statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

statistics of intense turbulent vorticity events we investigate statistical properties of vorticity fluctuations in fully developed turbulence which are known to exhibit a strong intermittent behavior taking as the starting point the navier stokes equations with a random force term correlated at large scales we obtain in the high reynolds number regime a closed analytical expression for the probability distribution function of an arbitrary component of the vorticity field the central idea underlying the analysis consists in the restriction of the velocity configurational phase space to a particular sector where the rate of strain and the rotation tensors can be locally regarded as slow and fast degrees of freedom respectively this prescription is implemented along the martin siggia rose functional framework whereby instantons and perturbations around them are taken into account within a steepest descent approach

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

mixing mir inequalities with two divisible coefficients this article is a polyhedral study of a generalization of the mixing set where two different divisible coefficients are allowed for the integral variables our results generalize earlier work on mixed integer rounding mixing and extensions they also directly apply to applications such as production planning problems involving lower bounds or start ups on production when these are modeled as mixed integer linear programs we define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed integer set we give a characterization of each of the maximal faces of the convex hull as well as a closed form description of its extreme points and rays and show how to separate over this set in mathcal o n log n finally we give several extended formulations of polynomial size and study conditions under which adding certain simple constraints on the integer variables preserves our main result

compact formulations as a union of polyhedra we explore one method for finding the convex hull of certain mixed integer sets the approach is to break up the original set into a small number of subsets find a compact polyhedral description of the convex hull of each subset and then take the convex hull of the union of these polyhedra the resulting extended formulation is then compact its projection is the convex hull of the original set and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation the approach is demonstrated on three different sets a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lot sizing and a single node flow model with divisible capacities that arises as a subproblem in network design

singularities and the distribution of density in the burgers adhesion model abstract we are interested in the tail behavior of the pdf of mass density within the one and d dimensional burgers adhesion model used e g to model the formation of large scale structures in the universe after baryon photon decoupling we show that large densities are localized near kurtoparabolic singularities residing on space time manifolds of codimension two d 2 or higher d 3 for smooth initial conditions such singularities are obtained from the convex hull of the lagrangian potential the initial velocity potential minus a parabolic term the singularities contribute universal power law tails to the density pdf when the initial conditions are random in one dimension the singularities are preshocks nascent shocks whereas in two and three dimensions they persist in time and correspond to boundaries of shocks in all cases the corresponding density pdf has the exponent 7 2 originally proposed by e et al phys rev lett 78 1997 1904 for the pdf of velocity gradients in one dimensional forced burgers turbulence we also briefly consider models permitting particle crossings and thus multi stream solutions such as the zel dovich approximation and the jeans vlasov poisson equation with single stream initial data they have singularities of codimension one yielding power law tails with exponent 3

lifted inequalities for 0 1 mixed integer programming superlinear lifting we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables through the lifting of continuous variables fixed at their upper bounds we introduce the concept of a superlinear inequality and show that in this case lifting is significantly simpler than for general inequalities we use the superlinearity theory together with the traditional lifting of 0 1 variables to describe families of facets of the mixed 0 1 knapsack polytope finally we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large

perturbation analysis for the matrix least squares problem a x b c let s and s be two sets of solutions to matrix least squares problem lsp a x b c and the perturbed matrix lsp a x b c respectively where a a a b b b c c c and a b c are all small perturbation matrices for any given x s we deduce general formulas of the least squares solutions x s that are closest to x under appropriated norms meanwhile we present the corresponding distances between them with the obtained results we derive perturbation bounds for the nearest least squares solutions at last a numerical example is provided to verify our analysis

a preliminary study of the burgers equation with symbolic computation a novel approach based on recursive symbolic computation is introduced for the approximate analytic solution of the burgers equation once obtained appropriate numerical values can be inserted into the symbolic solution to explore parametric variations the solution is valid for both inviscid and viscous cases covering the range of reynolds number from 500 to infinity whereas current direct numerical simulation dns methods are limited to reynolds numbers no greater than 4000 what further distinguishes the symbolic approach from numerical and traditional analytic techniques is the ability to reveal and examine direct nonlinear interactions between waves including the interplay between inertia and viscosity thus preliminary efforts suggest that symbolic computation may be quite effective in unveiling the anatomy of the myriad interactions that underlie turbulent behavior however due to the tendency of nonlinear symbolic operations to produce combinatorial explosion future efforts will require the development of improved filtering processes to select and eliminate computations leading to negligible high order terms indeed the initial symbolic computations present the character of turbulence as a problem in combinatorics at present results are limited in time evolution but reveal the beginnings of the well known saw tooth waveform that occurs in the inviscid case i e re future efforts will explore more fully developed 1 d flows and investigate the potential to extend symbolic computations to 2 d and 3 d potential applications include the development of improved subgrid scale sgs parameterizations for large eddy simulation les models and studies that complement dns in exploring fundamental aspects of turbulent flow behavior

