non isochronicity of the center at the origin in polynomial hamiltonian systems with even degree nonlinearities abstract in 2002 x jarque and j villadelprat proved that no center in a planar polynomial hamiltonian system of degree 4 is isochronous and raised a question is there a planar polynomial hamiltonian system of even degree which has an isochronous center in this paper we give a criterion for non isochronicity of the center at the origin of planar polynomial hamiltonian systems moreover the orders of weak centers are determined our results answer a weak version of the question proving that there is no planar polynomial hamiltonian system with only even degree nonlinearities having an isochronous center at the origin

asymptotic numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers this paper considers approximate solution methods for a one dimensional stefan problem describing solvent diffusion in glassy polymers similar to the classic stefan problem the region initially has zero thickness and so must be analysed carefully before performing a numerical computation a small time analysis gives the correct starting solution which is then incorporated into the second order accurate keller box finite difference scheme we also consider a detailed analysis of small and large time expansions as well as the large control parameter limit and show that our generalised approach enables us to obtain higher order terms than given in previous studies finally we apply the combined integral method cim to this problem which is a refinement of the popular heat balance integral method hbim and compare both the cim and asymptotic solutions to the numerical results

approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

an accurate finite difference method for ablation type stefan problems a recently derived numerical algorithm for one dimensional time dependent stefan problems is extended for the purpose of solving one phase ablation type moving boundary problems in tandem with the keller box finite difference scheme the so called boundary immobilization method is used an important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space the analysis also determines that the ablation front initially moves as the time raised to the power 3 2 hence it evolves considerably more slowly than the phase change front in the classical stefan problem with isothermal cooling

local bifurcations of critical periods for cubic lienard equations with cubic damping continuing chicone and jacobs work for planar hamiltonian systems of newton s type in this paper we study the local bifurcation of critical periods near a nondegenerate center of the cubic lienard equation with cubic damping and prove that at most 2 local critical periods can be produced from either a weak center of finite order or the linear isochronous center and that at most 1 local critical period can be produced from nonlinear isochronous centers

finite difference methods with increased accuracy and correct initialization for one dimensional stefan problems although the numerical solution of one dimensional phase change or stefan problems is well documented a review of the most recent literature indicates that there are still unresolved issues regarding the start up of a computation for a region that initially has zero thickness as well as how to determine the location of the moving boundary thereafter this paper considers the so called boundary immobilization method for four benchmark melting problems in tandem with three finite difference discretization schemes we demonstrate a combined analytical and numerical approach that eliminates completely the ad hoc treatment of the starting solution that is often used and is numerically second order accurate in both time and space a point that has been consistently overlooked for this type of moving boundary problem

applying the combined integral method to two phase stefan problems with delayed onset of phase change in this paper the combined integral method cim is applied to non classical two phase stefan problems with delayed onset of phase change this can occur if the phase change is caused by a heat flux or robin boundary condition the method requires choosing an approximating function typically a polynomial but it is not clear what should be used as the exponent in the highest order term previous studies have determined exponents either from exact solutions or from expansions valid over short time scales neither approach is satisfactory and can be very inaccurate for larger times we combined the heat balance and refined integral methods to determine this exponent as part of the solution process allowing the exponent to be time dependent from comparing the approximate solutions with numerical and exact analytical solutions whenever possible we show that the cim greatly improves the accuracy on standard heat balance integral methods without detracting from its simplicity

on the period function in a class of generalized lotka volterra systems in this note motivated by the recent results of wang et al wang et al local bifurcations of critical periods in a generalized 2d lv system appl math comput 214 2009 17 25 we study the behaviour of the period function of the center at the point 1 1 of the planar differential systemu u p 1 v q v email protected q u p 1 where p q email protected r with pq 0 and m 0 our aim is twofold firstly we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function secondly by taking advantage of the results of wang et al we show some properties of the bifurcation diagram of the period function and we make some comments for further research the differential system under consideration is a generalization proposed by farkas and noszticzius of the lotka volterra model

fast approximation schemes for convex programs with many blocks and coupling constraints this paper presents block coordinate descent algorithms for the approximate solution of large structured convex programming problems the constraints of such problems consist of k disjoint convex compact sets b k called blocks and m nonnegative valued convex block separable inequalities called coupling or resource constraints the algorithms are based on an exponential potential function reduction technique it is shown that feasibility as well as min mix resource sharing problems for such constraints can be solved to a relative accuracy varepsilon in o k ln m varepsilon 2 ln k iterations each of which solves k block problems to a comparable accuracy either sequentially or in parallel the same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration an extension to objective and constraint functions of arbitrary sign is also presented the above results yield fast approximatio

fast and simple approximation schemes for generalized flow we present fast and simple fully polynomial time approximation schemes fptas for generalized versions of maximum flow multicommodity flow minimum cost maximum flow and minimum cost multicommodity flow we extend and refine fractional packing frameworks introduced in fptas s for traditional multicommodity flow and packing linear programs our fptas s dominate the previous best known complexity bounds for all of these problems some by more than a factor of n2 where n is the number of nodes this is accomplished in part by introducing an efficient method of solving a sequence of generalized shortest path problems our generalized multicommodity fptas s are now as fast as the best non generalized ones we believe our improvements make it practical to solve generalized multicommodity flow problems via combinatorial methods

asymptotic solutions to the stefan problem with a constant heat source at the moving boundary in 1989 huppert formulated mathematical models of the melting and or freezing that occurs when a hot fluid begins to flow turbulently over a cold substrate he solved the resulting stefan problems numerically and presented certain asymptotic results this note revisits one of his problems and adds to the mathematical analysis of its asymptotic structure generalizations and relationships with other results are also noted

local bifurcations of critical periods in a generalized 2d lv system abstract in this paper we investigate a generalized two dimensional lotka volterra system which has a center we give an inductive algorithm to compute polynomials of periodic coefficients find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order we show that with appropriate perturbations at most two critical periods bifurcate from the center

non isochronicity of the center in polynomial hamiltonian systems abstract in 2002 jarque and villadelprat proved that planar polynomial hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial hamiltonian systems of even degree recently it was proved that a planar polynomial hamiltonian system is non isochronous if a quantity denoted by m 2 m 2 can be computed such that m 2 m 2 0 as a corollary of this criterion the open question was answered for those systems with only even degree nonlinearities in this paper we consider the case of m 2 m 2 0 and give a new criterion for non isochronicity applying the new criterion we also answer the open question for some cases in which some terms of odd degree are included

efficient algorithms for budget constrained markov decision processes discounted discrete time discrete state space discrete action space markov decision processes mdps form a classical topic in control game theory and learning and as a result are widely applied increasingly in very large scale applications many algorithms have been developed to solve large scale mdps algorithms based on value iteration are particularly popular as they are more efficient than the generic linear programming approach by an order of magnitude in the number of states of the mdp yet in the case of budget constrained mdps no more efficient algorithm than linear programming is known the theoretically slower running times of linear programming may limit the scalability of constrained mdps piratically while theoretically it invites the question of whether the increase is somehow intrinsic in this technical note we show that it is not and provide two algorithms for budget constrained mdps that are as efficient as value iteration denoting the running time of value iteration by vi and the magnitude of the input by u for an mdp with m expected budget constraints our first algorithm runs in time o poly m log u vi given a pre specified degree of precision for satisfying the budget constraints our second algorithm runs in time o log m poly log u 1 2 vi but may produce solutions that overutilize each of the m budgets by a multiplicative factor of 1 in fact one can substitute value iteration with any algorithm possibly specially designed for a specific mdp that solves the mdp quickly to achieve similar theoretical guarantees both algorithms restrict attention to constrained infinite horizon mdps under discounted costs

barrier subgradient method in this paper we develop a new affine invariant primal dual subgradient method for nonsmooth convex optimization problems this scheme is based on a self concordant barrier for the basic feasible set it is suitable for finding approximate solutions with certain relative accuracy we discuss some applications of this technique including fractional covering problem maximal concurrent flow problem semidefinite relaxations and nonlinear online optimization for all these problems the rate of convergence of our method does not depend on the problem s data

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

on the numerical solution of a stefan problem with finite extinction time in many phase change problems of practical interest it is important to know when a phase is depleted a quantity referred to as the extinction time however there are no numerical schemes that are able to compute this with any degree of rigour or formal accuracy in this paper we develop such a scheme for the one dimensional time dependent problem of an evaporating spherical droplet the keller box finite difference scheme is used in tandem with the so called boundary immobilization method an important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space in particular as regards resolving a square root singularity in the droplet radius as the extinction time is approached

reachability of maximal number of critical periods without independence abstract the reachability of the maximal number of critical periods bifurcated from a weak center is judged with independence of period quantities since the independence is a sufficient condition in this paper we give a method to judge the reachability without the independence our main idea is to consider how the number of zeros for derivative of the period function changes as a curve in the parameter space scans near a specific point and choose an appropriate curve to find the largest number of zeros we apply this method to two polynomial differential systems having zero dimensional isochronicity variety and nonzero dimensional one separately to show that the maximal number is reached and obtain greater numbers than those obtained by classical independence separately with our method the two examples give the possibility that more critical periods arise from an isochronous center than a finite order weak center an opposite fact to many known results

decomposition of algebraic sets and applications to weak centers of cubic systems there are many methods such as grobner basis characteristic set and resultant in computing an algebraic set of a system of multivariate polynomials the common difficulties come from the complexity of computation singularity of the corresponding matrices and some unnecessary factors in successive computation in this paper we decompose algebraic sets stratum by stratum into a union of constructible sets with sylvester resultants so as to simplify the procedure of elimination applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center we determine the order of weak centers and discuss the bifurcation of critical periods

a penalty function heuristic for the resource constrained shortest path problem abstract the resource constrained shortest path problem rcsp consists of finding the shortest path between two nodes of an assigned network with the constraint that traversing an arc of the network implies the consumption of certain limited resources in this paper we propose a new heuristic for the solution of the rcsp problem in medium and large scale networks it is based on the extension to the discrete case of the penalty function heuristic approach for the fast e approximate solution of difficult large scale continuous linear programming problems computational experience on test instances has shown that the proposed penalty function heuristic pfh is very effective in the solution of medium and large scale rcsp instances for all the tests reported it provides very good upper bounds in many cases the optimal solution in less than 26 iterations where each iteration requires only the computation of a shortest path

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

critical periods of perturbations of reversible rigidly isochronous centers abstract in this paper we discuss bifurcation of critical periods in an m th degree time reversible system which is a perturbation of an n th degree homogeneous vector field with a rigidly isochronous center at the origin we present period bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function this procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods applying our procedure to the case n m 2 we determine the maximum number of critical periods and their location to the case n m 3 we investigate the bifurcation of critical periods up to the first order in e and obtain the expression of the second period bifurcation function when the first one vanishes

an accurate numerical solution for the transient heating of an evaporating spherical droplet a recently derived numerical algorithm for one dimensional time dependent stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet the keller box finite difference scheme is used in tandem with the so called boundary immobilization method an important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space an issue not previously discussed in relation to this widely used scheme in addition we demonstrate that our solution is in close agreement with the solution obtained using an alternative numerical scheme that employs an analytic solution of the heat conduction equation inside the droplet for which the droplet radius was assumed to be a piecewise linear function of time the advantages of the new method are discussed

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

approximation algorithm for the mixed fractional packing and covering problem we propose an approximation algorithm based on the lagrangian or price directive decomposition method to compute an epsilon approximate solution of the mixed fractional packing and covering problem find x in b such that f x le 1 epsilon a g x ge 1 epsilon b where f x g x are vectors with m nonnegative convex and concave functions a and b are m dimensional nonnegative vectors and b is a convex set that can be queried by an optimization oracle we propose an algorithm that needs only o m epsilon 2 ln m epsilon 1 iterations or calls to the oracle the main contribution is that the algorithm solves the general mixed fractional packing and covering problem in contrast to pure fractional packing and covering problems and to the special mixed packing and covering problem with b rz n and runs in time independent of the so called width of the problem

weak centers and bifurcation of critical periods in reversible cubic systems abstract in this paper we investigate nonhomogeneous cubic differential systems with reversibility which guarantees that the systems have a center at the origin we apply the ritt wu method to process algebraic equations and inequalities using maple v 3 on a computer we give an inductive algorithm for computing the period coefficient polynomials we find the structure of solutions of systems of algebraic equations corresponding to isochronous centers and weak centers of every finite order and we derive conditions on the parameters under which the origin is an isochronous center or a weak center of finite order we show that with appropriate perturbations local bifurcation of critical periods will occur from weak centers of finite order and isochronous centers

a class of cubic differential systems with 6 tuple focus abstract this paper presents a class of cubic differential systems with the origin as a 6 tuple focus from which 6 limit cycles may be constructed for this class of differential systems the stability of the origin is given

application of standard and refined heat balance integral methods to one dimensional stefan problems the work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems these include standard test problems both with one and two phase changes which have exact solutions that enable us to test the accuracy of the approximate solutions we also consider situations where no analytical solution is available and compare these to numerical solutions it is popular to use a quadratic profile as an approximation of the temperature but we show that a cubic profile seldom considered in the literature is far more accurate in most circumstances in addition the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases we assess which integral method is better for various problems showing that it is largely dependent on the specified boundary conditions

on the accurate numerical solution of a two phase stefan problem with phase formation and depletion in this paper the keller box finite difference scheme is employed in tandem with the so called boundary immobilization method for the purposes of solving a two phase stefan problem that has both phase formation and phase depletion an important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to resolve the boundary condition discontinuities that are associated with the onset of phase formation and depletion in particular this allows the depletion time to be determined and the solution to be computed after depletion the method gives second order accuracy in both time and space for all variables throughout the entire computation

on the numerical solution of two phase stefan problems with heat flux boundary conditions a recently derived numerical algorithm for one dimensional one phase stefan problems is extended for the purpose of two phase moving boundary problems in which the second phase first appears only after a finite delay time this can occur if the phase change is caused by a heat flux boundary condition in tandem with the keller box finite difference scheme the so called boundary immobilization method is used an important component of the work is the use of variable transformations that must be built into the numerical algorithm to resolve the boundary condition discontinuity that is associated with the onset of phase change this allows the delay time until solidification begins to be determined and gives second order accuracy in both time and space

bifurcation of critical periods of polynomial systems abstract we describe a general approach to studying bifurcations of critical periods based on a complexification of the system and algorithms of computational algebra using this approach we obtain upper bounds on the number of critical periods of several families of cubic systems in some cases we overcome the problem of nonradicality of a relevant ideal by moving it to a subalgebra generated by invariants of a group of linear transformations

computational experience with a difficult mixed integer multicommodity flow problem the following problem arises in the study of lightwave networks given a demand matrix containing amounts to be routed between corresponding nodes we wish to design a network with certain topological features and in this network route all the demands so that the maximum load total flow on any edge is minimized as we show even small instances of this combined design routing problem are extremely intractable we describe computational experience with a cutting plane algorithm for this problem