chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

on the energy spectrum and parameter spaces of classicalcp n models it is known that a large class of smooth solutions ofcpn models can be constructed starting from holomorphic maps of an algebraic curve into complex projective spaces here we apply results from algebraic geometry to describe the energy spectrum and the parameter spaces for such models

tangent cones to discriminant loci for families of hypersurfaces a deformation of a variety with nonisolated hypersurface singularities such as a projective hypersurface or a theta divisor of an abelian variety determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation the basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation a detailed study of the method which emerged from interpreting andreotti mayer s work on theta divisors in terms of schlessinger s deformation theory of singularities is given along with examples applications and multiplicity formulas introduction this paper is about the geometry associated to deformations of nonisolated hypersurface singularities the basic principle is that a deformation of a hypersurface determines a projective model of the singular locus whose dual variety is the projectivized tangent cone to the discriminant this includes an infinitesimal form of bertini s theorem for a linear system of divisors and a converse to andreotti mayer s condition for double points of theta divisors now we describe the main results more precisely working in the category of complex analytic spaces let x be a space which can locally be written as a hypersurface of a nonsingular space and let x s be a deformation of x ir 1 0 then t1 the sheaf of first order deformations of x is a line bundle on sg x the singular locus of x and ir determines a kodaira spencer linear map to s k h0 t1 i e a linear system on the singular locus if x and s are nonsingular this linear system has no base locus and the corresponding morphism sg x pto s satisfies 0 p ir p tpx for p e sg x hence is called the gauss map of ir for a projective hypersurface of degree d x is the d fold veronese map restricted to the singular locus and for the theta divisor of an abelian variety 0 is the rational map assigning to a double point its quadric tangent cone assuming the restriction of ir to the critical locus c c x is proper consider the discriminant locus d ir c with its reduced analytic structure in addition to the above suppose finally that c is nonsingular of the expected codimension in x duality theorem there is a finite collection of 8ubvarietie8 zz of sg x which is intrinsic to the analytic space sg x and includes the irreducible components received by the editors march 31 1987 the contents of this paper were presented october 18 1986 by the first author in the special session on singularities and algebraic geometry at the charlotte north carolina meeting of the ams 1980 mathematics subject classification 1985 revision primary 14d15 14n05 secondary 32g11 53c57 d1988 american mathematical society 0002 9947 88 1 00 25 per page

comparison of numerical methods for the calculation of two dimensional turbulence the estimate derived by henshaw kreiss and reyna for the smallest scale present in solutions of the two dimensional incompressible navier stokes equations is employed to obtain convergent pseudospectral approximations these solutions are then compared with those obtained by a number of commonly used numerical methods if the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero the chopped solution differs considerably from the convergent solution even at early times in the case that the regular viscosity is replaced by a hyperviscosity term i e the square of the laplacian we also derive an estimate for the smallest scale present if the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution the hyperviscosity solution is also completely different from the convergent solution if we tune the hyperviscosity coefficient then the solutions are similar in amplitude or phase but not both the solution obtained by a second order difference method with twice the number of points as the pseudospectral model or a fourth order difference method with the same number of points as the pseudospectral model is essentially identical to the convergent solution this is reasonable since most of the energy of the solution is contained in the lower part of the spectrum

existence of periodic travelling wave solutions of non autonomous reaction diffusion equations with lambda omega type abstract periodic travelling wave solutions of reaction diffusion equations were studied by many authors the type reaction diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by kopell and howard in 1973 there are now similar systems which are investigated by means of autonomous dynamics in contrast there are few papers which are concerned with non autonomous cases for this reason we apply mawhin s continuation theorem to derive the existence of periodic travelling wave solutions for non autonomous systems and we describe the disappearance of periodic travelling wave solutions under special situations our main result is also illustrated by examples

singular initial data and uniform global bounds for the hyper viscous navier stokes equation with periodic boundary conditions abstract in the hyper viscous navier stokes equations of incompressible flow the operator a is replaced by a a b aa ba for real numbers a b with 1 and b 0 we treat here the case a 0 and equip a and hence a a b with periodic boundary conditions over a rectangular solid r n for initial data in l p with n 2p 1 2 we establish local existence and uniqueness of strong solutions generalizing a result of giga miyakawa for 1 and b 0 specializing to the case p 2 which holds a particular physical relevance in terms of the total energy of the system it is somewhat interesting to note that the condition n 4 1 2 is sufficient also to establish global existence of these unique regular solutions and uniform higher order bounds for the borderline case n 4 1 2 we generalize standard existing for n 3 folklore results and use energy techniques and gronwall s inequality to obtain first a time dependent h bound and then convert to a time independent global exponential h bound this is to be expected given that uniform bounds already exist for n 2 1 6 pp 78 79 and the folklore bounds already suggest that the n 4 1 2 cases for n 3 should behave as well as the n 2 case what is slightly less expected is that the n 3 cases are easier to prove and give better bounds e g the uniform bound for n 3 depends on the square of the data in the exponential rather than the fourth power for n 2 more significantly for n 4 1 2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform l2 estimate and are algebraic in terms of the uniform l2 bounds on the initial and forcing data the integer powers on the square of the data increase without bound as n 4 1 2 thus anticipating the exponential bound in the borderline case n 4 1 2 we prove our results for the case a 1 and b 0 the general case with a 0 and b 0 can be recovered by using norm equivalence we note that the hyperviscous navier stokes equations have both physical and numerical application

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

hopf bifurcation of reaction diffusion and navier stokes equations under discretization the long time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a hopf bifurcation is studied in this paper the framework includes reaction diffusion and incompressible navier stokes equations it is shown that the phase portrait of a supercritical hopf bifurcation is correctly represented by runge kutta time discretization in particular the bifurcation point and the hopf orbits are approximated with higher order a basic tool in the analysis is the reduction of the dynamics to a two dimensional center manifold a large portion of the paper is therefore concerned with studying center manifolds of the discretization

inertial manifolds for the hyperviscous navier stokes equations abstract we prove the existence of inertial manifolds for the incompressible hyperviscous navier stokes equations on the two or three dimensional torus u t u u u p f t x r t d div u 0 where d 2 or 3 and 3 2 since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions we employ the spatial averaging method introduced by mallet paret and sell in 26

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m

on the evolution of periodic plane waves in reaction diffusion systems of l w type lambda omega systems are a class of simple reaction diffusion equations with a limit cycle in the reaction kinetics the author considers the solution of the system given by lambda r lambda 0 r p omega r omega 0 r p on a semi infinite spatial domain with initial data decaying exponentially across the domain numerical evidence is presented showing that this initial condition induces a wave front moving across the domain with periodic plane waves behind the front these periodic waves can move in either direction depending on the parameter values the author uses intuitive criteria to derive an expression for the speed of the advancing front and by reducing the system to ordinary differential equations for similarity solutions the amplitude and speed of the periodic plane waves are determined the speed of these periodic waves varies continuously with the initial data perturbation theory is then used to obtain an analytical approximation to the solutions these anal

multiscale fractional step schemes for the numerical simulation of the rotating shallow water flows with complex periodic topography in this paper we study several multiscale fractional step schemes for the numerical solution of the rotating shallow water equations with complex topography we consider the case of periodic boundary conditions f plane model spatial discretization is obtained using a fourier spectral galerkin method for the schemes presented in this paper we consider two approaches the first approach multiscale schemes is based on topography scale separation and the numerical time integration is function of the scales the second approach is based on a splitting of the operators and the time integration method is function of the operator considered fractional step schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme with these multiscale fractional step schemes the objective is to propose new schemes giving numerical results similar to those obtained using only one uniform fine grid n nn n and a time step t but with a cpu time near the cpu time needed when using only one coarse grid n1 n1n1 n1 n1 t t t

towards large eddy and direct simulation of complex turbulent flows abstract recent advances in the methodology for direct numerical simulation of turbulent flows and some of the current applications are reviewed it is argued that high order finite difference schemes yield solutions with comparable accuracy to the spectral methods with the same number of degrees of freedom the effects of random inflow conditions on the downstream evolution of turbulence are discussed

a refined brill noether theory over hurwitz spaces let f c rightarrow mathbb p 1 be a degree k genus g cover the stratification of line bundles l in mathrm pic d c by the splitting type of f l is a refinement of the stratification by brill noether loci w r d c we prove that for general degree k covers these strata are smooth of the expected dimension in particular this determines the dimensions of all irreducible components of w r d c for a general k gonal curve there are often components of different dimensions extending results of pflueger adv math 312 46 63 2017 and jensen and ranganathan brill noether theory for curves of a fixed gonality arxiv 1701 06579 2017 the results here apply over any algebraically closed field

existence and uniqueness of solutions for the navier stokes equations with hyperdissipation abstract the navier stokes equations with hyperdissipation are used to numerically simulate turbulent flows while numerical aspects of this model are well understood its analytic properties have not been thoroughly explored in this article we prove the existence and uniqueness of strong solutions to these equations on t 2 with periodic boundary conditions our proof extends the methods of cannone 1995 and cannone and meyer 1995 to the discrete setting and relies on littlewood paley theory and techniques from multiple fourier series

multigrid algorithms for a vertex centered covolume method for elliptic problems we analyze v cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem as an application we study the convergence of multigrid algorithms for a covolume method or a vertex centered finite volume element method for variable coefficient elliptic problems on polygonal domains as in standard finite element methods the v cycle algorithm with one pre smoothing converges with a rate independent of the number of levels various types of smoothers including point or line jacobi and gauss seidel relaxation are considered

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

length scale of interaction in spatiotemporal chaos extensive systems have no long scale correlations and behave as a sum of their parts various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction diffusion networks information about network size boundary condition or abnormalities in network topology gets scrambled in spatiotemporal chaos and the attenuation of information provides such characteristic length scales space time information flow associated with the recovery of spatiotemporal chaos from finite perturbations a concept somewhat opposite to the paradigm of lyapunov exponents defines another characteristic length scale high precision computational studies of asymptotic spatiotemporal chaos in the complex ginzburg landau system and transient spatiotemporal chaos in the gray scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction preliminary studies demonstrate the relevance of these length scales for stable chaos

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

complex ginzburg landau equation in the presence of walls and corners we investigate the influence of walls and corners with dirichlet and neumann boundary conditions in the evolution of two dimensional autooscillating fields described by the complex ginzburg landau equation analytical solutions are found and arguments provided to show that dirichlet walls introduce strong selection mechanisms for the wave pattern corners between walls provide additional synchronization mechanisms and associated selection criteria the numerical results fit well with the theoretical predictions in the parameter range studied

chern schwartz macpherson classes and the euler characteristic of degeneracy loci and special divisors this concept overlaps a large family of interesting varieties for example the varieties of special divisors studied in section 3 several authors have worked out explicit formulas for the euler characteristic of dr 9 in terms of different cohomological and numerical invariants under the assumption that x is nonsingular and v is appropriately general for instance if 0 is a section of a vector bundle then the formulas for the euler characteristic x do fp were given by hirzebruch h and navarro aznar n if dr is a curve or a surface in a nonsingular x some explicit formulas for x dr v were given by harris and tu h t in terms of the chern classes of e f and x but under the extra assumption dr 1 v 0 which implies that dr q is nonsingular in loc cit the authors also posed the problem of finding a general formula for x dr o if such exists under the assumption dr 1 0 or even stronger without this assumption the first problem was solved positively by the second named author prl proposition 5 7 by the use of polynomials universally supported on degeneracy

direct limits of infinite dimensional lie groups compared to direct limits in related categories let g be a lie group which is the union of an ascending sequence g1 g2 of lie groups all of which may be infinite dimensional we study the question when g limgn in the category of lie groups topological groups smooth manifolds respectively topological spaces full answers are obtained for g the group diffc m of compactly supported c diffeomorphisms of a compact smooth manifold m and for test function groups cc m h of compactly supported smooth maps with values in a finite dimensional lie group h we also discuss the cases where g is a direct limit of unit groups of banach algebras a lie group of germs of lie group valued analytic maps or a weak direct product of lie groups

multilevel schemes for the shallow water equations in this paper we study a number of multilevel schemes for the numerical solution of the shallow water equations new schemes and new perspectives of known schemes are examined we consider the case of periodic boundary conditions spatial discretization is obtained using a fourier spectral galerkin method for the time integration two strategies are studied the first one is based on scale separation and we choose the time scheme explicit or semi implicit as a function of the spatial scales multilevel schemes the second approach is based on a splitting of the operators and we choose the time integration method as a function of the operator considered multistep or fractional schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme and with the semi implicit scheme leap frog scheme with crank nicholson scheme for the gravity terms both computed with a similar mesh the drawback of the explicit reference scheme being the numerical stability constraint on the time step and the drawback of the semi implicit scheme being the dispersive error the aim with the new schemes is to obtain schemes with less dispersive error than the semi implicit scheme and with better stability properties than the explicit reference scheme the numerical results obtained show that the schemes proposed allow one to reduce the dispersive error and to increase the numerical stability at reduced cost

secondary structures in a one dimensional complex ginzburg landau equation with homogeneous boundary conditions experiments in extended systems such as the counter rotating couette taylor flow or the taylor dean flow system have shown that patterns with vanishing amplitude may exhibit periodic spatio temporal defects for some range of control parameters these observations could not be interpreted by the complex ginzburg landau equation cgle with periodic boundary conditions we have investigated the one dimensional cgle with homogeneous boundary conditions we found that in the benjamin feir stable region the basic wave train bifurcates to state with periodic spatio temporal defects the numerical results match the observations quite well we have built a new state diagram in the parameter plane spanned by the criticality or equivalently the linear group velocity and the nonlinear frequency detuning

ulrich sheaves and higher rank brill noether theory abstract we show that the existence of an ulrich sheaf on a projective variety x p n is equivalent to the solution of a possibly higher rank brill noether problem for a curve on x that is rarely general in moduli in addition we exhibit a large family of curves for which this brill noether problem admits a solution and we show that existence of an ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor

shadowing for nonautonomous parabolic problems with applications to long time error bounds shadowing results for time discretizations of nonautonomous parabolic evolution equations z t a t z t f t are given the analysis is performed in an abstract banach space setting of uniformly sectorial operators a t under natural assumptions on the operators a t it is shown that the backward euler discretization of a smooth solution obtained with constant stepsize h is ch shadowed by a true solution of the system and vice versa every smooth trajectory is ch shadowed by a suitable backward euler discretization this validates numerical computations over long time intervals it is further explained how these results extend to higher order runge kutta methods

linked symplectic forms and limit linear series in rank 2 with special determinant abstract we generalize the prior linked symplectic grassmannian construction applying it to prove smoothing results for rank 2 limit linear series with fixed special determinant on chains of curves we apply this general machinery to prove new results on nonemptiness and dimension of rank 2 brill noether loci in a range of degrees

multigrid methods for parabolic distributed optimal control problems multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated accuracy properties of finite difference approximation are discussed and validated two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies semicoarsening and standard coarsening the resulting multigrid algorithms show robustness with respect to changes of the value of v the weight of the cost of the control is sufficiently small fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters results of numerical experiments are reported that demonstrate sharpness of fourier analysis estimates a multigrid algorithm that solves optimal control problems with box constraints on the control is considered

linear series with cusps and n fold points a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim vr h x d dim v 1 the linear series has a cuspofordere atapoint p if dim frch0 xs e l p dim z l linear series with cusps and n fold points are shown to exist if certain inequalities are satisfied the dimensions of the families of linear series with cusps are determined for general curves introduction throughout this paper we work over the complex numbers c let x be a smooth projective curve of genus g a gd on x is a linear series of degree d and dimension r on x i e a pair v s consisting of a line bundle s of degree d on x and an r l dimensional subspace vof h x s when p g r d g r 1 g r d o then x has gd s and the family of gd s on x forms a projective scheme gd x of dimension p g r d furthermore if x is general in moduli then gd x is smooth of dimension p g r d acgh definitic n if d is an effective divisor of degree n 2 on x we say that a gd v s has an n fold point along d if dim v n h x 2 d r we say that a gd has a cusp of order e at a point p if it has an e l fold point along the divisor e 1 p the aim of this paper is to prove the following i if g n p g r d n 1 r n o and p g r 1 d n o then there exists a gd on x with an n fold point along some divisor d of degree n ii if p g r d n 1 r o p g r 1 d n 0 and d is any divisor of degree n on x then there exists a gd on x with an n fold point along d iii if p g r d er 1 0 and p g r 1 d e 1 o then there exists a gd on x with a cusp of order e at some point p iv if x is general in moduli then the family of gd s on x with a cusp of order e has dimension p g r d er 1 if it is nonempty assertion iv has been proved independently by marc coppens c it is easy to see that the hypotheses p g r 1 d n 0 in assertions i and ii and p g r 1 d e 1 0 in assertion iii are necessary if x is general in moduli received by the editors january 20 1987 1980 mclthemcltlcs subject clclsslficcltion 1985 revision primary 14h10 1987 american mathematical society 0002 9947 87 1 00 25 per page

vanishing theorems for ample vector bundles the main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety in particular we extend classical theorems of griffiths and le potier to the whole dolbeault cohomology prove a variant of an uncorrect conjecture of sommese and answer a question of demailly as an application we prove conjectures of debarre and kim for branched coverings of grassmannians and extend a well known barth lefschetz type theorem for branched covers of projective spaces due to lazarsfeld we also obtain new restriction theorems for certain degeneracy loci