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

instanton theory of burgers shocks and intermittency a lagrangian approach to burgers turbulence is carried out along the lines of the field theoretical martin siggia rose formalism of stochastic hydrodynamics we derive from an analysis based on the hypothesis of unbroken galilean invariance the asymptotic form of the probability distribution function of negative velocity differences the origin of burgers intermittency is found to rely on the dynamical coupling between shocks identified to instantons and noncoherent background fluctuations which char22 then char22 cannot be discarded in a consistent statistical description of the flow

phase diagram of optimal paths we show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media such as directed percolation invasion percolation and directed and nondirected spanning polymers we also introduce a simple and efficient algorithm which solves the d dimensional model numerically in o n 1 df d steps where df is the fractal dimension of the path using extensive simulations in two dimensions we identify the phase boundaries of the directed polymer universality class a new strong disorder phase occurs where the optimum paths are self affine with parameter dependent scaling exponents furthermore the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines

computation of system balancing transformations and other applications of simultaneous diagonalization algorithms an algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization the algorithm requires no squaring up or unnecessary matrix products various algorithmic aspects are discussed in detail a key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product other contragredient transformation applications are also described it is further shown that a similar approach may be taken involving the generalized singular value decomposition to the classical simultaneous diagonalization problem these svd based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems

universality classes for self similarity of noiseless multidimensional burgers turbulence and interface growth the present work is devoted to the evolution of random solutions of the unforced burgers and kpz equations in d dimensions in the limit of vanishing viscosity we consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell we show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function three classes of initial distribution leading to self similar evolution are identified a distributions with a power law tail b compactly supported potential c stretched exponential tails in class c we find that the mean potential mean height of the surface increases logarithmically with time and the turbulence energy e t mean square gradient of the surface decays as 1 t times a logarithmic correction in classes a and b we find that the changes in the mean potential and energy have a power law time dependence in class c the roughness of the surface measured by its mean square gradient may either decrease or increase with time we discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the burgers turbulence

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

the 0 1 knapsack problem with a single continuous variable constraints arising in practice often contain many 0 1 variables and one or a small number of continuous variables existing knapsack separation routines cannot be used on such constraints here we study such constraint sets and derive valid inequalities that can be used as cats for such sets as well for more general mixed 0 1 constraints specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable called the mixed 0 1 knapsack problem first different classes of facet defining inequalities are derived based on restriction and lifting the order of lifting particularly of the continuous variable plays an important role secondly we show that the flow cover inequalities derived for the single node flow set consisting of are flows into and out of a single node with binary variable lower and upper bounds on each are can be obtained from valid inequalities for the mixed 0 1 knapsack problem thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0 1 constraints initial computational results on a variety of problems are presented

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

a note on the componentwise perturbation bounds of matrix inverse and linear systems in this note we present sharp bounds for the componentwise perturbation of matrix inverse and linear systems especially for nonsingular m matrix we also use matrix derivatives to deduce the matrix componentwise condition number and present some numerical tests to show our results

an iteration method to solve multiple constrained least squares problems abstract in this paper we propose an iteration method to solve the multiple constrained least squares matrix problem we first transform the multiple constrained least squares matrix problems into the multiple constrained matrix optimal approximation problem and then we use the idea of dykstra s algorithm to derive the basic iterative pattern we observe that we only need to solve multiple single constrained least squares matrix problems at each iteration step of the proposed algorithm we give a numerical example to illustrate the effectiveness of the proposed method to solve the original problems also we give an example to illustrate that the method proposed by escalante and li to solve the single constrained least squares matrix problem is not correct

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

a study of the lot sizing polytope the lot sizing polytope is a fundamental structure contained in many practical production planning problems here we study this polytope and identify facet defining inequalities that cut off all fractional extreme points of its linear programming relaxation as well as liftings from those facets we give a polynomial time combinatorial separation algorithm for the inequalities when capacities are constant we also report computational experiments on solving the lot sizing problem with varying cost and capacity characteristics

mixing polyhedra with two non divisible coefficients we consider the mixed integer set x s x y in mathbb r times mathbb z n times mathbb z m s a 1 x j geq b j forall j in n 1 s a 2 y j geq d j j in n 2 where n 1 1 ldots n n 2 1 ldots m and a 1 a 2 in mathbb z setminus 0 this set may arise in a relaxation of mixed integer problems such as lot sizing problems we decompose x into a small number of subsets whose convex hull description is trivial the convex hull of x is equal to the closure of the convex hull of the union of those polyhedra using a projection theorem from balas discret appl math 89 3 44 1998 we obtain a compact characterization of the facets of the convex hull of x then by studying the projection cone we characterize all the facet defining inequalities of the convex hull of x in the space of the original variables each of those inequalities is either a mixed mir inequality gunluk and pochet in math programm 90 429 457 2001 or it is based on a directed cycle on a special bipartite graph when a 1 and a 2 are relative prime the convex hull of x is described by the mixed mir inequalities

decaying turbulence in the generalised burgers equation we consider the generalised burgers equation frac partial u partial t f u frac partial u partial x nu frac partial 2 u partial x 2 0 t geqq 0 x in s 1 where f is strongly convex and is small and positive we obtain sharp estimates for sobolev norms of u upper and lower bounds differ only by a multiplicative constant then we obtain sharp estimates for the dissipation length scale and the small scale quantities which characterise the decaying burgers turbulence i e the structure functions and the energy spectrum the proof uses a quantitative version of an argument by aurell et al j fluid mech 238 467 486 1992 note that we are dealing with decaying as opposed to stationary turbulence thus our estimates are not uniform in time however they hold on a time interval t1 t2 where t1 and t2 depend only on f and the initial condition and do not depend on the viscosity these results allow us to obtain a rigorous theory of the one dimensional burgers turbulence in the spirit of kolmogorov s 1941 theory in particular we obtain two results which hold in the inertial range on one hand we explain the bifractal behaviour of the moments of increments or structure functions on the other hand we obtain an energy spectrum of the form k 2 these results remain valid in the inviscid limit

on solutions of matrix equation axb cyd f in this paper the matrix equation with two unknown matrices x y of form axb cyd f is discussed by applying the canonical correlation decomposition ccd of matrix pairs we obtain expressions of the least squares solutions of the matrix equation and sufficient and necessary conditions for the existence and uniqueness of the solutions we also derive a general form of the solutions we also study the least squares hermitian skew hermitian solutions of equation axah cych f

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

ranks of least squares solutions of the matrix equation axb c for a complex matrix equation axb c we solve the following two problems 1 the maximal and minimal ranks of least square solution x to axb c and 2 the maximal and minimal ranks of two real matrices x 0 and x 1 in least square solution x x 0 ix 1 to axb c we also give a necessary and sufficient condition for matrix equations a ix ib i c i i 1 2 to have a common least square solution

polyhedra for lot sizing with wagner whitin costs we examine the single item lot sizing problem with wagner whitin costs over ann period horizon i e pt ht pt 1 fort 1 n 1 wherept ht are the unit production and storage costs in periodt respectively so it always pays to produce as late as possible

network formulations of mixed integer programs we consider mixed integer sets described by system of linear inequalities in which the constraint matrix a is totally unimodular the right hand side is arbitrary vector and a subset of the variables is required to be integer we show that the problem of checking nonemptiness of a set of this type is np complete even in the case in which the linear system describes mixed integer network flows with half integral requirement on the nodes this is in contrast to the case in which a is totally unimodular and contains at most two nonzeros per row in this case we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual network matrix with an integral right hand side vector the size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set when this number is polynomial in the dimension of a the extended formulation is of polynomial size if in addition the corresponding list of fractional parts can be computed efficiently then our result provides a polynomial algorithm for the optimization problem over these sets we show that there are instances for which this list is of exponential size and we also give conditions under which it is short and can be efficiently computed finally we show that these results provide a unified framework leading to polynomial size extended formulations for several generalizations of mixing sets and lot sizing sets studied in the last few years

perturbation of the least squares problem consider the least squares solutions to ax b ax b where a a a and b b b are perturbed from a b respectively in an earlier paper the author deduced the perturbation bounds for the solution when b r a in this paper we extend the results to cover more general cases when b r a and a is not of full rank

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

a primal all integer algorithm based on irreducible solutions this paper introduces an exact primal augmentation algorithm for solving general linear integer programs the algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities we prove that various versions of our algorithm are finite it is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively an implementation of the presented algorithms is given computational results for a number of hard 0 1 integer programs from the miplib demonstrate the practical power of the method

the perturbation of consistent least squares problems abstract suppose that the linear system ax b is consistent and a a a b b b are perturbed from a b respectively where a and are m n matrices b and b are m dimensional vectors let a f e a 2 b 2 e b 2 and a 2 a 2 be the condition number of a it is shown that if e 1 then the least squares solution x of a x b is close to a solution x of ax b even when a is not of full rank and rank rank a

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

probability distribution functions for the random forced burgers equation statistical properties of solutions of the random forced burgers equation have been a subject of intensive studies recently see refs 1 6 of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces 1 it is also hoped that the field theoretic techniques developed for the burgers equation will eventually be useful for understanding more complex phenomena such as turbulence in this paper we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced burgers equation we will consider the problem

a dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos capturing all scales of random modes on independent grids in stochastic computations or uncertainty quantification methods the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion the size of this system increases drastically when the number of independent random variables and or order of polynomial chaos expansions increases this is invariably the case for large scale simulations and or problems involving steep gradients and other multiscale features such features are variously reflected on each solution component or random uncertainty mode requiring the development of adaptive methods for their accurate resolution in this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space refinement on physical space that allows all scales of each solution component to be refined independently of the rest we exemplify this using the convection diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method thus we establish a new elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

the sequential knapsack polytope in this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible the corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

probability densities in strong turbulence abstract in this work we using mellin s transform combined with the gaussian large scale boundary condition calculate probability densities pdfs of velocity increments p r u r velocity derivatives p u r and the pdf of the fluctuating dissipation scales q re where re is the large scale reynolds number the resulting expressions strongly deviate from the log normal pdf p l r u r often quoted in the literature it is shown that the probability density of the small scale velocity fluctuations includes information about the large integral scale dynamics which is responsible for the deviation of p r u r from p l r u r an expression for the function d h of the multifractal theory free from spurious logarithms recently discussed in u frisch m martins afonso a mazzino v yakhot j fluid mech 542 2005 97 is also obtained

multidimensional potential burgers turbulence we consider the multidimensional generalised stochastic burgers equation in the space periodic setting frac partial mathbf u partial t nabla f mathbf u cdot nabla mathbf u nu delta mathbf u nabla eta quad t geq 0 mathbf x in mathbb t d mathbb r mathbb z d under the assumption that u is a gradient here f is strongly convex and satisfies a growth condition is small and positive while is a random forcing term smooth in space and white in time for solutions u of this equation we study sobolev norms of u averaged in time and in ensemble each of these norms behaves as a given negative power of these results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence namely the averages of the increments and of the energy spectrum these quantities behave as a power of the norm of the relevant parameter which is respectively the separation l in the physical space and the wavenumber k in the fourier space our bounds do not depend on the initial condition and hold uniformly in nu we generalise the results obtained for the one dimensional case in 10 confirming the physical predictions in 4 30 note that the form of the estimates does not depend on the dimension the powers of nu mathbf k ell are the same in the one and the multi dimensional setting

componentwise perturbation theory for linear systems with multiple right hand sides existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right hand sides and to a general class of componentwise measure of perturbations involving holder p norms it is shown that for a system of order n with r right hand sides the componentwise backward error can be computed by finding the minimum p norm solutions to n underdetermined linear systems and an explicit expression is obtained in the case r 1 a perturbation bound is derived and from this the componentwise condition number is obtained to within a multiplicative constant applications of the results are discussed to invariant subspace computations quasi newton methods based on multiple secant equations and an inverse ode problem

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar