high accuracy numerical methods for the gardner ostrovsky equation the gardner ostrovsky equation also known as the extended rotation modified korteweg de vries kdv equation describes weakly nonlinear internal oceanic waves under the influence of earth rotation high accuracy numerical methods are needed to follow with precision the long time evolution of the solutions of this equation with the additional difficulty that the numerical methods have to conserve very accurately several invariants of the solutions including the mass and the energy of the waves finite difference methods traditionally used for the solution of the kdv equation fails to preserve accurately these invariants for large times in this paper we show that this difficulty can be overcome by using a high accuracy finite difference hafd numerical method we present a strong stability preserving finite difference scheme and compare its performance using some relevant examples with those of two recently published numerical methods for solving this kind of equation a simpler second order finite difference method and a pseudospectral numerical scheme that enforces the conservation of energy the numerical comparison shows that the three methods have similar accuracy for short times but the simpler finite difference scheme is less accurate in preserving the three invariants affecting the numerical accuracy of the solution as time goes on on the contrary the hafd method presented here preserves the invariants with even better accuracy than the pseudospectral scheme but with a much lower computational cost in addition the numerical implementation of the hafd method is as easy as that of the simpler finite difference method being both much simpler than the intricate energy conservation pseudospectral scheme these advantages makes the hafd method presented here very appropriate for solving numerically this type of equations particularly for studying long time wave propagation

new conservative schemes with discrete variational derivatives for nonlinear wave equations new conservative finite difference schemes for certain classes of nonlinear wave equations are proposed the key tool there is discrete variational derivative by which discrete conservation property is realized a similar approach for the target equations was recently proposed by furihata but in this paper a different approach is explored where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties applications for the nonlinear klein gordon equation and the so called good boussinesq equation are presented numerical examples reveal the good performance of the new schemes

minimum fill in inapproximability and almost tight lower bounds abstract given an n n sparse symmetric matrix with m nonzero entries performing gaussian elimination may turn some zeroes into nonzero values so called fill ins the minimum fill in problem asks whether it is possible to perform the elimination with at most k fill ins we exclude the existence of polynomial time approximation schemes for this problem assuming p np and the existence of 2 o n 1 time approximation schemes for any positive assuming the exponential time hypothesis we also give a 2 o k 1 2 n o 1 parameterized lower bound all these results come as corollaries of a new reduction from vertex cover to the minimum fill in problem which might be of its own interest all previous reductions for similar problems start from some kind of graph layout problems and hence have limited use in understanding their fine grained complexity

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

parabolic projective functors in type a abstract we classify projective functors on the regular block of rocha caridi s parabolic version of the bgg category o in type a in fact we show that in type a the restriction of an indecomposable projective functor from o to the parabolic category is either indecomposable or zero as a consequence we obtain that projective functors on the parabolic category o in type a are completely determined up to isomorphism by the linear transformations they induce on the level of the grothendieck group which was conjectured by stroppel in 39

robust global optimization with polynomials we consider the optimization problems maxz minx k p z x and minx k maxz p z x where the criterion p is a polynomial linear in the variables z the set can be described by lmis and k is a basic closed semi algebraic set the first problem is a robust analogue of the generic sdp problem maxz p z whereas the second problem is a robust analogue of the generic problem minx k p x of minimizing a polynomial over a semi algebraic set we show that the optimal values of both robust optimization problems can be approximated as closely as desired by solving a hierarchy of sdp relaxations we also relate and compare the sdp relaxations associated with the max min and the min max robust optimization problems

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

the unique games conjecture integrality gap for cut problems and embeddability of negative type metrics into l 1 in this article we disprove a conjecture of goemans and linial namely that every negative type metric embeds into e1 with constant distortion we show that for an arbitrarily small constant 0 for all large enough n there is an n point negative type metric which requires distortion at least log log n 1 6 to embed into e1 surprisingly our construction is inspired by the unique games conjecture ugc establishing a previously unsuspected connection between probabilistically checkable proof systems pcps and the theory of metric embeddings we first prove that the ugc implies a super constant hardness result for the nonuniform s parsest c ut problem though this hardness result relies on the ugc we demonstrate nevertheless that the corresponding pcp reduction can be used to construct an integrality gap instance for s parsest c ut towards this we first construct an integrality gap instance for a natural sdp relaxation of u nique g ames then we simulate the pcp reduction and translate the integrality gap instance of u nique g ames to an integrality gap instance of s parsest c ut this enables us to prove a log log n 1 6 integrality gap for s parsest c ut which is known to be equivalent to the metric embedding lower bound

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

analytical and numerical studies of weakly nonlocal solitary waves of the rotation modified korteweg de vries equation abstract a century ago the korteweg de vries kdv equation was derived as a model for weakly nonlinear long waves propagating down a channel when cross channel and depth variations are sufficiently weak in this article we study the steadily translating coherent structures of a generalization of this equation the rotation modified kdv equation which applies when coriolis forces are significant u t uu x u xxx x 2 u 0 where x is the down channel coordinate this is also called the ostrovosky equation the rmkdv solitary waves are weakly nonlocal due to the radiation of long waves where the radiation is proportional to the small parameter because of its simplicity the rmkdv coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter micropterons we extend the matched asymptotic expansions of hunter to third order we also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem new modes with multiple kdv like peaks flanked by small quasi sinusoidal oscillations are shown to be well described by matched asymptotics too

the parameterized complexity of the k biclique problem given a graph g and an integer k the k b iclique problem asks whether g contains a complete bipartite subgraph with k vertices on each side whether there is an f k vgvo 1 time algorithm solving k b iclique for some computable function f has been a longstanding open problem we show that k b iclique is w 1 hard which implies that such an f k vgvo 1 time algorithm does not exist under the hypothesis w 1 fpt from parameterized complexity theory to prove this result we give a reduction which for every n vertex graph g and small integer k constructs a bipartite graph h e lc r e in time polynomial in n such that if g contains a clique with k vertices then there are k k 1 2 vertices in l with n 1 k common neighbors otherwise any k k 1 2 vertices in l have at most k 1 e common neighbors an additional feature of this reduction is that it creates a gap on the right side of the biclique such a gap might have further applications in proving hardness of approximation results assuming a randomized version of exponential time hypothesis we establish an f k vgvo sk time lower bound for k biclique for any computable function f combining our result with the work of bulatov and marx 2014 we obtain a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems

a combinatorial approach to functorial quantum sl k knot invariants this paper contains a categorification of the frak sl k link invariant using parabolic singular blocks of category cal o our approach is intended to be as elementary as possible providing essentially combinatorial arguments for the main results of sussan the justification that our combinatorial arguments and steps are correct uses non combinatorial geometric and representation theoretic results e g the kazhdan lusztig and soergel s theorems we take these results as granted and use them like axioms called it facts in the text we first construct an exact functor valued invariant of webs or special trivalent graphs labelled with 1 2 k 1 k satisfying the moy relations afterwards we extend it to the frak sl k invariant of links by passing to the derived categories the approach of khovanov using foams appears naturally in this context more generally we expect that our approach provides a representation theoretic interpretation of the frak sl k homology based on foams and the kapustin li formula from mackaay stosic and vaz conjecturally this implies that the khovanov rozansky link homology is obtained from our invariant by restriction

conservative numerical schemes for the ostrovsky equation the ostrovsky equation describes gravity waves under the influence of coriolis force it is known that solutions of this equation conserve the l 2 norm and an energy function that is determined non locally in this paper we propose four conservative numerical schemes for this equation a finite difference scheme and a pseudospectral scheme that conserve the norm and the same types of schemes that conserve the energy a numerical comparison of these schemes is also provided which indicates that the energy conservative schemes perform better than the norm conservative schemes

projective injective modules serre functors and symmetric algebras we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra in our approach projective injective modules that is modules which are both

simple transitive 2 representations of small quotients of soergel bimodules in all finite coxeter types but i2 12 i2 18 and i2 30 we classify simple transitive 2 representations for the quotient of the 2 category of soergel bimodules over the coinvariant algebra which

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

categorification of the temperley lieb category tangles and cobordisms via projective functors to each generic tangle projection from the three dimensional real vector space onto the plane we associate a derived endofunctor on a graded parabolic version of the bernstein gel fand category mathcal o we show that this assignment is up to shifts invariant under tangle isotopies and reidemeister moves and defines therefore invariants of tangles the occurring functors are defined via so called projective functors the first part of the paper deals with the indecomposability of such functors and their connection with generalised temperley lieb algebras which are known to have a realisation via decorated tangles the second part of the paper describes a categorification of the temperley lieb category and proves the main conjectures of bfk moreover we describe a functor from the category of 2 cobordisms into a category of projective functors

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

constant factor lasserre integrality gaps for graph partitioning problems partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem called balanced separator that arises in many settings for this problem and variants such as the uniform sparsest cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge there are large gaps between the known approximation algorithms and nonapproximability results while no constant factor approximation algorithms are known even apx hardness is not known either in this work we prove that for balanced separator and uniform sparsest cut semidefinite programs from the lasserre hierarchy which are the most powerful relaxations studied in the literature have an integrality gap bounded away from 1 even for omega n levels of the hierarchy this complements recent algorithmic results in guruswami and sinop proceedings of the 52nd annual symposium on foundations of computer science focs

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

isotypic faithful 2 representations of mathcal j simple fiat 2 categories we introduce the class of isotypic 2 representations for finitary 2 categories and the notion of inflation of 2 representations under some natural assumptions we show that isotypic 2 representations are equivalent to inflations of cell 2 representations

expected length of the longest probe sequence in hash code searching an investigation ts made of the expected value of the maximum number of accesses needed to locate any element m a hashing file under various colhston resoluuon schemes this differs from usual worst case considerations winch for hashmg would be the largest sequence of accesses for the worst possible file asymptotic expressxons of these expected values are found for full and partly full tables for the open addressing scheme with a clustering free model these values are found to be 0 6315 x n for a full table and logan for a partly full table where n ts the number of records m is the size of the table and a n m for the open addressing scheme which reorders the insertions to minimize the worst case the lower bounds in n 1 077 and a in l a are found for full and partly full tables respectively fmauy for the separate chaining or direct chaining method both expected values are found to be f t n these results show that for these schemes the actual behawor of the worst case in hash tables is quite good on the average

local and global well posedness for the ostrovsky equation abstract we consider the initial value problem for 0 1 t u x 3 u x 1 u uu x 0 x t r where u is a real valued function and are real numbers such that 0 and x 1 f i 1 f this equation differs from korteweg de vries equation in a nonlocal term nevertheless we obtained local well posedness in x s f h s r x 1 f l 2 r s 3 4 using techniques developed in c e kenig g ponce l vega well posedness of the initial value problem for the korteweg de vries equation j amer math soc 4 1991 323 347 for the case 0 we also obtain a global result in x 1 using appropriate conservation laws

a conservative fourth order stable finite difference scheme for the generalized rosenau kdv equation in both 1d and 2d abstract in the present work a conservative fourth order stable finite difference scheme is proposed to solve the generalized rosenau kdv equation in both 1d and 2d the existence uniqueness and mass and energy conservations of the numerical solution are proved using the discrete energy method the new scheme is convergent with o 2 h 4 and unconditionally stable numerical experiments are carried out to show that the scheme is efficient and reliable

average value of solutions for the bipartite boolean quadratic programs and rounding algorithms we consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem bbqp with m n variables a closed form formula is developed to compute the average objective function value a of all solutions in o m n time however computing the median objective function value of the solutions is shown to be np hard also we show that any solution with objective function value no worse than a dominates at least 2 m n 2 solutions and this bound is the best possible further we show that such a solution can be identified in o m n time and hence the domination ratio of this algorithm is at least 1 4 we then show that for any fixed natural numbers a and b such that a b 1 no polynomial time approximation algorithm exists for bbqp with domination ratio larger than 1 2 1 m n unless p np it is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than a one of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value a new integer programming formulation of bbqp is also given and computational results with our rounding algorithms are reported

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

numerical integration of the ostrovsky equation based on its geometric structures we consider structure preserving numerical schemes for the ostrovsky equation which describes gravity waves under the influence of coriolis force this equation has two associated invariants an energy function and the l 2 norm it is widely accepted that structure preserving methods such as invariants preserving and multi symplectic integrators generally yield qualitatively better numerical results in this paper we propose five geometric integrators for this equation energy preserving and norm preserving finite difference and galerkin schemes and a multi symplectic integrator based on a newly found multi symplectic formulation a numerical comparison of these schemes is provided which indicates that the energy preserving finite difference schemes are more advantageous than the other schemes

inapproximability results for maximum edge biclique minimum linear arrangement and sparsest cut we consider the minimum linear arrangement problem and the uniform sparsest cut problem so far these two notorious np hard graph problems have resisted all attempts to prove inapproximability results we show that they have no polynomial time approximation scheme unless np complete problems can be solved in randomized subexponential time furthermore we show that the same techniques can be used for the maximum edge biclique problem for which we obtain a hardness factor similar to previous results but under a more standard assumption

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

ruling out ptas for graph min bisection dense k subgraph and bipartite clique assuming that np not subseteq cap epsilon 0 bptime 2 n epsilon we show that graph min bisection dense k subgraph and bipartite clique have no polynomial time approximation scheme ptas we give a reduction from the minimum distance of code mdc problem starting with an instance of mdc we build a quasi random probabilistically checkable proof pcp that suffices to prove the desired inapproximability results in a quasi random pcp the query pattern of the verifier looks random in a certain precise sense among the several new techniques we introduce the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials as is important for our purpose the certificate itself happens to be another polynomial and it can be checked probabilistically by reading a constant number of its values

kazhdan lusztig cells and the murphy basis let h be the iwahori hecke algebra associated with s n the symmetric group on n symbols this algebra has two important bases the kazhdan lusztig basis and the murphy basis we establish a precise connection between the two bases allowing us to give for the first time purely algebraic proofs for a number of fundamental properties of the kazhdan lusztig basis and lusztig s results on the a function

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

new approximation techniques for some linear ordering problems we describe logarithmic approximation algorithms for the np hard graph optimization problems of minimum linear arrangement minimum containing interval graph and minimum storage time product this improves upon the best previous approximation bounds of even naor rao and schieber j acm 47 2000 pp 585 616 for these problems by a factor of omega log log n we use the lower bound provided by the volume w of a spreading metric for each of the ordering problems above as defined by even et al in order to find a solution with cost at most a logarithmic factor times w for these problems we develop a divide and conquer strategy where the cost of a solution to a problem at a recursive level is c plus the cost of a solution to the subproblems at this level and where the spreading metric volume on the subproblems is less than the original volume by omega c log n ensuring that the resulting solution has cost o log n times the original spreading metric volume we note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems for planar graphs we combine a structural theorem of klein plotkin and rao proceedings of the 25th acm symposium on theory of computing 1993 pp 682 690 with our new recursion technique to show that the spreading metric cost volumes are within an o log log n factor of the cost of an optimal solution for the minimum linear arrangement and the minimum containing interval graph problems

integrating tabu search and vlsn search to develop enhanced algorithms a case study using bipartite boolean quadratic programs abstract the bipartite boolean quadratic programming problem bbqp is a generalization of the well studied boolean quadratic programming problem the model has a variety of real life applications however empirical studies of the model are not available in the literature except in a few isolated instances in this paper we develop efficient heuristic algorithms based on tabu search very large scale neighborhood vlsn search and a hybrid algorithm that integrates the two the computational study establishes that effective integration of simple tabu search with vlsn search results in superior outcomes and suggests the value of such an integration in other settings complexity analysis and implementation details are provided along with conclusions drawn from experimental analysis in addition we obtain solutions better than the best previously known for almost all medium and large size benchmark instances

a multilevel analysis of the lasserre hierarchy abstract this paper analyzes the relation between different orders of the lasserre hierarchy for polynomial optimization pop although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying pop other problems require sequentially solving the second or higher orders until a solution is found for these cases and assuming that the lower order semidefinite programming relaxation has been solved we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation we can prove feasibility in the higher order of the hierarchy of the points obtained using the operators as well as convergence to the optimal as the relaxation order increases furthermore the operators are simple and inexpensive for problems where the projection over the feasible set is easy to calculate for example integer 0 1 and 1 1 pops our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators in particular we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point we use this technique to solve quadratic integer 0 1 problems as well as max cut and integer partition problems

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

category o gradings and translation functors abstract in this article we consider a graded version of category o we reprove some results of beilinson et al j amer math soc 9 1996 473 527 using a different approach furthermore we define a graded version of translation functors and duality this provides the construction of various graded modules on the other hand we describe how to get modules which are not gradable

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur