pattern formation in the brusselator system in the paper we deal with a reaction diffusion system well known as the brusselator model and some improved results for the steady states of this model are presented we first give an a priori estimates positive upper and lower bounds of positive steady states then we obtain the non existence and existence of positive non constant steady states as the parameters and b are varied which means some certain conditions under which the pattern formation occurs or not

compactications defined by arrangements ii locally symmetric varieties of type iv we define a new class of completions of locally symmetric varieties of type iv which interpolates between the baily borel compactification and mumford s toric compactifications an arithmetic arrangement in a locally symmetric variety of type iv determines such a completion canonically this completion admits a natural contraction that leaves the complement of the arrangement untouched the resulting completion of the arrangement complement is very much like a baily borel compactification it is the proj of an algebra of meromorphic automorphic forms when that complement has a moduli space interpretation then what we get is often a compactification obtained by means of geometric invariant theory we illustrate this with several examples moduli spaces of polarized k3 and enriques surfaces and the semi universal deformation of a triangle singularity we also discuss the question when a type iv arrangement is definable by an automorphic form

turing patterns in the lengyel epstein system for the cima reaction the first experimental evidence of the turing pattern was observed by de kepper and her associates 1990 on the cima reaction in an open unstirred gel reactor almost 40 years after turing s prediction lengyel and epstein characterized this famous experiment using a system of reaction diffusion equations in this paper we report some fundamental analytic properties of the lengyel epstein system our result also indicates that if either of the initial concentrations of the reactants the size of the reactor or the effective diffusion rate are not large enough then the system does not admit nonconstant steady states a priori estimates are fundamental to our approach for this nonexistence result the degree theory was combined with the a priori estimates to derive existence of nonconstant steady states

the moduli space of cubic fourfolds via the period map we characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space it follows then that the geometric invariant theory git compactification of the moduli space of cubic fourfolds is isomorphic to the looijenga compactification associated to this arrangement this paper builds on and is a natural continuation of our previous work on the git compactification of the moduli space of cubic fourfolds

lattices and k3 surfaces of degree 6 we relate a class of degenerations of k3 surfaces of degree 6 to certain lattices via hodge theory

modular forms and special cubic fourfolds abstract we study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ade singularities in this paper we show that the generating series of the degrees of such divisors is a level three modular form

the cohomology rings of moduli spaces of bundles over riemann surfaces the cohomology of the moduli space ji n d of semistable bundles of coprime rank n and degree d over a fixed riemann surface m of genus g 2 has been much studied over the last two decades and a great deal is known about it it is known that the cohomology has no torsion 2 the betti numbers have been computed using various different methods 2 5 11 and a set of generators for the cohomology ring is known 2 the purpose of this article is to provide a method of deciding when an expression in these generators is a relation and in particular when the rank n is two to prove two conjectures about the relations between the generators due to mumford and to newstead and ramanan newstead and ramanan conjectured that certain relations hold among the generators whereas mumford conjectured that a certain set of relations is a complete set in fact the conjecture of newstead and ramanan has two parts the first of which was proved by gieseker in 1982 4 the second part of their conjecture which is the part examined in this article was given an independent proof recently by thaddeus 19 who has a very different method of obtaining information about the relations in the cohomology ring the computation of the betti numbers of the moduli spaces i n d in 2 stimulated a general method of computing betti numbers of quotients in the sense of mumford s geometric invariant theory 14 of nonsingular complex projective varieties by complex reductive group actions see 8 9 10 similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces i n d also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients the first section provides background material on the cohomology of the moduli spaces i n d in the second section mumford s conjecture is proved when the rank n is two the method could in principle be used to decide whether the conjecture is true for rank n 2 but appears to become significantly more complicated in the general case the third section gives a method for deciding when an expression in the generators is a relation and the fourth

hopf bifurcation and turing instability of 2 d lengyel epstein system with reaction diffusion terms in this paper we study 2 d lengyel epstein l e reaction diffusion system with homogeneous neumann boundary condition by employing stability theory hopf bifurcation theorem and turing s theory we present some sufficient conditions ensuring the equilibrium point of system to be stable and derive conditions on the parameters so that spatial homogenous hopf bifurcation and turing instability occur these conditions obtained have important leading significance in applications of l e system finally we show some numerical examples to verity the theoretical analysis

