optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

impacts of noise on a class of partial differential equations abstract this paper is concerned with effects of noise on the solutions of partial differential equations we first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations then we prove that noise could induce singularities finite time blow up of solutions finally we show that a stochastic allen cahn equation does not have finite time singularities and the unique solution exists globally

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

on domination problems for permutation and other graphs abstract there is an increasing interest in results on the influence of restricting np complete graph problems to special classes of perfect graphs as e g permutation graphs it was shown that several problems restricted to permutation graphs are solvable in polynomial time 2 3 4 6 7 14 16 in this paper we give 1 i an algorithm with time bound o n 2 for the weighted independent domination problem on permutation graphs which is an improvement of the o n 3 solution given in 7 2 ii a polynomial time solution for the weighted feedback vertex set problem on permutation graphs 3 iii an investigation of weighted dominating clique problems for several graph classes including an np completeness result for weakly triangulated graphs as well as polynomial time bounds

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

a basis deficiency allowing variation of the simplex method for linear programming h one of the most important and fundamental concepts in the simplex methodology basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real world linear programming problems which are usually highly degen erate to dodge this difficulty we first generalize the basis to allow the deficient case characterized as one that has columns fewer than rows of the coefficient matrix variations of the primal and dual simplex procedures are then made and used to form a two phase method based on such a basis the number of whose columns varies dynamically in the solution process generally speaking the more degenerate a problem to be handled is the fewer columns the basis will have so thii renders the possibility of efficiently solving highly degenerate problems we also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments 1998 elsevier science ltd all rights reserved

convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh we study here the discretisation of the nonlinear hyperbolic equationu t div vf u 0 in 3 with given initial conditionu 0 u 0 in 2 wherev is a function from 2 to 2 such that divv 0 andf is a given nondecreasing function from to an explicit euler scheme is used for the time discretisation of the equation and a triangular mesh for the spatial discretisation under a usual stability condition we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation

stochastic conservation laws weak in time formulation and strong entropy condition abstract this article is an attempt to complement some recent developments on conservation laws with stochastic forcing in a pioneering development feng and nualart 8 have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition however the authors formulation of entropy inequalities are weak in space but strong in time in the absence of a priori path continuity for the solutions we take a critical outlook towards this formulation and offer an entropy formulation which is weak in time and weak in space

efficient algorithms for the minimum weighted dominating clique problem on permutation graphs abstract given a graph g v e with real weights assigned to its vertices a clique of g that also dominates its vertex set v is called a dominating clique dc of g given a permutation graph g with all its vertices having nonnegative weights and its permutation representation the problem addressed in this paper is that of finding any minimum weight dc of g we improve the existing o v 2 algorithm for this problem to o v log v the space complexity of our algorithm is o v we also present a v processor o log v time o v log v space parallel erew pram algorithm for this problem

a projective simplex method for linear programming linear programming problems with quite square coefficient matrix form a wide range of problems that are not amenable to existing algorithms the method proposed in this paper attacks such problems from the dual side alternatively arranging computations of the simplex method using the qr factorization in each iteration its tableau version handles an n m n 1 tableau rather than the m 1 n 1 conventional tableau where m and n are the numbers of rows and columns of the coefficient matrix in contrast to the simplex method where two m m systems are solved per iteration the new approach solves a single s s s n m system only it allows nonbasis deficiency and hence could reduce computational work dramatically a favorable complexity analysis is given for one of its implementations against its conventional counterpart in addition a new crash heuristic having a clear geometrical meaning towards an optimal basis is developed to provide good input we also report numerical results obtained from our trials to give an insight into the method s behavior

a projective simplex algorithm using lu decomposition abstract recently we proposed a so called projective simplex method which is amenable to linear programming problems with quite square coefficient matrix since it is based on qr decomposition however the method is not a suitable choice for large and sparse problems unless n m is far less than m where m and n are the numbers of rows and columns of the coefficient matrix respectively to dodge this flaw in this paper we propose a method using lu decomposition in contrast to the simplex method in which an m 1 n 1 tableau is used its tableau version handles an n m n 1 tableau in each iteration its revised version solves a single n m n m system only compared with the two m m systems solved in the revised simplex method a complexity analysis establishes its superiority over an implementation of the simplex method in the case of the coefficient matrix being not too flat of particular interest might be the introduction of deficient nonbasis via exploiting dual degeneracy to reduce computational work further an lu decomposition based crash heuristic is furnished to provide good input computational results are also reported to give an insight into its interesting and distinctive behavior

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

a dual approach to primal degeneracy the revised primal simplex algorithm in its simplest form has no defence against degeneracy various forms of the perturbation method are usually effective but most offer no guarantee of avoiding all degeneracy and can lead to numerical difficulties this paper presents a method that avoids cycling and circling by taking a dual approach

a primal deficient basis simplex algorithm for linear programming abstract the standard basis which plays a fundamental role in simplex methodology was recently extended to include a deficient case with fewer columns than rows by taking advantage of primal degeneracy computational results have been favorable with dense implementations in this paper we propose a primal simplex algorithm using sparse lu factors of deficient bases amenable to real world linear programming problems which are often degenerate or even highly degenerate the algorithm would solve them with potentially improved stability compared to the simplex algorithm the proposed algorithm has been implemented and tested on a set of 50 netlib test problems as well as a set of 15 much larger real world problems including 8 kennington and 5 bpmpd problems it significantly outperformed minos 5 3 in terms of both iteration counts and run time in particular these results reveal that there is no inevitable correlation between an algorithm s inefficiency and degeneracy contradicting common belief

weighted connected k domination and weighted k dominating clique in distance hereditary graphs a graph is distance hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph this paper presents efficient algorithms for solving the weighted connected k domination and the weighted k dominating clique problems in distance hereditary graphs

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

martingale and weak solutions for a stochastic nonlocal burgers equation on finite intervals abstract this work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal burgers equation on a one dimensional bounded domain the existence of a martingale solution is shown by using a galerkin approximation prokhorov s theorem and skorokhod s embedding theorem the same galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal burgers equation on a bounded domain

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

a revised dual projective pivot algorithm for linear programming we revise the dual projective pivot algorithm using sparse rectangular lu factors in each iteration the proposed algorithm solves at most three triangular systems while simplex algorithms solve four instead of the classical basis it uses a so called pseudobasis a rectangular matrix having fewer columns than rows thereby solving smaller linear systems with a potentially improved stability compared to simplex algorithms most importantly it generates good search directions at a low cost we report encouraging computational results on a set of 50 netlib standard test problems as well as a set of 15 much larger real world problems a code named rdppa 1 10 based on the proposed algorithm outperformed minos 5 51 significantly in terms of both iterations and run time in particular it appears that a high proportion of degenerate iterations need not imply many total iterations contradicting the common belief

densities for sdes driven by degenerate alpha stable processes in this work by using the malliavin calculus under hormander s condition we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated brownian motions moreover in a special degenerate case we also obtain the smoothness of the density in particular we obtain the existence of smooth heat kernels for the following fractional kinetic fokker planck nonlocal operator mathscr l alpha b delta alpha 2 mathrm v mathrm v cdot nabla x b x mathrm v cdot nabla mathrm v qquad x mathrm v in mathbb r d where alpha in 0 2 and b mathbb r d times mathbb r d to mathbb r d is smooth and has bounded derivatives of all orders

stochastic scalar conservation laws we introduce a notion of stochastic entropic solution a la kruzkov but with ito s calculus replacing deterministic calculus this results in a rich family of stochastic inequalities defining what we mean by a solution a uniqueness theory is then developed following a stochastic generalization of l1 contraction estimate an existence theory is also developed by adapting compensated compactness arguments to stochastic setting we use approximating models of vanishing viscosity solution type for the construction while the uniqueness result applies to any spatial dimensions the existence result in the absence of special structural assumptions is restricted to one spatial dimension only

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

regularization by 12 laplacian and vanishing viscosity approximation of hjb equations abstract we investigate the regularizing effect of adding small fractional laplacian with critical fractional exponent 1 2 to a general first order hjb equation our results include some regularity estimates for the viscosity solutions of such perturbations making the solutions classically well defined most importantly we use these regularity estimates to study the vanishing viscosity approximation to first order hjb equations by 1 2 laplacian and derive an explicit rate convergence for the vanishing viscosity limit

stochastic nonlocal conservation laws on whole space abstract after introducing an entropy formulation for stochastic nonlocal conservation laws via an integral formula we develop a uniqueness theory by using a doubling variables technique and an existence theory by using a vanishing viscosity method we further consider the continuous dependence of the entropy solution on initial data

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

cascading divide and conquer a technique for designing parallel algorithms techniques for parallel divide and conquer are presented resulting in improved parallel algorithms for a number of problems the problems for which improved algorithms are given include segment intersection detection trapezoidal decomposition and planar point location efficient parallel algorithms are algo given for fractional cascading three dimensional maxima two set dominance counting and visibility from a point all of the algorithms presented run in o log n time with either a linear or a sublinear number of processors in the crew pram model

scalar conservation laws with stochastic forcing we show that the cauchy problem for a randomly forced periodic multi dimensional scalar first order conservation law with additive or multiplicative noise is well posed it admits a unique solution characterized by a kinetic formulation of the problem which is the limit of the solution of the stochastic parabolic approximation

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems