partially implicit schemes for the numerical solutions of some non linear differential equations as numerical methods are one of the main tools used to discover the non linear behaviour characteristics of continuous dynamical systems a study on the use of numerical schemes which delineate the true physics from numerical artefact is desirable as a step in that direction partially implicit rational schemes based on first order methods and a second order method are derived and analysed from the point of view of dynamical systems for a scalar autonomous ordinary differential equation the second order method is developed by taking a linear combination of first order methods though implicit in nature the methods are applied explicitly the approach adopted is extended to obtain numerical schemes for the solution of a class of reaction diffusion partial differential equations the methods are tested on differential equations with cubic and quadratic reaction terms it is found that for certain sets of initial conditions the first order partially implicit schemes converge monotonically to the correct fixed points for all time steps unlike the euler method and the second order method in other words for the same steady state solution the associated domain of attraction of the discrete problem and its continuous counterpart are exactly the same however for other initial conditions it is found that the domains of attraction of the second order method mimic more closely the domains of attraction of the continuous differential equations the comparative study of the various explicit runge kutta methods of orders 2 6 shows that it is the discretized form with its specific algebraic form recursive relation which are the source of spurious steady state solutions and numerical chaos from these methods it is found however that the use in general of fifth order runge kutta methods is seen to be preferred to other runge kutta methods as they have an extended stability range in particular the fifth order lawson method which has a far superior range of stability than the other runge kutta methods

irregular dynamics and homoclinic orbits to hamiltonian saddle centers we consider 4 dimensional real analytic hamiltonian systems with a saddle center equilibrium related to a pair of real and a pair of imaginary eigenvalues and a homoclinic orbit to it we find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium we also consider one parameter families of reversible 4 dimensional hamiltonian systems we prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points we also present similar results for systems with heteroclinic orbits to saddle center equilibria 1997 john wiley sons inc

a stochastic dynamic model to evaluate the influence of economy and well being on unemployment control abstract this paper presents a stochastic dynamic mathematical model to study the evolution of the unemployment rate and other relevant related variables in a country this model is composed by three basic interrelated subsystems demographic economic and well being ones a key aspect of this model is that it considers three un well being variables simultaneously human development index gender empowerment index and gender differentiation index these variables involve key concepts for human development as health education economy and female labor with this model the most outstanding variables found in the literature in relation with unemployment control can be used to design strategies and scenarios to reduce the unemployment rate in the future the model has been fitted for the case of spain in the 2002 2014 period the largest one with information about all the variables involved in the model finally several tentative scenarios and strategies have been tested to reduce the unemployment rate in spain in the horizon of year 2025 and the corresponding forecast evolution of the gender differentiation index national income per capita public debt and the ratio between public debt and gross domestic product are shown

whitney extension theorems for convex functions of the classes c1 and c1 ω let c be a subset of n not necessarily convex f c r be a function and g c n be a uniformly continuous function with modulus of continuity we provide a necessary and sufficient condition on f g for the existence of a convex function f cc1 n such that f f on c and f g on c with a good control of the modulus of continuity of f in terms of that of g on the other hand assuming that c is compact we also solve a similar problem for the class of c1 convex functions on n with a good control of the lipschitz constants of the extensions namely lip f g finally we give a geometrical application concerning interpolation of compact subsets k of n by boundaries of c1 or c1 1 convex bodies with prescribed outer normals on k

a faster combinatorial approximation algorithm for scheduling unrelated parallel machines we consider the problem of scheduling n independent jobs on m unrelated parallel machines without preemption job i takes processing time p i j on machine j and the total time used by a machine is the sum of the processing times for the jobs assigned to it the objective is to minimize makespan the best known approximation algorithms for this problem compute an optimum fractional solution and then use rounding techniques to get an integral 2 approximation in this paper we present a combinatorial approximation algorithm that matches this approximation quality it is much simpler than the previously known algorithms and its running time is better this is the first time that a combinatorial algorithm always beats the interior point approach for this problem our algorithm is a generic minimum cost flow algorithm without any complex enhancements tailored to handle unsplittable flow it pushes unsplittable jobs through a two layered bipartite generalized network defined by the scheduling problem in our analysis we take advantage from addressing the approximation problem directly in particular we replace the classical technique of solving the lp relaxation and rounding afterwards by a completely integral approach we feel that this approach will be helpful also for other applications

global surfaces of section in non regular convex energy levels of hamiltonian systems in this paper we prove the existence of global sections of disk type in non regular and strictly convex energy levels of integrable and near integrable hamiltonian systems with two degrees of freedom this extends a result of hofer et al in ann math 2 148 1 197 289 1998 where the same statement is true provided the energy level is regular