probabilistic approach to the lame equations of linear elastostatics abstract a probabilistic approach to systems of partial differential equations is developed on the basis of the well known feynman kac and bismut formulas providing explicit probabilistic representations of the solutions and of their derivatives of scalar differential equations some numerical examples are also included in particular the lame equations of elastostatics are solved and the results are compared with some known exact analytic solutions to demonstrate the efficiency of the approach

the stringy e function of the moduli space of rank 2 bundles over a riemann surface of genus 3 we compute the stringy e function or the motivic integral of the moduli space of rank 2 bundles over a riemann surface of genus 3 in doing so we answer a question of batyrev about the stringy e functions of the git quotients of linear representations

probabilistic solutions of the helmholtz equation solutions of the schrodinger and helmholtz equations are represented as mathematical expectations of functionals on paths in a complex space formulas obtained for the helmholtz equation appear as direct improvements of the ray method approximations to the exact solution presented numerical results confirm the efficiency of the suggested description of wave propagation phenomena

a path integral time domain method for electromagnetic scattering a new full wave time domain formulation for the electromagnetic field is obtained by means of a path integral the path integral propagator is derived via a state variable approach starting with maxwell s differential equations in tensor form a numerical method for evaluating the path integral is presented and numerical dispersion and stability conditions are derived and numerical error is discussed an absorbing boundary condition is demonstrated for the one dimensional 1 d case it is shown that this time domain method is characterized by the unconditional stability of the path integral equations and by its ability to propagate an electromagnetic wave at the nyquist limit two numerical points per wavelength as a consequence the calculated fields are not subject to numerical dispersion other advantages in comparison to presently popular time domain techniques are that it avoids time interval interleaving and it does not require the methods of linear algebra such as basis function selection or matrix methods

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

three dimensional dissipative structures in reaction diffusion systems abstract we study analytically and numerically the tridimensional pattern selection problem for reaction diffusion systems qualitative agreement is found with the recent experimental results

stability and equivariant maps consider a linearized action of a reductive algebraic group on a projective algebraic varietyx over an algebraically closed field in this situation mumford 1 defined the concept of stability for points ofx given an equivariant morphismy x we introduce a suitable linearization of the action ony and relate stability iny to stability inx in particular we prove a relative hilbert mumford theorem which says that stability inx andy can be tested simultaneously by 1 parameter subgroups it is hoped that this result will have applications to moduli problems in the casey x is a blowing up of a sheaf of ideals inx we give a simple explicit description of the properly stable and semi stable loci iny our relative hilbert mumford theorem is used here in an essential way we also consider the following resolution problem introduced by kirwan 2 the equivariant mapy x is called a stable resolution if it is an isomorphism over the properly stable locus inx and every point ofy is either unstable or properly stable kirwan 2 gave a canonical procedure for constructing a stable resolution over the complex numbers we show that a stable resolution can be obtained by resolving the singularities of a naturally defined subvariety ofx this gives an alternative non canonical procedure for constructing stable resolutions

random walk approach to wave propagation in wedges and cones two and three dimensional helmholtz equations in wedge shaped and conical domains are addressed by the random walk method the solutions of the dirichlet problems in such domains are represented as mathematical expectations of specified functionals on trajectories of multidimensional random motions whose radial components run in a complex space while the angular components remain real valued this technique is applied to the sommerfeld problem of diffraction by a semi infinite screen which is explicitly solved here in the probabilistic form the numerical results confirm the efficiency of the random walk approach to the analysis of wave propagation

intersection cohomology of quotients of nonsingular varieties let m p be a nonsingular projective variety acted on by a connected complex reductive group g kc via a homomorphism g gl n 1 which is the complexification of a homomorphism k u n 1 geometric invariant theory git gives us a recipe to form a quotient m m g of the set of semistable points and many interesting spaces in algebraic geometry are constructed in this manner mfk94 but often this quotient is singular and hence intersection cohomology with middle perversity is an important topological invariant the purpose of this paper is to present a way to compute the middle perversity intersection cohomology of the singular quotients the choice of an embedding m p provides us with a moment map for the k action and the morse stratification of m with respect to its norm square is kequivariantly perfect with the unique open dense stratum m kir84 hence one can compute the betti numbers for the equivariant cohomology h k m as well as the cup product at least in principle see also kir92 when g acts locally freely on m we get an orbifold m g m g and h k m h m g ih m g

