existence and stability of stationary profiles of the lw scheme in this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws when a difference scheme introduces artificial numerical diffusion for example the lax friedrichs scheme we experience smearing of the shocks whereas when a scheme introduces numerical dispersion for example the lax wendroff scheme we experience oscillations which decay exponentially fast on both sides of the shock in his dissertation gray jennings studied approximation by monotone schemes these contain artificial viscosity and are first order accurate they are known to be contractive in the sense of any lp norm jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes here we study similar questions for the lax wendroff scheme without artificial viscosity this is a nonmonotone second order accurate scheme we prove existence of a one parameter family of stationary profiles we also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm the proof relies on studying the linearized lax wendroff scheme

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

equivalences of derived catgories of sheaves on smooth stacks we extend orlov s representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities

decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws this paper is to study the decay rate for perturbations of stationary discrete shocks for the lax friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method if the summation of the initial perturbation over j is small and decays at the algebraic rate as j then the solution approaches the stationary discrete shock profiles at the corresponding rate as n this rate seems to be almost optimal compared with the rate in the continuous counterpart proofs are given by applying the weighted energy integration method to the integrated scheme of the original one the selection of the time dependent discrete weight function plays a crucial role in this procedure

a projection method for convex constrained monotone nonlinear equations with applications in this paper we present a projection method to solve monotone nonlinear equations with convex constraints this method can be viewed as an extension of cg descent method which is one of the most effective conjugate gradient methods for solving unconstrained optimization problems because of derivative free and low storage the proposed method can be used to solve large scale nonsmooth monotone nonlinear equations its global convergence is established under some appropriate conditions preliminary numerical results show that the proposed method is effective and promising moreover we also successfully use the proposed method to solve the sparse signal reconstruction in compressive sensing

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

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

derived categories of quadric fibrations and intersections of quadrics abstract we construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the clifford algebras on the base corresponding to this quadric fibration generalizing the kapranov s description of the derived category of a single quadric as an application we verify that the noncommutative algebraic variety p s 2 w b 0 where b 0 is the universal sheaf of even parts of clifford algebras is homologically projectively dual to the projective space p w in the double veronese embedding p w p s 2 w using the properties of the homological projective duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

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

on the stability of finite element methods for shock waves this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form 1 1 u t f u x u xx x 2 r i t 0 u 2 r i u delta 0 u 0 delta we treat systems which are strictly hyperbolic such systems possess a smooth travelling wave solution which is called a viscous p shock wave solution u x t oe x gamma oet x sigma1 oe x u sigma provided that the shock strength ffl j ju gamma u gamma j is small 19 the constant states u sigma and the wave speed oe are related by the rankine hugoniot condition 1 3a f u gamma gamma f u oe u gamma gamma u

a conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing abstract cg descent is a state of the art algorithm to solve large scale unconstrained minimization problems however research activities on cg descent in some other scenarios are relatively fewer in this paper by combining with the projection method of solodov and svaiter we extend cg descent to solve large scale nonlinear convex constrained monotone equations the proposed method does not require the jacobian information even though it does not store any matrix at each iteration it thus has the potential to solve large scale non smooth problems under some mild conditions we show that the proposed method converges globally primary numerical results illustrate that the proposed method works quite well moreover we also extend this method to solve the l 1 norm regularized problems to decode a sparse signal in compressive sensing performance comparisons show that the proposed method is practical efficient and competitive with the compared ones

nonlinear stability of discrete shocks for systems of conservation laws in this paper we study the asymptotic nonlinear stability of discrete shocks for the lax friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the lp norm for all p 1 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the lax friedrichs scheme for the single shock solution of system of hyperbolic conservation laws if the riemann solution corresponding to the given far field states is a superposition of m single shocks from each characteristic family we show that the corresponding multiple discrete shocks are nonlinearly stable in l p p 2 these results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

spectral residual method without gradient information for solving large scale nonlinear systems of equations a fully derivative free spectral residual method for solving largescale nonlinear systems of equations is presented it uses in a systematic way the residual vector as a search direction a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior the global convergence analysis of the combined scheme is presented an extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well known newton krylov methods for largescale problems is also presented

