock ignition calculations for a hydrogen air system agree well with other methods

the maximum angle condition is not necessary for convergence of the finite element method we show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems this condition is only sufficient in fact finite element approximations may converge even though some dihedral angles of simplicial elements tend to

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

moduli spaces of stable sheaves on abelian surfaces in this paper we consider basic problems on moduli spaces of stable sheaves on abelian surfaces our main assumption is the primitivity of the associated mukai vector we determine the deformation types albanese maps bogomolov factors and their weight 2 hodge structures we also discuss the deformation types of moduli spaces of stable sheaves on k3 surfaces

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

on the non degeneracy property of the longest edge trisection of triangles the longest edge le trisection of a triangle t is obtained by joining the two equally spaced points of the longest edge of t with the opposite vertex in this paper we prove that for any given triangle t with smallest interior angle t 0 if the minimum interior angle of the three triangles obtained by the le trisection of t into three new triangles is denoted by t 1 then t 1 email protected c 1 where c email protected 3arctan 3 5 3 1403 moreover we show empirical evidence on the non degeneracy property of the triangular meshes obtained by iterative application of the le trisection of triangles if t n denotes the minimum angle of the triangles obtained after n iterative applications of the le trisection then t n t c where c is a positive constant independent of n an experimental estimate of c 6 7052025350 is provided

on the bisection method for triangles let uvw be a triangle with vertices u v and w it is bisected as follows choose a longest edge say vw of uvw and let a be the midpoint of vw the uvw gives birth to two daughter triangles uva and uwa continue this bisection process forever we prove that the infinite family of triangles so obtained falls into finitely many similarity classes and we obtain sharp estimates for the longest jth generation edge

bridgeland s stabilities on abelian surfaces in this paper we shall study the structure of walls for bridgeland s stability conditions on abelian surfaces in particular we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the picard number 1

stability conditions via spherical objects an object in the bounded derived category db x of coherent sheaves on a complex projective k3 surface x is spherical if it is rigid and simple although spherical objects form only a discrete set in the moduli stack of complexes they determine much of the structure of x and db x here we show that a stability condition on db x is determined by the stability of spherical objects

a categorical invariant for cubic threefolds abstract we prove a categorical version of the torelli theorem for cubic threefolds more precisely we show that the non trivial part of a semi orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class