combinatorial interior point methods for generalized network flow problems we present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on wallacher and zimmermann s combinatorial interior point method for the minimum cost network flow problem the algorithms have features of both a combinatorial algorithm and an interior point method they work towards optimality by iteratively reducing the value of a potential function while maintaining interior point solutions at each iteration flow is augmented along a generalized circulation which is computed by solving a tvpi two variables per inequality system the algorithms run in time where m and n are respectively the number of arcs and nodes in the graph and l is the length of the input data

a numerical ode solver that preserves the fixed points and their stability in this paper we provide a one step predictor corrector method for numerically solving first order differential initial value problems with two fixed points the method preserves the stability behaviour of the fixed points which results in an efficient integrator for this kind of problem some numerical examples are provided to show the good performance of the method

periodic orbits and non integrability of henon heiles systems we apply the averaging theory of second order to study the periodic orbits for a generalized henon heiles system with two parameters which contains the classical henon heiles system two main results are shown the first result provides sufficient conditions on the two parameters of these generalized systems which guarantee that at any positive energy level the hamiltonian system has periodic orbits these periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy the second result shows that for the non integrable henon heiles systems in the sense of liouville arnol d which have the periodic orbits analytically found with averaging theory cannot exist any second first integral of class in particular for any second first integral of class we prove that the classical henon heiles system and many generalizations of it are not integrable in the sense of liouville arnol d moreover the tools we use for studying the periodic orbits and the non liouville arnol d integrability can be applied to hamiltonian systems with an arbitrary number of degrees of freedom

stability of explicit runge kutta methods abstract the classical theory of stability of explicit runge kutta methods is concerned with lipschitzian problems it is not useful for stable problems with large lipschitz constants the classical theory of absolute stability considers some very special problems of this kind the problems treated arise when a general problem is linearized it is hoped that the behavior in the case of the special problem provide guidelines as to the behavior in the case of a general problem a synthesis is proposed here which responds to this unsatisfactory state of affairs in the classical theory

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

a criterion for non integrability based on poincare s theorem abstract the non integrability of a two degrees of freedom hamiltonian is investigated by a method based on a well known theorem of poincare it is proved that the perturbed system is non integrable for values of the perturbative parameter in an open interval around zero if on a dense set of resonant tori of the unperturbed system the average value of the perturbative function evaluated along the periodic orbits on each torus depends on the particular orbit an application to the separable quartic oscillator with a quadratic perturbation is made and it is shown that if the perturbation is non separable in the same coordinates the system is non integrable

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

on the number of isolating integrals in perturbed hamiltonian systems with n or 3 degrees of freedom we consider a perturbed hamiltonian system with n or 3 degrees of freedom of the form h h0 epsilon h1 and show that the properties of the average value of the perturbing function h1 along the periodic orbits of the unperturbed integrable part h0 supply criteria for non integrability which restrict the allowed total number of independent integrals of motion

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

new algorithms for generalized network flows this paper of which a preliminary version appeared in istcs 92 is concerned with generalized network flow problems in a generalized network each edgee u v has a positive flow multiplier a e associated with it the interpretation is that if a flow ofx e enters the edge at nodeu then a flow ofa e x e exits the edge atv the uncapacitated generalized transshipment problem ugt is defined on a generalized network where demands and supplies real numbers are associated with the vertices and costs real numbers are associated with the edges the goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand and the sum over the edges of the product of the cost and the flow is minimized adler and cosares operations research 39 1991 955 960 reduced the restricted uncapacitated generalized transshipment problem where only demand nodes are present to a system of linear inequalities with two variables per inequality the algorithms presented by the authors in siam journal on computing to appear result in a faster algorithm for restricted ugt generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges i e upper bounds on the amount of flow the goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized we present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of ugt the algorithm can be used to find an optimal flow or an approximation thereof when used to find a constant factor approximation the algorithm is not only more efficient than previous algorithms but also strongly polynomial it is believed to be the first strongly polynomial approximation algorithm for generalized circulation the existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm

periodic orbits and nonintegrability of generalized classical yang mills hamiltonian systems the averaging theory of first order is applied to study a generalized yang mills system with two parameters two main results are proved first we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy and these families are given up to first order in a small parameter second we prove that for the nonintegrable classical yang mills hamiltonian systems in the sense of liouville arnold which have the isolated periodic orbits found with averaging theory cannot exist in any second first integral of class c1 this is important because most of the results about integrability deals with analytic or meromorphic integrals of motion