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

an extension theorem for convex functions of class c1 1 on hilbert spaces abstract let h be a hilbert space e h be an arbitrary subset and f e r g e h be two functions we give a necessary and sufficient condition on the pair f g for the existence of a convex function f c 1 1 h such that f f and f g on e we also show that if this condition is met f can be taken so that lip f lip g we give a geometrical application of this result concerning interpolation of sets by boundaries of c 1 1 convex bodies in h finally we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous

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

on the stability of double homoclinic loops we consider 2 degrees of freedom hamiltonian systems with an involutive symmetry and a pair of orbits bi asymptotic homoclinic to a saddle center equilibrium related to pairs of pure real and pure imaginary eigenvalues i we show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one dimensional scattering problem and we also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

the conley zehnder index and the saddle center equilibrium abstract we consider hamiltonian systems with two degrees of freedom we suppose the existence of a saddle center equilibrium in a strictly convex component s of its energy level moser s normal form for such equilibriums and a theorem of hofer wysocki and zehnder are used to establish the existence of a periodic orbit in s with several topological properties we also prove the explosion of the conley zehnder index of any periodic orbit that passes close to the saddle center equilibrium

the existence of hypersurfaces of constant gauss curvature with prescribed boundary in this paper we are concerned with the problem of finding hypersurfaces of constant gauss kronecker curvature k hypersurfaces in r n 2 with prescribed boundary given a disjoint collection 1 m of closed smooth embedded n 1 dimensional submanifolds of r decide whether there exist immersed k hypersurfaces m in r with m locally this problem reduces to questions concerning monge ampere type equations and we seek solutions for which the resulting equation is elliptic this means that we must confine ourselves to the class of locally strictly convex hypersurfaces i e those whose principal curvatures are all positive such hypersurfaces locally lie on one side of their tangent planes at any point but need not do so globally as they have non empty boundary finding hypersurfaces with prescribed curvature and boundary has been a major challenge in geometric analysis because of the highly nonlinear nature of the problem and the lack of variational methods beginning around 1980 some success was achieved due to breakthroughs in the theory of monge ampere equations and general fully nonlinear equations but only for hypersurfaces which are globally graphs of functions over domains with geometric restrictions e g strictly convex domains this was the case even for k hypersurfaces where the only general existence results were consequences of the existence theory for monge ampere equations see 6 20 23 and was restricted to strictly convex domains this means that the resulting surfaces must be simply connected graphs a very strong restriction geometrically the first idea that a more general result was possible came in the paper of hoffmanrosenberg spruck 18 and subsequently such a general result was developed in 12 and 11 in these papers the authors proved an essentially optimal existence theorem for monge ampere equations in domains of arbitrary geometry and thus the limit of

conservative dynamics unstable sets for saddle center loops we consider two degree of freedom hamiltonian systems with a saddle center loop namely an orbit homoclinic to a saddle center equilibrium related to pairs of pure real and pure imaginary i eigenvalues we study the topology of the sets of orbits that have the saddle center loop as their and limit set a saddle center loop as a periodic orbit is a closed loop in phase space and the above sets are analogous to the unstable and stable manifolds respectively of a hyperbolic periodic orbit

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

the convex hull property and topology of hypersurfaces with nonnegative curvature abstract we prove that in euclidean space any nonnegatively curved compact smoothly immersed hypersurface lies outside the convex hull of its boundary provided the boundary satisfies certain required conditions this gives a convex hull property dual to the classical one for surfaces with nonpositive curvature a version of this result in the nonsmooth category is obtained as well we show that our boundary conditions determine the topology of the surface up to at most two choices the proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure a noncollapsing convergence theorem and a gluing procedure

a stochastic dynamical social model involving a human happiness index abstract this paper presents a new human happiness index built through five dimensions development freedom solidarity justice and peace these five dimensions are evaluated through quantitative variables obtained from the human development reports world data bank and eurostat the new happiness index has been built following the guidelines set by the human development reports of the un for the construction of quality indices and it has been compared on a set of 13 eu countries with the overall life satisfaction index which is used by the un moreover the new index has been included in a dynamic mathematical model through the demographic rates to study the evolution of the population the obtained model has been calibrated for the period 2004 2009 and validated for the period 2010 2015 for the case of spain finally the model has been used to maximize the happiness index in spain for the period 2016 2030 with the conclusion that to achieve this purpose it is necessary to invest in education research and development

topology of riemannian submanifolds with prescribed boundary we prove that a smooth compact submanifold of codimension 2 immersed in r n 3 bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces this settles a question of guan and spruck related to a problem of yau analogous results for complete fillings of arbitrary riemannian submanifolds are obtained as well on the other hand we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular our proofs employ among other methods a relative version of nash s isometric embedding theorem and the theory of alexandrov spaces with curvature bounded below including the compactness and stability theorems of gromov and perelman