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

high resolution conjugate filters for the simulation of flows this paper proposes a hermite kernel realization of the conjugate filter oscillation reduction cfor scheme for the simulation of fluid flows the hermite kernel is constructed by using the discrete singular convolution dsc algorithm which provides a systematic generation of low pass filter and its conjugate high pass filters the high pass filters are utilized for approximating spatial derivatives in solving flow equations while the conjugate low pass filter is activated to eliminate spurious oscillations accumulated during the time evolution of a flow as both low pass and high pass filters are derived from the hermite kernel they have similar regularity time frequency localization effective frequency band and compact support fourier analysis indicates that the cfor hermite scheme yields a nearly optimal resolution and has a better approximation to the ideal low pass filter than previous cfor schemes thus it has better potential for resolving natural high frequency oscillations from a shock extensive one and two dimensional numerical examples including both incompressible and compressible flows with or without shocks are employed to explore the utility test the resolution and examine the stability of the present cfor hermite scheme small ratio of point perwavelength ppw is achieved in advancing a wavepacket and resolving a shock entropy wave interaction the present results for the advection of an isentropic vortex compare very favorably to those in the literature

a fokker planck control framework for multidimensional stochastic processes an efficient framework for the optimal control of probability density functions pdfs of multidimensional stochastic processes is presented this framework is based on the fokker planck equation that governs the time evolution of the pdf of stochastic processes and on tracking objectives of terminal configuration of the desired pdf the corresponding optimization problems are formulated as a sequence of open loop optimality systems in a receding horizon control strategy many theoretical results concerning the forward and the optimal control problem are provided in particular it is shown that under appropriate assumptions the open loop bilinear control function is unique the resulting optimality system is discretized by the chang cooper scheme that guarantees positivity of the forward solution the effectiveness of the proposed computational framework is validated with a stochastic lotka volterra model and a noised limit cycle model

analytic mappings between lb spaces and applications in infinite dimensional lie theory we give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces this tool is then used to construct new classes of real and complex infinite dimensional lie groups the group diffgerm k x of germs of analytic diffeomorphisms around a compact set k in a banach space x and the group s n n gn where the gn are banach lie groups

the surface c c on jacobi varieties and 2nd order theta functions preprint enviat per a la seva publicacio en una revista cientifica acta mathematica volume 157 1986 1 22 https projecteuclid org euclid acta 1485890436

codimension and connectedness of degeneracy loci over local rings we deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules f m rightarrow n over a local noetherian ring a mathfrak m we show for instance that the expected determinantal bounds on the dimension of the t th degeneracy locus of f hold if f in mathfrak m mathrm hom m n and that this degeneracy locus is connected in the expected dimension provided hat a is a domain

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

continuity of bilinear maps on direct sums of topological vector spaces abstract we prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces as a first application we get a new proof for the fact due to hirai et al 2001 that the map f c c r n c c r n c c r n taking a pair of test functions to their convolution is continuous the criterion also allows an open problem by k h neeb to be solved if e is a locally convex space regard the tensor algebra t e j n 0 t j e as the locally convex direct sum of projective tensor powers of e we show that t e is a topological algebra if and only if every sequence of continuous seminorms on e has an upper bound in particular if e is metrizable then t e is a topological algebra if and only if e is normable also t e is a topological algebra if e is dfs or k

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

the existence of spiral waves in an oscillatory reaction diffusion system rotating waves are proven to exist on the unit disk for an oscillatory reaction diffusion equation with neuman boundary conditions the method of proof relies on a two parameter shooting argument for the ensemble frequency and the radial derivative of the magnitude numerical solutions indicate that the waves are stable if the diffusion is sufficiently small it is also shown that these solutions cease to exist for large diffusion the origin of the rotating waves is discussed key words reaction diffusion equations spiral waves oscillatory media

algebraic cycles on prym varieties this article proposes a generalization of tautological rings introduced by beauville and moonen for jacobians the main result is that under certain hypotheses the special subvarieties of prym varieties are algebraically equivalent and their classes belong to the tautological ring