the kirwan map equivariant kirwan maps and their kernels consider a hamiltonian action of a compact lie group k on a compact symplectic manifold we find descriptions of the kernel of the kirwan map corresponding to a regular value of the moment map k we start with the case when k is a torus t we determine the kernel of the equivariant kirwan map defined by goldin in go corresponding to a generic circle s t and show how to recover from this the kernel of t as described by tolman and weitsman in to we in the situation when the fixed point set of the torus action is finite similar results have been obtained in our previous papers je je ma for a compact nonabelian lie group k we will use the non abelian localization formula of je ki1 and je ki2 to establish relationships some of them obtained by tolman and weitsman in to we between ker k and ker t where t k is a maximal torus in the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of t

constructing constrained invariant sets in multiscale continuum systems we present a method that we name the constrained invariant manifold method a visualization tool to construct stable and unstable invariant sets of a map or flow where the invariant sets are constrained to lie on a slow invariant manifold the construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural mechanical system consisting of a pendulum coupled to a viscoelastic rod additionally we extend the step and stagger method d sweet h nusse and j yorke phys rev lett 86 2261 2001 to calculate a delta pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the lyapunov exponents of the saddle

elliptic genus of phases of n 2 theories we discuss an algebro geometric description of witten s phases of n 2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met for landau ginzburg phase one recovers elliptic genus of lg models proposed in physics literature in early 1990s for certain transitions between phases we derive invariance of elliptic genus from an equivariant form of mckay correspondence for elliptic genus as special cases one obtains landau giznburg calabi yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybrid cy correspondences

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

diffusive stability of spatially periodic solutions of the brusselator model applying the lyapunov schmidt reduction approach introduced by mielke and schneider in their analysis of the fourth order scalar swift hohenberg equation we carry out a rigorous small amplitude stability analysis of turing patterns for the canonical second order system of reaction diffusion equations given by the brusselator model our results confirm that stability is accurately predicted in the small amplitude limit by the formal ginzburg landau amplitude equations rigorously validating the standard weakly unstable approximation and the eckhaus criterion

geometric and algebraic reduction for singular momentum maps this paper concerns the reduction of singular constraint sets of symplectic manifolds it develops a geometric reduction procedure as well as continues the work of sniatycki and patrick on reduction a la dirac the relationships among the dirac geometric and sniatycki weinstein algebraic reduction procedures are studied primary emphasis is placed on the case where the constraints are given by the vanishing of a singular momentum map associated to the hamiltonian action of a compact lie group specifically an explicit local normal form is given for the momentum map which is used to show that these various reductions are all well defined and a necessary and sufficient condition for them to agree is derived related conditions are investigated and shown to be sufficient in the case of torus actions numerous examples are computed illustrating the results some discussion and examples are given for the noncompact case

symmetries and pattern selection in rayleigh benard convection abstract this paper describes the process of pattern selection between rolls and hexagons in rayleigh benard convection with reflectional symmetry in the horizontal midplane this symmetry is a consequence of the boussinesq approximation provided the boundary conditions are the same on the top and bottom plates all possible local bifurcation diagrams assuming certain non degeneracy conditions are found using only group theory the results are therefore applicable to other systems with the same symmetries rolls hexagons or a new solution regular triangles can be stable depending on the physical system rolls are stable in ordinary rayleigh benard convection the results are compared to those of buzano and golubitsky 1 without the midplane reflection symmetry the bifurcation behavior of the two cases is quite different and a connection between them is established by considering the effects of breaking the reflectional symmetry finally the relevant experimental results are described

the index formula for the moduli of g bundles on a curve we prove the formulae conjectured by the first author for the index of k theory classes over the moduli stack of algebraic g bundles on a smooth projective curve the formulae generalize e verlinde s for line bundles and have witten s integrals over the moduli space of stable bundles as their large level limits as an application we prove the newstead ramanan conjecture on the vanishing of high chern classes of certain moduli spaces of semi stable g bundles

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

on steady state solutions of the brusselator type system abstract in this article we shall be concerned with the following brusselator type system u 1 b 1 u b u m v in v a 2 u u m v in under the homogeneous neumann boundary conditions this system was recently investigated by m ghergu in nonlinearity 21 2008 2331 2345 here r n n 1 is a smooth and bounded domain and a b m and are positive constants when m 2 this system corresponds to the well known stationary brusselator model which has received extensive studies analytically as well as numerically in the present work we derive some further results for the general system our conclusions show that there is no non constant positive steady state for large a while small a may produce non constant positive steady states if 1 n 3 and 1 m 3 we particularly determine the asymptotic behavior of non constant positive steady states as a converges to zero thereby solving an open problem left in ghergu work

random walk methods and wave diffraction this paper develops an approach to problems of wave diffraction that combines the physical clarity of the ray method with the versatility of direct numerical methods first it addresses scalar problems with general linear first order boundary conditions and then it considers problems formulated in adjacent domains with imposed interface conditions we start by following closely the scheme of the ray method but instead of looking for approximate expressions for the amplitudes of the liouville decomposition we obtain their exact representations as the mathematical expectations of some functionals on the space of brownian trajectories the obtained solutions provide direct improvements of the ray method approximations to the exact solutions and they are shown to admit efficient numerical evaluations

interconnected turing patterns in three dimensions we study numerically the turing pattern in three dimensions in a fitzhugh nagumo type reaction diffusion system we have found that interconnected periodic domain structures such as a gyroid fddd and perforated lamellar structures appear in three dimensions which never exist in lower dimensions the stability analysis of these structures is also performed by means of a mode expansion

rational intersection cohomology of quotient varieties given a linear action of a complex reductive group g on a complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory 18 from the construction of this quotient it is not immediately obvious how its topology is related to the topology of x and the group action however when x is nonsingular it is possible at least if one technical condition is satisfied to compute the intersection betti numbers of x g in terms of the rational cohomology of certain nonsingular linear sections z of x and the classifying spaces of certain reductive subgroups h of g see 15 the technical condition is that there should exist at least one point of x which is properly stable for the action of g this is satisfied in most interesting examples such as those coming from moduli problems the sections of x and subgroups of g which occur have a simple description in terms of the representation of g which induces the linear action on x it thus seems reasonable to ask what happens when x is singular the aim of this paper is to show that the procedure for computing the intersection betti numbers of x g can be generalised to the singular case see 2 25 and 2 28 however now one needs to know not only the intersection cohomology of the various sections z of x but also how they are positioned relative to the singularities of x in the special case when they meet the singularities transversely then one has exactly the same formulas as when x is nonsingular except that ordinary cohomology is replaced by intersection cohomology throughout the situation which it is most important to understand is that in which every semistable point is stable and this case is considered in detail in w 2 after the necessary preliminaries have been covered in w 1 some brief remarks are made about the more general case when one only assumes that there exist some stable points a simple example is worked out 2 27 which shows that the formulas can also be used in reverse to compute the intersection betti numbers of x from knowledge of the intersection betti numbers of the quotient x g even simply from the knowledge that these vanish in dimensions greater than dim x g

pattern formation in a predator prey model with spatial effect abstract in this paper spatial dynamics in the beddington deangelis predator prey model with self diffusion and cross diffusion is investigated we analyze the linear stability and obtain the condition of turing instability of this model moreover we deduce the amplitude equations and determine the stability of different patterns numerical simulations show that this system exhibits complex dynamical behaviors in the turing space we find three types of typical patterns one is the coexistence of hexagon patterns and stripe patterns the other two are hexagon patterns of different types the obtained results well enrich the finding in predator prey models with beddington deangelis functional response

on the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface let j be an abelian surface with a generic ample line bundle open image in new window for n 1 the moduli space mj 2 0 2n of open image in new window 1 semistable sheaves f of rank 2 with chern classes c1 f 0 c2 f 2n is a singular projective variety endowed with a holomorphic symplectic structure on the smooth locus in this paper we show that there does not exist a crepant resolution of mj 2 0 2n for n 2 this certainly implies that there is no symplectic desingularization of mj 2 0 2n for n 2

a compactification of open varieties in this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors this is done by showing that blowing up along an arrangement of subvarieties can be carried out important examples such as ulyanov s configuration spaces and complements of arrangements of linear subspaces in projective spaces etc are covered intersection ring and nonrecursive hodge polynomials are computed furthermore some general structures arising from the blowup process are also described and studied

simple and superlattice turing patterns in reaction diffusion systems bifurcation bistability and parameter collapse abstract we use equivariant bifurcation theory to investigate pattern selection at the onset of a turing instability in a general two component reaction diffusion system the analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion both simple periodic patterns stripes squares hexagons and rhombs and superlattice patterns are considered the latter correspond to patterns that have structure on two disparate length scales the short length scale is dictated by the critical wave number from linear theory while the periodicity of the pattern is on a larger scale analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction diffusion system using perturbation theory we show that no matter how complicated the reaction kinetics might be the nonlinear reaction terms enter the analysis through just four parameters moreover for hexagonal problems all patterns bifurcate unstably unless a particular degeneracy condition is satisfied and at this degeneracy we find that the number of effective system parameters drops to two allowing a complete characterization of the possible bifurcation results at this degeneracy for example we find that rhombs squares and superlattice patterns always bifurcate unstably we apply these general results to some specific model equations including the lengyel epstein cima model to investigate the relative stability of patterns as a function of system parameters and to numerically test the analytical predictions

embedding pointed curves in k3 surfaces we analyze morphisms from pointed curves to k3 surfaces with a distinguished rational curve such that the marked points are taken to the rational curve perhaps with specified cross ratios this builds on work of mukai and others characterizing embeddings of curves into k3 surfaces via non abelian brill noether theory our study leads naturally to enumerative problems which we solve in several specific cases these have applications to the existence of sections of del pezzo fibrations with prescribed invariants

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

the moduli of curves of genus six and k3 surfaces we prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type iv by giving a period map to the moduli space of some lattice polarized k3 surfaces

turing patterns with o 3 symmetry abstract we perform an explicit center manifold reduction for the general steady state bifurcation of a chemical reaction diffusion system in an arbitrary compact domain in particular we focus on those systems with two chemical species in domains with o 3 symmetry we illustrate with the brusselator on the spherical shell and produce explicit computations of the solution branches and their stabilities for the low l bifurcations because of the phenomenon of parameter collapse we can create universal bifurcation diagrams for even l this work is an extension of that done in physica d 132 1999 339 362

teichmuller theory and critically finite endomorphisms abstract we present a systematic way to generate critically finite endomorphisms of p n these maps arise in the context of teichmuller theory specifically in thurston s topological characterization of rational maps the dynamical objects for the endomorphisms correspond to central objects from thurston s theorem our theorems build infinitely many of these endomorphisms in fact a large number of examples of critically finite endomorphisms of p n found in the literature arise from this construction

gauged gromov witten theory for small spheres we relate the genus zero gauged gromov witten invariants of a smooth projective variety for sufficiently small area with equivariant gromov witten invariants as an application we deduce a gauged version of abelianization for gromov witten invariants in the small area chamber in the symplectic setting we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map

the intersection cohomology of the satake compactification of mathcal a _g for g le 4 we completely determine the intersection cohomology of the satake compactifications mathcal a 2 mathrm sat mathcal a 3 mathrm sat and mathcal a 4 mathrm sat except for ih 10 mathcal a 4 mathrm sat we also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the satake compactification in these genera as a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved

desingularizations of the moduli space of rank 2 bundles over a curve let x be a smooth projective curve of genus g 3 and m0 be the moduli space of rank 2 semistable bundles over x with trivial determinant there are three desingularizations of this singular moduli space constructed by narasimhan ramanan nr78 seshadri ses77 and kirwan kir86b respectively the relationship between them has not been understood so far the purpose of this paper is to show that there is a morphism from kirwan s desingularization to seshadri s which turns out to be the composition of two blow downs in doing so we will show that the singularities of m0 are terminal and the plurigenera are all trivial as an application we compute the betti numbers of the cohomology of seshadri s desingularization in all degrees this generalizes the result of bs90 which computes the betti numbers in low degrees another application is the computation of the stringy e function see bat98 for definition of m0 for any genus g 3 which generalizes the result of kie03

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

an example of symmetry breaking to heteroclinic cycles is orbitally asymptotically stable suppose now that we consider a small system symmetry breakingperturbation of 1 1 z f z g z 1 2 where is small and g is only 2 equivariant where 2 1 is a lie subgroup when is sufficiently small normal hyperbolicity guarantees that there isa perturbed flow invariant manifold x

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

intersection homology and torus actions in the bialynicki birula decomposition of x see 15 6 10 13 the same formula holds for appropriate integers my when c is replaced by a torus t c r in 9 it is shown that the formula 0 1 is valid even when x is singular provided that the bialynicki birula decomposition is good the aim of this paper is to generalize 0 1 to the case when x is singular in a different way which involves replacing ordinary homology by intersection homology with respect to the middle perversity however only rational coefficients are considered when x is nonsingular its intersection homology and ordinary homology coincide but when x is singular its intersection homology behaves better in many respects than its ordinary cohomology it is shown that when x is singular just as when x is nonsingular the rational intersection homology of x is determined by the action of t on an arbitrarily small neighborhood of the fixed point set xt as might be expected the formula 0 1 does not carry over directly when intersection homology replaces ordinary homology the terms hi 2m f z appearing in the right hand side are replaced by hypercohomology groups of certain complexes of sheaves over the f which depend upon how the f meet the singularities of x see they y orem 2 3 1 of this paper contains a review of a proof of 0 1 for rational coefficients which uses equivariant morse theory in 2 it is shown how this proof can be extended to apply to singular varieties when intersection homology is used the

pattern formation in the three dimensional reaction diffusion systems existing group theoretic analysis of pattern formation in three dimensions t k callahan e knobloch symmetry breaking bifurcations on cubic lattices nonlinearity 10 1997 1179 1216 is used to make specific predictions about the formation of three dimensional patterns in two models of the turing instability the brusselator model and the lengyel epstein model spatially periodic patterns having the periodicity of the simple cubic sc face centered cubic fcc or body centered cubic bcc lattices are considered an efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae sc fcc double diamond hexagonal prism bcc and bcci states both models possess a special wavenumber k at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams this property is generic for two species chemical reaction diffusion models with a single activator and inhibitor 1999 elsevier science b v all rights reserved

equivariant intersection cohomology of semi stable points the main result of the paper is that the equivariant intersection cohomology of the semi stable points on a complex projective variety for the action of a complex reductive group may be determined from the equivariant intersection cohomology of the semi stable points for the action of a maximal torus it extends the work of brion who considered the smooth case using equivariant cohomology equivariant intersection cohomology is a theory due to brylinski and the second author as an application a surprising relation between the intersection cohomology of chow hypersurfaces is established in the last section of the paper in this paper we show that the equivariant intersection cohomology of the semi stable points on a complex projective variety for the action of a complex reductive group may be determined from the equivariant intersection cohomology of the semi stable points for the action of a maximal torus this extends the work of brion see br who considered the smooth case using equivariant cohomology the equivariant intersection cohomology we employ is due to brylinski and the

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

probabilistic computation of poiseuille flow velocity fields velocity fields for poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first passage times of random walks in the interior of the cross sectional domain d of the pipe this method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest these examples include a tube with fractal cross section and open channel flows the calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description e g torsional rigidity of rods membrane deflection the example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums

morphological transitions and bistability in turing systems it is well known that in two dimensions turing systems produce spots stripes and labyrinthine patterns and in three dimensions lamellar and spherical structures or their combinations are observed in this paper we study transitions between these states in both two and three dimensions first we derive the regions of stability for different patterns using nonlinear bifurcation analysis then we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects the method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction diffusion system

hecke correspondence stable maps and the kirwan desingularization we prove that the moduli space m0 0 n 2 of stable maps of degree 2 to the moduli space n of rank 2 stable bundles of flxed odd determinant ox ix over a smooth projective curve x of genus g 3 has two irreducible components which intersect transversely one of them is kirwan s partial desingularization e mx of the moduli space mx of rank 2 semistable bundles with determinant isomorphic to ox yix for some y 2 x the other component is the partial desingularization of the git quotient phom sl 2 w pgl 2 for a vector bundle w r 1 l i2 ix of rank g over the jacobian of x we also show that the hilbert scheme h the chow scheme c of conics in n and m0 0 n 2 are related by explicit contractions