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

mathbf h principles for hypersurfaces with prescribed principle curvatures and directions we prove that any compact orientable hypersurface with boundary immersed resp embedded in euclidean space is regularly homotopic resp isotopic to a hypersurface with principal directions which may have any prescribed homotopy type and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant further we construct regular homotopies resp isotopies which control the principal curvatures and directions of hypersurfaces in a variety of ways these results which we prove by holonomic approximation establish some h principles in the sense of gromov and generalize theorems of gluck and pan on embedding and knotting of positively curved surfaces in 3 space

existence of chaos in the plane r 2 and its application in macroeconomics the existence of devaney li yorke and distributional chaos in the plane r 2 the chaos in a continuous dynamical system generated by euler equation branching all combinations of hyperbolic singular points of both branches admit chaos the chaotic set of solutions associated with chaotic subset of r 2 exists new overall macroeconomic equilibrium model called is lm qy ml model the existence of chaos in an economy the devaney li yorke and distributional chaos in the plane r 2 can occur in the continuous dynamical system generated by euler equation branching euler equation branching is a type of differential inclusion x f x g x where f g x r n r n are continuous and f x g x in every point x x stockman and raines 2010 defined the so called chaotic set in the plane r 2 whose existence leads to the existence of devaney li yorke and distributional chaos in this paper we follow up on stockman and raines 2010 and we show that chaos in the plane r 2 is always admitted for hyperbolic singular points in both branches not lying in the same point in r 2 but the chaos existence is also caused by a set of solutions of euler equation branching we research this set of solutions in the second part we create the new overall macroeconomic equilibrium model called is lm qy ml model this model is based on the fundamental macroeconomic equilibrium model called is lm model we model an inflation effect an endogenous money supply an economic cycle etc in addition to the original is lm model we research the dynamical behaviour of this new is lm qy ml model and show when a chaos exists with relevant economic interpretation

local convexity and nonnegative curvature gromov s proof of the sphere theorem an immersed hypersurface s in a riemannian manifold m will be called e convex for some e 0 if all principal curvatures have the same sign and absolute value at least e can one characterize all compact e convex immersed hypersurfaces of a complete manifold m if m is euclidean n space n 3 this problem is solved by a theorem of hadamard 14 generalized by hopf 15 s is embedded and bounds a convex n disk see 6 6 6 3 if m is a flat space form s is no longer necessarily embedded but it still bounds an immersed convex n disk the aim of this paper is to show the same fact for a manifold m of curvature k 0

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

makespan minimization on unrelated parallel machines with simple job intersection structure and bounded job assignments abstract let there be a set j of n jobs and a set m of m parallel machines where each job j j takes p i j z time units on machine m i and assume p i j implies job j j cannot be scheduled on machine m i in makespan minimization on unrelated parallel machines r c m a x the goal is to schedule each job non preemptively on a machine so as to minimize the makespan a job intersection graph g j j e j is an unweighted undirected graph where there is an edge j j j j e j if there is a machine m i such that both p i j and p i j in this paper we consider two variants of r c m a x where there are a small number of eligible jobs per machine first we prove that there are no approximation algorithms with approximation ratios less than 3 2 for r c m a x when restricted to instances with job intersection graphs belonging to some graph classes such as diamondless graphs and planar graphs unless p np second we match this lower bound by presenting a 3 2 approximation algorithm for r c m a x restricted to instances with diamondless job intersection graphs and furthermore show that when g j is triangle free r c m a x is solvable in polynomial time for r c m a x restricted to instances when every machine can process at most l jobs we give an approximation algorithm with approximation ratios 3 2 and 5 3 for l 3 and l 4 respectively a polynomial time algorithm when l 2 and prove that it is np hard to approximate the optimum solution within a factor less than 3 2 when l 3 in the special case where every p i j p j called the restricted assignment problem and there are only two job lengths p j we present a 2 1 l 1 approximation algorithm when l 3 in addition we give a 2 approximation algorithm for the so called unrelated graph balancing problem for l 3 in the case where j is partitioned into b sets called bags and no two jobs from the same bag can be scheduled on the same machine

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

the convex hull property of noncompact hypersurfaces with positive curvature we prove that in euclidean space r n 1 every metrically complete positively curved immersed hypersurface m with compact boundary m lies outside the convex hull of m provided that m is embedded on the boundary of a convex body and n 2 for n 2 on the other hand we construct examples which contradict this property

tangent cones and regularity of real hypersurfaces we characterize c embedded hypersurfaces of r as the only locally closed sets with continuously varying flat tangent cones whose measuretheoretic multiplicity is at most m 3 2 it follows then that any topological hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is c in the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface in particular any convex real analytic hypersurface x r is c furthermore if x is real algebraic strictly convex and unbounded then its projective closure is a c hypersurface as well which shows that x is the graph of a function defined over an entire hyperplane