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

equivalences of twisted k3 surfaces we prove that two derived equivalent twisted k3 surfaces have isomorphic periods the converse is shown for k3 surfaces with large picard number it is also shown that all possible twisted derived equivalences between arbitrary twisted k3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a k3 surface

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

a refined derived torelli theorem for enriques surfaces we prove that two general enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their kuznetsov components are equivalent we apply the same techniques to give a new simple proof of a conjecture by ingalls and kuznetsov relating the derived categories of the blow up of general artin mumford quartic double solids and of the associated enriques surfaces

an efficient implementation of merrill s method for sparse or partially separable systems of nonlinear equations when merrill s method without an extra dimension is applied to solving sparse or partially separable systems of nonlinear equations the computational efficiency can be further improved i e a larger piece of linearity can be traversed in one step by using a suitable linear system one of the linear systems is updated and the corresponding technique is shown to update information about the linear system and the large piece in the implementation of the method some numerical results of the method support the claim that it is efficient also some mistakes from a previous paper in which the main technique exploiting sparsity was proposed are corrected

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

a globally convergent bfgs method for nonlinear monotone equations without any merit functions r since 1965 there has been significant progress in the theoretical study on quasi newton methods for solving nonlinear equations especially in the local convergence analysis however the study on global convergence of quasi newton methods is relatively fewer especially for the bfgs method to ensure global convergence some merit function such as the squared norm merit function is typically used in this paper we propose an algorithm for solving nonlinear monotone equations which combines the bfgs method and the hyperplane projection method we also prove that the proposed bfgs method converges globally if the equation is monotone and lipschitz continuous without differentiability requirement on the equation which makes it possible to solve some nonsmooth equations an attractive property of the proposed method is that its global convergence is independent of any merit function we also report some numerical results to show efficiency of the proposed method

l 1 stability of stationary discrete shocks the nonlinear stability in the l sup p norm p ge 1 of stationary weak discrete shocks for the lax friedrichs scheme approximating general m times m systems of nonlinear hyperbolic conservation laws is proved provided that the summations of the initial perturbations equal zero the result is proved by using a both weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme 13 refs

convergence rates to the discrete travelling wave for relaxation schemes this paper is concerned with the asymptotic convergence of nu merical solutions toward discrete travelling waves for a class of relaxation nu merical schemes approximating the scalar conservation law it is shown that if the initial perturbations possess some algebraic decay in space then the nu merical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time provided the sums of the initial perturbations for the u component equal zero a polynomially weighted l 2 norm on the perturba tion of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate

convergence rates to discrete shocks for nonconvex conservation laws this paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the lax friedrichs scheme approximating non convex scalar conservation laws we assume that the discrete initial data tend to constant states as j rightarrow pm infty respectively and that the riemann problem for the corresponding hyperbolic equation admits a stationary shock wave if the summation of the initial perturbation over infty j is small and decays with an algebraic rate as j rightarrow infty then the perturbations to discrete shocks are shown to decay with the corresponding rate as n rightarrow infty the proof is given by applying weighted energy estimates a discrete weight function which depends on the space time variables for the decay rate and the state of the discrete shocks in order to treat the non convexity plays a crucial role

the large time behavior of the scalar genuinely nonlinear lax friedrichs scheme the lax friedrichs scheme approximating the scalar genuinely nonlinear conservation law u sub t f sub x u 0 where f u is say strictly convex double dot f dot a sub asterisk 0 is studied the divided differences of the numerical solution at time t do not exceed 2 t dot a sub asterisk to the 1 this one sided lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case in particular it is independent of the initial amplitude in sharp contrast to liner problems it guarantees the entropy compactness of the scheme in this case as well as providing a quantitive insight into the large time behavior of the numerical computation

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

a conjugate gradient projection method for solving equations with convex constraints abstract in this paper we propose a derivative free iterative method for a class of equations with convex constraints appearing in a variety of the practical problems such as compressing sensing fluid mechanics plasma physics nonlinear optics and solid state in the iteration our search direction can be viewed as an extension of a modified three term cg method by an appropriate line search and the projection step our method is convergent to the solution our method inherits the advantages of cg method and projection method and thus is suitable for solving large scale non smooth problem under the assumption that f is lipschitz continuous and satisfies a weaker condition of monotonicity our method is globally convergent numerical results show the efficiency of the proposed method