renormalization group analysis of differential equations subject to slowly modulated perturbations abstract the application of renormalization group rg theory to the asymptotic analysis of differential equations is considered it is found that there is a class of small structural perturbations whose effects cannot be systematically treated using the gell mann low rg approach applied in this context guided by a reinterpretation of the calculational procedure employed in this approach whose motivation is provided by the unnecessarily complicated nature of its standard interpretation we formulate a generalized perturbative analysis and an rg approach which naturally and systematically treat equations subject to perturbations of this class this formulation of rg theory is demonstrated with a number of examples for which the gell mann low formulation fails to provide a systematic theoretical framework for one representative example it is found that the wilson rg formulation also fails the implications of this failure are discussed

an o n lg k 2 n 2 time and o k 2 n k space algorithm for certain np complete problems for a subset of np complete problems schroeppel and shamir 1981 described a general purpose algorithm with o 2n2 time and o 2n4 space complexity in this note we try to combine their idea with the selection in x y problem this approach yields a family of algorithms solving a subset of np complete problems in o nlg k 2n2 time and o k 2nk space

scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

optimal multi way number partitioning the np hard number partitioning problem is to separate a multiset s of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized the classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan the elapsed time to complete the schedule is minimized the two way number partitioning decision problem is one of the original 21 problems that richard karp proved np complete it is also one of garey and johnson s six fundamental np complete problems and the only one based on numbers this article explores algorithms for solving multi way number partitioning problems optimally we explore previous algorithms as well as our own algorithms which fall into three categories sequential number partitioning snp a branch and bound algorithm binary search improved bin completion bsibc a bin packing algorithm and cached iterative weakening ciw an iterative weakening algorithm we show experimentally that for large random numbers snp and ciw are state of the art algorithms depending on the values of n and k both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime

faster exact solutions for some np hard problems this paper considers a number of np complete problems and provides faster algorithms for solving them the solutions are based on a recursive partitioning of the problem domain and careful elimination of some of the branches along the search without actually checking them the time complexity of the proposed algorithms is of the form o 2en for constant 0 e 1 where n is the output size of the problem in particular such algorithms are presented for the exact sat and exact hitting set problems with e 0 3212 and for the exact 3sat problem with e 0 2072 both algorithms improve on previous ones proposed in the literature

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

on the structure of stationary solutions of the navier stokes equations this paper is a supplementary section to 1 we show that without any additional hypothesis the main result in 1 theorem 1 can be considerably strengthened

renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

stability of moving fronts in the ginzburg landau equation we use renormalization group ideas to study stability of moving fronts in the ginzburg landau equation in one spatial dimension in particular we prove stability of the real fronts under complex perturbations this extends the results of aronson and weinberger to situations where the maximum principle is inapplicable and constitutes a step in proving the general marginal stability hypothesis for the ginzburg landau equation

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

long time asymptotics of non degenerate non linear diffusion equations we study the long time asymptotics of prototypical non linear diffusion equations specifically we consider the case of a non degenerate diffusivity function that is a non negative polynomial of the dependent variable of the problem we motivate these types of equations using einstein s random walk paradigm leading to a partial differential equation in non divergence form on the other hand using conservation principles leads to a partial differential equation in divergence form a transformation is derived to handle both cases then a maximum principle on both an unbounded and a bounded domain is proved in order to obtain bounds above and below for the time evolution of the solutions to the non linear diffusion problem specifically these bounds are based on the fundamental solution of the linear problem the so called aronson s green function having thus sandwiched the long time asymptotics of solutions to the non linear problems between two fundamental solutions of the linear problem we prove that unlike the case of degenerate diffusion a non degenerate diffusion equation s solution converges onto the linear diffusion solution at long times select numerical examples support the mathematical theorems and illustrate the convergence process our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes such as in the flow of particulate materials that have been discussed in the applied physics literature

combinatorial models for coalgebraic structures abstract we introduce a convenient category of combinatorial objects known as cell sets on which we study the properties of the appropriate free abelian group functor we obtain a versatile generalization of the notion of incidence coalgebra giving rise to an abundance of coalgebras hopf algebras and comodules all of whose structure constants are positive integers with respect to certain preferred bases our category unifies and extends existing constructions in algebraic combinatorics providing proper functorial descriptions it is inspired in part by the notion of cw complex and is also geared to future applications in algebraic topology and the theory of formal group laws

diffusive stability of oscillations in reaction diffusion systems we study nonlinear stability of spatially homogeneous oscillations in reaction diffusion systems assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase we prove that spatially localized perturbations decay algebraically with the diffusive rate t n 2 in space dimension n we also compute the leading order term in the asymptotic expansion of the solution and show that it corresponds to a spatially localized modulation of the phase our approach is based on a normal form transformation in the kinetics ode which partially decouples the phase equation at the expense of making the whole system quasilinear stability is then obtained by a global fixed point argument in temporally weighted sobolev spaces corresponding author arnd scheel

non linear stability of modulated fronts for the swift hohenberg equation we consider front solutions of the swift hohenberg equation t u 1 x 2 2 u 2 u u 3 these are traveling waves which leave in their wake a periodic pattern in the laboratory frame using renormalization techniques and a decomposition into bloch waves we show the non linear stability of these solutions it turns out that this problem is closely related to the question of stability of the trivial solution for the model problem t u x t x 2 u x t 1 tanh x ct u x t u x t p with p 3 in particular we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front

on the minimum gap between sums of square roots of small integers let k and n be positive integers n k define r n k to be the minimum positive value of a1 ak b1 bk where a1 a2 ak b1 b2 bk are positive integers no larger than n define r n k to be logr n k it is important to find tight bounds for r n k and r n k in connection to the sum of square roots problem a famous open problem in computational geometry the current best lower bound and upper bound are far apart in this paper we prove an upper bound of 2o n logn for r n k which is better than the best known result o 22klogn whenever n cklogk for some constant c in particular our result implies an algorithm subexponential in k i e with time complexity 2o k logn o 1 to compare two sums of k square roots of integers of value o klogk we then present an algorithm to find r n k exactly in nk o k time and in n k 2 o k space as an example we are able to compute r 100 7 exactly in a few hours on a single pc the numerical data indicate that the root separation bound is very far away from the true value of r n k

pattern propagation in nonlinear dissipative systems abstract we discuss the problem of pattern selection in situations where a stable nonuniform state of a nonlinear dissipative system propagates into an initially unstable homogeneous region our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known and we point out that these known properties are consistent with a marginal stability hypothesis that has been suggested in the theory of dendritic crystal growth we then describe a more general interpretation of the marginal stability hypothesis and finally present numerical evidence for its validity from three different pattern forming models

a parallel time hardware tradeoff t h o 2 sup n 2 for the knapsack problem a parallel algorithm for solving the knapsack problem on a single instruction multiple data machine with shared memory is presented the shared memory allows concurrent reading while concurrent writing is forbidden the knapsack problem is of size n which the algorithm solves in time t o n 2 sup n 2 sup epsilon when p o 2 sup n 2 sup 1 epsilon 0

asymptotic behavior of global solutions of a semilinear parabolic problem abstract we study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem u t u u p x r n t 0 u 0 u 0 x r n t 0 it is known that the nonnegative solution u t of this problem blows up in finite time for 1 p 1 2 n moreover if p 1 2 n and the norm of u 0 is small enough the problem admits a global solution in this work we use the entropy method to obtain the decay rate of the global solution u t

first order approximation for quadratic dispersive equations by the renormalization group approach in this paper we study the application of the renormalization group method to a system of partial differential equations together with constructing a long time first order approximation of the solution of the system we compare the difference between the true solution and the approximate formation with the help of dispersive estimates in lp space

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

asymptotics of solutions of nonlinear parabolic equations abstract this paper is concerned with the large time behaviour of solutions to the cauchy problem of the following nonlinear parabolic equations u t u f u d x u d x 2 u u r n u t x t 0 u 0 x x r n n 1 under the optimal growth conditions on the smooth nonlinear function f u dxu dx2u we obtain the global existence results to the above cauchy problem the influence of the nonlinear function f u dxu dx2u on the large time behaviour of the global smooth solution u t x is also studied

complexity of selection in x y let x and y be two sorted n vectors and a x y be an n n matrix with sorted rows and columns such that aij xi yj let 1 k n2 vyskoc 1987 claimed that selecting the k 1 st element of a could be done in o logn time if the kth element is known in this note we prove that this result is not exact by showing that o n is a lower bound for the problem under vyskoc s hypothesis we also describe an o n algorithm and conclude by showing how the same algorithm can be used for searching on such matrices