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

a tropical proof of the brill noether theorem we produce brill noether general graphs in every genus confirming a conjecture of baker and giving a new proof of the brill noether theorem due to griffiths and harris over any algebraically closed field

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

the severi bound on sections of rank two semistable bundles on a riemann surface let c be a compact riemann surface of genus g 1 for a line bundle l on c of degree d with 0 d 2g 2 the classical theorem of clifford states that the dimension h 0 l of the space of global sections of l satisfies h 0 l d 2 1 it is also quite easy to list the cases which give equality apart from the trivial bundle and the canonical bundle these exist only on hyperelliptic curves this theorem can be extended to a semistable bundle e on c of rank n and degree d with 0 d n 2g 2 when it states that h 0 e d 2 n now let e be a semistable bundle of rank 2 in this case the authors obtain a more refined result depending on the segre invariant s e which is defined as the minimum value of dege l degl for l a line subbundle of e in particular the semistability condition becomes s e 0 the main theorem is that if dege d and s d 4g 4 s then h 0 e d s 2 where 1 or 2 note that d s is always even in a previous paper e arrondo and the second author conjectured this inequality with 1 with a small number of exceptions similar to those for rank 1 however this is not correct and 2 is sometimes necessary the authors call the bundles which attain this bound severi bundles the number is obtained as follows we define polynomials k r n n the krawtchouk polynomials by means of the generating function r k r n n z r 1 z n 1 z n n let r d s 2 1 then we can take 1 if k r g 2g s 0 and 2 otherwise the connection with krawtchouk polynomials is intriguing since these are used in coding theory and recent work of t johnsen has established links between coding theory and semistable bundles of rank 2 on riemann surfaces these links make essential use of the segre invariant s the key to the proof of the inequality is the following lemma let e be a bundle of rank 2 and degree d on a curve of genus g having at least r 1 g independent sections if k r g 2g 2r 2 d 0 then there is a nonzero section of e vanishing at r points this is proved using a computation of chern classes formal properties of k r n n and the porteous formula the authors give examples to show that for any allowable values of g s d the bound 1 can be attained in fact on a hyperelliptic curve a good deal is known about the vanishing of k r n n thus providing a restricted list of candidates for the invariants of severi bundles but these values do not all arise from actual bundles the severi bundles with d s 0 are all known and the authors compute those with d s 2 and d s 4 in particular the bundles obtained by i grzegorczyk ulam quart 3 1996 no 2 41 ff approx 3 pp electronic are severi bundles with g 5 s 4 d 8 the authors also obtain sharp bounds for the numbers of independent sections of non semistable bundles

the brill noether curve and prym tyurin varieties we prove that the jacobian of a general curve c of genus g 2a 1 with a ge 2 can be realized as a prym tyurin variety for the brill noether curve w 1 a 2 c as consequence of this result we are able to compute the class of the sum of secant divisors of the curve c embedded with a complete linear series g a 1 3a 2

limit linear series and ranks of multiplication maps we develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus 1 curves we use this approach to prove a purely elementary criterion for proving cases of the maximal rank conjecture and then apply the criterion to several ranges of cases giving a new proof of the case of quadrics and also treating several families in the case of cubics our proofs do not require restrictions on direction of approach so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails

solution of lambda omega systems theta schemes and multigrid methods the numerical solution of a time dependent reaction diffusion lambda omega system discretized by theta schemes and finite differences is considered stability and accuracy of finite difference theta schemes for this system are established to solve the time implicit evolution equations a nonlinear multigrid method is applied the convergence properties of this solver are investigated considering a linearized lambda omega model

the connectedness of symmetric and skew symmetric degeneracy loci even ranks a degeneracy locus is the set of points where a vector bundle map has rank at most a given integer such a set is symmetric or skew symmetric according as whether the vector bundle map is symmetric or skew symmetric we prove a connectedness result first conjectured by fulton and lazarsfeld for skew symmetric degeneracy loci and for symmetric degeneracy loci of even ranks

accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system the finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered stability of the finite difference optimality system and optimal order error estimates in the discrete l2 norm and in the discrete h1 norm under minimum smoothness requirements on the exact solution are proved sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local fourier analysis a multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system