deciding linear inequalities by computing loop residues v r pratt has shown that the real and integer feastbdlty of sets of linear mequallues of the form x y c can be decided quickly by examining the loops m certain graphs pratt s method is generahzed first to real feaslbdlty of mequahues m two variables and arbitrary coefficients and ultimately to real feaslbdlty of arbitrary sets of hnear mequahtles the method is well suited to apphcatlons m program verification

a strongly polynomial algorithm for a special class of linear programs we extend the list of linear programming problems that are known to be solvable in strongly polynomial time to include a class of lps which contains special cases of the generalized transshipment problem the result is facilitated by exploiting some special properties associated with leontief substitution systems and observing that a feasible solution to the system ax b x 0 in which no variable appears in more than two equations can be found in strongly polynomial time for b belonging to some set

embedded runge kutta formulae with stable equilibrium states the equilibrium theory of hall and higham 1988 can be used to determine whether a runge kutta algorithm will perform smoothly when stability restricts the stepsize in this paper we show that current high quality order 4 5 pairs do not behave well in this respect and we determine the extent to which the overall quality must be compromised in order for the equilibrium conditions to be satisfied three new formulae are presented and their properties are compared with those of existing formulae

new aspects of integrability of generalized henon heiles systems the class of so called henon heiles systems is slightly broadened by allowing for the existence of non standard hamiltonians the extra parameter in the equations of motion is shown to give rise to a generalization of the three known integrability cases in addition three degenerate cases are detected characterized by a partial decoupling of the equations for these cases the author still obtains two independent first integrals but their involutiveness can only be understood in terms of a nonstandard poisson structure

proof of non integrability for the henon heiles hamiltonian near an exceptional integrable case abstract in this paper we describe an analytical method for establishing non integrability in hamiltonian systems which are small perturbations of integrable ones we illustrate the method with an application to a system of henon heiles type

an efficient variable step size rational falkner type method for solving the special second order ivp in this paper firstly a rational one parameter family of falkner type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations the proposed family of methods has second algebraic order of convergence imposing that the principal term of the local truncation error of the proposed family vanishes we get an expression for the free parameter at the grid point xn yn by substituting this value of the free parameter in the family a new rational third order method is obtained further by combining the third order method with any member of the second order family their variable step size formulation as an embedded pair is considered some numerical experiments are given to illustrate the performance and efficiency of the proposed methods

a unified approach for the development of k step block falkner type methods for solving general second order initial value problems in odes a family of k step block multistep methods where the main formulas are of falkner type is proposed for the direct integration of the general second order initial value problem where the differential equation is of the general form y f x y y the two main falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k 1 points the main characteristics of the methods are discussed through their formulation in vector form each method is formulated as a group of 2 k simultaneous formulas over k non overlapping intervals in this way the method produces the approximation of the solution simultaneously at k points on these intervals as in other block methods there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods the resulting family is efficient and competitive compared with other existing methods in the literature as may be seen from the numerical results

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

control of local error stabilizes integrations codes for the solution of the initial value problem for a system of ordinary differential equations adjust their step sizes so as to keep the estimated local errors smaller than a given tolerance and at the same time solve the problem as efficiently as possible step sizes providing the desired accuracy may be too large for stability yet this does not cause the popular codes to become unstable the theory explaining why this is so holds only in very restricted circumstances this paper broadens and deepens the theory for explicit runge kutta methods

solving first order initial value problems by using an explicit non standard a stable one step method in variable step size formulation this paper presents the construction of a new family of explicit schemes for the numerical solution of initial value problems of ordinary differential equations odes the one parameter family is constructed by considering a suitable rational approximation to the theoretical solution resulting a family with second order convergence and a stable imposing that the principal term in the local truncation error vanishes we obtain an expression for the parameter value in terms of the point xn yn on each step with this approach the resulting method has third order convergence maintaining the characteristic of a stability finally combining this last method with other of order two in order to get an estimation for the local truncation error an implementation in variable step size has been considered the proposed method can be used in a wide range of problems for solving numerically a scalar ode or a system of first order odes several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature

an e relaxation method for separable convex cost generalized network flow problems cover title the extended abstract of this article appeared in the proceedings of the 5th international ipco conference vancouver june 1996 p 1

on the periodic orbits and the integrability of the regularized hill lunar problem the classical hill s problem is a simplified version of the restricted three body problem where the distance of the two massive bodies say primary for the largest one and secondary for the smallest one is made infinity through the use of hill s variables the levi civita regularization takes the hamiltonian of the hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the coriolis force and the sun action polynomials of degree 4 and 6 respectively in this paper we study periodic orbits of the planar hill problem using the averaging theory moreover we provide information about the c1 integrability or non integrability of the regularized hill lunar problem