stability of cahn hilliard fronts we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity c 1999 john wiley sons inc

on space efficient algorithms for certain np complete problems some recent results claimed the existence of a class of algorithms for certain np complete problems with running time o n1g k 2n 2 and storage requirements o k 2n k for 2 k n in this note we show that those results do not hold implying that an algorithm with time o n 2n 2 and space o 2n 4 is still the best known solution for such class of np complete problems

on a certain renormalization group method in this paper a mathematical study of the renormalization group method recently introduced by chen goldenfeld and oono phys rev e 54 376 1997 phys rev lett 73 1311 1994 nato adv study inst ser ser b 284 375 1991 is given for the case of autonomous nonlinear systems of differential equations we also observe that the approximation results obtained by this method by chen goldenfeld and ono nato adv study inst ser ser b 284 375 1991 are valid over long time intervals moreover a connection between this method and the classical poincare dulac normal forms and the averaging method is briefly discussed

renormalisation group theory for two problems in linear continuum mechanics abstract we consider the problems of 1 the stress field in an infinite wedge when a moment is applied at the tip and 2 the flow of an inviscid incompressible fluid past an infinite wedge both problems are linear but nevertheless exhibit anomalous dimensions for wedge angles larger than a critical value c in 1 it is the stress field and in 2 it is the velocity potential which can have anomalous power law behaviour in distance from the tip we use these problems as simple examples to illustrate how partial differential equations can be solved using the renormalisation group

a t o 2 n 2 s o 2 n 4 algorithm for certain np complete problems in this paper we develop a general purpose algorithm that can solve a number of np complete problems in time t o 2 n 2 and space s o 2 n 4 the algorithm can be generalized to a fam

universal self similarity of porous media equation with absorption the critical exponent case abstract in this paper we study the large time behavior of non negative solutions to the cauchy problem of u t u m u q in r n 0 where m 1 and q q c m 2 n is a critical exponent for non negative initial value u x 0 u 0 x l 1 r n we show that the solution converges if u 0 x 1 x k is bounded for some k n to a unique fundamental solution of u t u m independent of the initial value with additional logarithmic anomalous decay exponent in time as t

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

hopf bifurcation and exchange of stability in diffusive media we consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis for all values of the bifurcation parameter and that a pair of imaginary eigenvalues crosses the imaginary axis for a reaction diffusion convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations prove the occurrence of a hopf bifurcation and the nonlinear stability of the bifurcating time periodic solutions again with respect to spatially localized perturbations

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

diffusive mixing of periodic wave trains in reaction diffusion systems abstract we consider reaction diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u 0 k x t k that are parameterized by the wave number k we prove stable diffusive mixing of the asymptotic states u 0 k x k as x with different phases at infinity for solutions that initially converge to these states as x the proof is based on bloch wave analysis renormalization theory and a rigorous decomposition of the perturbations of these wave solutions into a phase mode which shows diffusive behavior and an exponentially damped remainder depending on the dispersion relation the asymptotic states mix linearly with a gaussian profile at lowest order or with a nonsymmetric non gaussian profile given by burgers equation which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation

nonlinear diffusion through large complex networks containing regular subgraphs transport through generalized trees is considered trees contain the simple nodes and supernodes either well structured regular subgraphs or those with many triangles we observe a superdiffusion for the highly connected nodes while it is brownian for the rest of the nodes transport within a supernode is affected by the finite size effects vanishing as n for the even dimensions of space d 2 4 6 the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat kernel expansion

an o 2 0 304n algorithm for solving maximum independent set problem a faster algorithm for finding a maximum independent set in a graph is presented the algorithm is an improved version of the one by tarjan and trojanowski 7 a technique to further accelerate this algorithm is also described

solutions without phase slip for the ginsburg landau equation we consider the ginsburg landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx if these solutions are not very different then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips i e which does not go through zero amplitude

renormalization group theory and variational calculations for propagating fronts we study the propagation of uniformly translating fronts into a linearly unstable state both analytically and numerically we introduce a perturbative renormalization group rg approach to compute the change in the propagation speed when the fronts are perturbed by structural modification of their governing equations this approach is successful when the fronts are structurally stable and allows us to select uniquely the numerical experimentally observable propagation speed for convenience and completeness the structural stability argument is also briefly described we point out that the solvability condition widely used in studying dynamics of nonequilibrium systems is equivalent to the assumption of physical renormalizability we also implement a variational principle due to hadeler and rothe which provides a very good upper bound and in some cases even exact results on the propagation speeds and which identifies the transition from linear to nonlinear marginal stability as parameters in the governing equation are varied

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

almost global existence and transient self similar decay for poiseuille flow at criticality for exponentially long times abstract we consider nonlinear diffusion equations with critical exponent as tu x2u u3 with x r for small initial data in l1 l it is well known that almost all solutions of this system explode in finite time however we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time moreover before explosion the l norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments as an application we consider the long time transient self similar decay to unstable poiseuille flow at criticality for exponentially long times this together with a subcritical bifurcation and short time transient amplification is a principal obstruction in all attempts to measure the critical reynolds number for this experiment more and more precisely

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

polynomial average time satisfiability problems a class of random conjunctive normal form cnf predicates is characterized by v the number of variables from which literals may be formed giving 2v literals p v the probability that a given literal is in a random clause and t v the number of random independently selected clauses in a random predicate determining satisfiability on such a class using backtracking takes polynomial average time under the following conditions 1 limv vp v 0 and t v ln2 v ln v l p v and 2 lim vp v lim p v 0 and t v ln2 ep v exp 2vp v for any fixed 0 additional cases are also covered an analysis by goldberg showed that problems for which p v is constant can be solved in polynomial average time by a simplified davis putnam procedure we give an algorithm for solving problems with t v ln lnv ln3 in polynomial worst case time thus all problem sets for which p v or t v grows extremely rapidly or extremely slowly can be solved in polynomial average time we also show that backtracking requires exponential average time when limvp v 0 and t v ln2 v ln v 1 p v and when limvp v limv vp v 0 and t v ln2 ge ep v exp 2vp v

average polynomial time complexity of some np complete problems abstract we define clique n n to be the problem clique restricted to the set of graphs g v e such that e n v in this paper it is proved that when n n n e e a rational number 0 clique n n is an np complete problem and under the assumption of a uniform distribution on the set of instances clique n n can be solved by a deterministic algorithm in average and almost everywhere polynomial time some cases of nonuniform distributions on the set of instances for which the usual problem clique is solved in average polynomial time are also considered

large time behavior for the fast diffusion equation with critical absorption we study the large time behavior of nonnegative solutions to the cauchy problem for a fast diffusion equation with critical zero order absorption partial t u delta u m u q 0 quad quad hbox in 0 infty times real n with m c n 2 n 1

group analysis and renormgroup symmetries an original regular approach to constructing special type symmetries for boundary value problems namely renormgroup symmetries is presented different methods of calculating these symmetries based on modern group analysis are described an application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model

self similar decay of localized perturbations in the integral boundary layer equation abstract the integral boundary layer equation ible arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane the trivial solution of the ible is linearly at best marginally stable i e it has essential spectrum at least up to the imaginary axis here we show that in the stable case this trivial solution is in fact nonlinearly stable with a burgers like self similar decay of localized perturbations the proof uses renormalization theory and the fact that in the stable case burgers equation is the amplitude equation for long small amplitude waves in the ible

self similar decay of spatially localized perturbations of the nusselt solution for the inclined film problem the nusselt solution for the flow of a viscous incompressible fluid with a free surface down an inclined plane is at best marginally stable i e the linearization has essential spectrum at least up to the imaginary axis nevertheless using a renormalization group approach here we establish the stability of the nusselt solution in the full nonlinear system in case of linear stability by proving the self similar decay of spatially localized perturbations the asymptotic decay for t to infty is similar to the dynamics of localized perturbations of the trivial solution in the burgers equation on the real line which is the amplitude equation of the problem

renormalization group analysis of nonlinear diffusion equations with time dependent coefficients and marginal perturbations in this paper we use a renormalization group rg method to study the long time asymptotics of nonlinear diffusion equations with time dependent diffusion coefficients and nonlinearities which are marginal or critical with respect to the rg operator these equations describe the time evolution of the average concentration of a passive scalar being advected by a random velocity field we prove that besides the expected diffusive behavior there is an extra logarithmic correction which is the imprint of the critical nonlinearity

an asymptotic formula for marginal running coupling constants and universality of loglog corrections given a two loop function for multiple marginal coupling constants we derive an asymptotic formula for the running coupling constants driven to an infrared fixed point it can play an important role in universal loglog corrections to physical quantities

global large time self similarity of a thermal diffusive combustion system with critical nonlinearity we study the initial value problem of the thermal diffusive combustion system u 1 t u 1 x x u 1 u 2 2 u 2 t du 2 xx u 1 u 2 2 x is an element of r 1 for non negative spatially decaying initial data of arbitrary size and for any positive constant d we show that if the initial data decay to zero sufficiently fast at infinity then the solution u 1 u 2 converges to a self similar solution of the reduced system u 1 t u 1 xx u 1 u 2 2 u 2 t du 2 xx in the large time limit in particular u 1 decays to zero like o t 1 2 delta where delta 0 is an anomalous exponent depending on the initial data and u 2 decays to zero with normal rate o t 1 2 the idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self similarity of the solutions in the large time limit c 1996 academic press inc

a complete anytime algorithm for number partitioning abstract given a set of numbers the two way number partitioning problem is to divide them into two subsets so that the sum of the numbers in each subset are as nearly equal as possible the problem is np complete based on a polynomial time heuristic due to karmarkar and karp we present a new algorithm called complete karmarkar karp ckk that optimally solves the general number partitioning problem and significantly outperforms the best previously known algorithms for large problem instances for numbers with twelve significant digits or less ckk can optimally solve two way partitioning problems of arbitrary size in practice for numbers with greater precision ckk first returns the karmarkar karp solution then continues to find better solutions as time allows over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time rather than building a single solution one element at a time or modifying a complete solution ckk constructs subsolutions and combines them together in all possible ways this approach may be effective for other np hard problems as well

diffusive mixing of stable states in the ginzburg landau equation the ginzburg landau equation on the real line has spatially periodic steady states of the form with and for we construct solutions which converge for all t 0 to the limiting pattern as these solutions are stable with respect to sufficiently small perturbations and behave asymptotically in time like where is uniquely determined by the boundary conditions this extends a previous result of brk92 by removing the assumption that should be close to zero the existence of the limiting profile is obtained as an application of the theory of monotone operators and the long time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term

a multiscale asymptotic analysis of time evolution equations on the complex plane using an appropriate norm on the space of entire functions we extend to the complex plane the renormalization group method as developed by bricmont et al the method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems the time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical n vector model we show that for initial conditions belonging to a certain frechet space of entire functions of exponential type the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions

renormalization group derivation of phase equations abstract phase equations describing the evolution of large scale modulation of spatially periodic patterns in two dimensional systems are derived by employing the renormalization group method a general formula for phase diffusion coefficients is given under certain conditions

renormalization theoretic analysis of non equilibrium phase transitions i the becker doring equations with power law rate coefficients we study in detail the application of renormalization theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes we investigate the becker d oring equations originally formulated to describe and analyse non equilibrium phase transitions and more recently generalized to describe a wide range of physicochemical problems in this paper we analyse how the systematic coarse graining renormalization of the becker d oring system of equations affects the aggregation and fragmentation rate coefficients we consider the case of power law size dependent cluster rate coefficients which we show lead to only three classes of system that require analysis coagulation dominated systems fragmentation dominated systems and those where coagulation and fragmentation are exactly balanced we analyse the late time asymptotics associated with each class

analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations abstract for singular perturbation problems the renormalization group rg method of chen goldenfeld and oono phys rev e 49 1994 4502 4511 has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system it has been applied to a variety of problems traditionally analyzed using disparate methods including the method of multiple scales boundary layer theory the wkbj method the poincare lindstedt method the method of averaging and others in this article we show how the rg method may be used to generate normal forms for large classes of ordinary differential equations first we apply the rg method to systems with autonomous perturbations and we show that the reduced or amplitude equations generated by the rg method are equivalent to the classical poincare birkhoff normal forms for these systems up to and including terms of o 2 where is the perturbation parameter this analysis establishes our approach and generalizes to higher order second we apply the rg method to systems with nonautonomous perturbations and we show that the reduced or amplitude equations so generated constitute time asymptotic normal forms which are based on kbm averages moreover for both classes of problems we show that the main coordinate changes are equivalent up to translations between the spaces in which they are defined in this manner our results show that the rg method offers a new approach for deriving normal forms for nonautonomous systems and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves finally we establish how well the solution to the rg equations approximates the solution of the original equations on time scales of o 1

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

the validity of phase diffusion equations and of cahn hilliard equations for the modulation of pattern in reaction diffusion systems abstract so called phase diffusion equations and cahn hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern it is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system the proof is given for an abstract reaction diffusion system as original system

selection stability and renormalization abstract we illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem this consideration leads us to a renormalization group study of the problem the study illustrates two very general conclusions 1 singular perturbations in applied mathematics are best understood as renormalized perturbation methods and 2 amplitude equations are renormalization group equations

optimal algorithms for generalized searching in sorted matrices we present a set of optimal and asymptotically optimal sequential and parallel algorithms for the problem of searching on an m n sorted matrix in the general case when m n our two sequential algorithms have a time complexity of o m log 2nm which is shown to be optimal our parallel algorithm runs in o log log mlog log m log 2nm1 z time using m log log mlog log m processors on a common crcw pram where 0 z 1 is a monotonically decreasing function on m which is asymptotically work optimal the two sequential algorithms differ mainly in the ways of matrix partitioning one uses row searching and the other applies diagonal searching the parallel algorithm is based on some non trivial matrix partitioning and processor allocation schemes all the proposed algorithms can be easily generalized for searching on a set of sorted matrices

diffusive stability against nonlocalized perturbations of planar wave trains in reaction diffusion systems abstract planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction in this paper we investigate the stability of planar wave trains in reaction diffusion systems we establish nonlinear diffusive stability against perturbations that are bounded along a line in r 2 and decay exponentially in the distance from this line our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation we also consider perturbations that are fully localized and establish nonlinear stability with better decay rates suggesting a trade off between spatial localization of perturbations and temporal decay rate our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations the nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme instead we track the perturbed solution in two different coordinate systems

phase turbulence in the complex ginzburg landau equation via kuramoto sivashinsky phase dynamics we study the complex ginzburg landau initial value problem for a complex field u c with r we consider the benjamin feir linear instability region we show that for all and for all initial data u 0 sufficiently close to 1 up to a global phase factor e i 0 0 r in the appropriate space there exists a unique spatially periodic solution of space period l 0 these solutions are small even perturbations of the traveling wave solution and s have bounded norms in various l p and sobolev spaces we prove that apart from corrections whenever the initial data satisfy this condition and that in the linear instability range the dynamics is essentially determined by the motion of the phase alone and so exhibits phase turbulence indeed we prove that the phase satisfies the kuramoto sivashinsky equation for times while the amplitude 1 2 s is essentially constant

new algorithms for exact satisfiability the exact satisfiability problem is to determine if a cnf formula has a truth assignment satisfying exactly one literal in each clause exact 3 satisfiability is the version in which each clause contains at most three literals in this paper we present algorithms for exact satisfiability and exact 3 satisfiability running in time o 20 2325n and o 20 1379n respectively the previously best algorithms have running times o 20 2441n for exact satisfiability methods oper res 43 1981 419 431 and o 20 1626n for exact 3 satisfiability annals of mathematics and artificial intelligence 43 1 2005 173 193 and zapiski nauchnyh seminarov pomi 293 2002 118 128 we extend the case analyses of these papers and observe that a formula not satisfying any of our cases has a small number of variables for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time

anomalous scaling for three dimensional cahn hilliard fronts we prove the stability of the one dimensional kink solution of the cahn hilliard equation under d dimensional perturbations for d 3 we also establish a novel scaling behavior of the large time asymptotics of the solution the leading asymptotics of the solution is characterized by a length scale proportional to t1 3 instead of the usual t1 2 scaling typical to parabolic problems 2004 wiley periodicals inc

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable

asymptotics for nonlinear integral equations with a generalized heat kernel using renormalization group technique in this paper we employ the renormalization group method to study the long time asymptotics of solutions to a class of nonlinear integral equations with a generalized heat kernel the nonlinearities are classified and studied according to its role in the asymptotic behavior here we prove that the behavior in the limit as t goes to infinity remains unchanged when compared with the one in the linear case if the nonlinearities are the ones classified as irrelevant in the renormalization group sense