elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

multiple positive solutions of superlinear elliptic problems with sign changing weight we study the existence of multiple positive solutions for a superlinear elliptic pde with a sign changing weight our approach is variational and relies on classical critical point theory on smooth manifolds a special care is paid to the localization of minimax critical points

efficient random walks in the presence of complex two dimensional geometries this paper details an efficient algorithm for particles undergoing random walks in the presence of complex two dimensional solid boundaries the scheme is developed for the simulation of vortex diffusion using the random vortex method both vortex blobs and sheets are handled the relevant modifications for using the algorithm with the vorticity redistribution technique are also discussed the algorithm is designed to be used in the framework of an existing fast multipole implementation a measure for the geometric complexity of a body is introduced and the algorithm s efficiency is studied as various parameters are changed for bodies of varying complexity

isolating segments fixed point index and symbolic dynamics ii homoclinic solutions we present a topological method for detecting homoclinic solutions in nonautonomous ode s as an application we show that there exist infinitely many geometrically distinct homoclinic solutions to the trivial 0 solution in the planar systemz z 1 ei t z 2 for 0 0 495

a fast two dimensional panel method a new two dimensional panel technique has been developed to solve laplacian flows which eliminates the edge effect present in traditional panel methods such a method is very useful for applications where the velocity induced by the panels is required at arbitrary locations particle based flow solvers are a prime example the method however requires considerably more computational effort in this paper the method is modified to improve computational efficiency by adapting the fast multipole algorithm for the panel method significant improvement in computational efficiency is obtained while ensuring that the edge effects are eliminated

a precise calculation of bifurcation points for periodic solution in nonlinear dynamical systems this paper presented a precise yet efficient algorithm to determine bifurcation points of periodic solutions in nonlinear dynamical systems period doubling and symmetry breaking points were obtained to an accuracy of 10 11 the bifurcation points can be directly determined without a tedious trail and error process calculations and analysis on bifurcations of periodic solutions has played a pivotal role in nonlinear dynamics researches this paper presents an efficient and precise approach based on the incremental harmonic balance ihb method to exactly determine the critical points at which periodic solutions exhibit bifurcations the presented method is based on a fact that some zero harmonic coefficients vary as non zeros during the bifurcation of a periodic solution equating one of these coefficients to an infinitesimal quantity the bifurcation value can be determined by the ihb method with the control parameter varying numerical examples show that the critical values for the control parameter where a symmetry breaking or period doubling happens can be calculated very accurately importantly each bifurcation value can be directly obtained without a tedious trail and error process

new transformations of cauchy matrices and trummer s problem abstract we show some new expressions for a cauchy matrix which enable us to simplify the solution of trummer s problem both in the general case and in the case where the input cauchy matrix is fixed for the problem whereas the input vector varies

isolating segments fixed point index and symbolic dynamics iii applications abstract we prove the existence of complicated dynamics in some class of periodic systems on the plane the proof is based on the continuation of the considered system to the model one

positive solutions with a complex behavior for superlinear indefinite odes on the real line abstract we show the existence of infinitely many positive solutions defined on the real line for the nonlinear scalar ode u a t a t u 3 0 where a is a periodic sign changing function and the parameter 0 is large such solutions are characterized by the fact of being either small or large in each interval of positivity of a in this way we find periodic solutions having minimal period arbitrarily large and bounded non periodic solutions exhibiting a complex behavior the proof is variational exploiting suitable natural constraints of nehari type

using hierarchical cartesian grids with multigrid acceleration this paper describes the use of a multigrid strategy on hierarchical cartesian grids for the simulation of small amplitude inviscid waves in a tank quadtree grid generation by recursive subdivision is described and the grid storage system discussed along with the associated grid operations such as neighbour finding and identification of empty cells a multigrid strategy using a novel global approach is discussed with respect to its application to quadtree grids a comparative study is presented of the performance of multigrid parameters such as number of iterations at each level grid visiting schedules and different prolongation operators it is shown that quadtree grids are ideally suited to a multigrid strategy when the solution of the elliptic laplace s equation is sought with savings of the order of 90 in cpu time with respect to a non multigrid solution copyright 2006 john wiley sons ltd

chaotic dynamics in a vertically driven planar pendulum as a first application we prove the presence of chaotic dynamics for a vertically driven planar pendulum the theory of topological horseshoes and linked twist maps developed so far along with phase plane analysis are used to show the presence of chaotic dynamics we propose in the general setting of topological spaces a definition of two dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells for these maps we prove some theorems on the existence of fixed points periodic points and sequences of iterates which are chaotic in a suitable manner our results motivated by the study of the poincare map associated to some nonlinear equations extend and improve some recent work

remark on complicated dynamics of some planar system abstract in this paper we study the complicated dynamics generated by the planar periodic system z z 1 e i t z 2 for 0 0 495 we prove the existence of infinitely many geometrically distinct periodic solutions with some special properties

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

isolating segments fixed point index and symbolic dynamics abstract an extension of the recently introduced srzednicki wojcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented as an application of the method we construct a topological model for the planar equation z 1 e i t z 2 z z c 1 and we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to eq 1 for 0

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

planar nonautonomous polynomial equations ii coinciding sectors abstract we give a few sufficient conditions for the existence of periodic solutions of the equation z j 0 n a j t z j where a j s are complex valued we prove the existence of one up to two periodic solutions and heteroclinic ones

a vorticity based method for incompressible unsteady viscous flows a novel approach is presented based on the integral form of the vorticity formulation in which the vorticity transport equation is solved by using the cell centred finite volume method while the velocities needed at the centre of each control volume are calculated by a modified biot savart formula in conjunction with a fast summation algorithm the vorticity and mass conservation in the flow are guaranteed during the calculation by virtue of the finite volume approach and the method of implementing the boundary conditions at the body surface as an example both the early stage development and long term evolution of the flow around an impulsively started circular cylinder are computed using the method the present results are compared with other numerical and experimental results for the same flow problem and show good agreement

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

topological entropy for local processes abstract the aim of this article is to formalize definition of chaos in terms of topological entropy for dynamics of processes described by nonautonomous differential equations we state a formal definition of topological entropy in this setting and provide tools for estimation of its value its upper or lower bounds in terms of poincare sections

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

chaotic dynamics in periodically forced asymmetric ordinary differential equations abstract using a topological approach we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ode u b u a u p t the choice of the equation is motivated by the studies about the dancer fucik spectrum and the lazer mckenna suspension bridge model

a new diffusion procedure for vortex methods a new method is proposed for simulating diffusion in vortex methods for two dimensional incompressible flows the method resolves length scales up to the spacing of the vortices the grid free nature of vortex methods is fully retained and the distribution of the vortices can be irregular it is shown for the stokes equations that in principle the method can have any order of accuracy it also conserves circulation linear and angular momentum the method is based on exchanging a conserved quantity between arbitrary computational points this suggests that extensions to more general flows may be possible for the two dimensional incompressible flows studied circulation is exchanged between vortices to simulate diffusion the amounts of circulation exchanged must satisfy a linear system of equations based on stability considerations the exchanged amounts should further be positive a procedure to find a solution to this problem is formulated using linear programming techniques to test the method the decay processes of a single point vortex and of a counterrotating pair of point vortices are computed current limitations of the method are discussed

positive periodic solutions of second order nonlinear equations with indefinite weight multiplicity results and complex dynamics we prove the existence of a pair of positive t periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic like dynamics for the scalar second order ode u a t g u 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight of the form a t a t with 0 and large

periodic points of the planar area preserving poincare map inside isolating segment abstract the aim of this note is to study the structure of the set of periodic points of the area preserving poincare map p associated to t periodic local planar process we will use the approach based on the method of periodic isolating segments isolating chains and the fixed point index theory the number of the applications to the processes generated by periodic in time non autonomous planar ode s are given

unstable polarized del pezzo surfaces we provide new examples of k unstable polarized smooth del pezzo surfaces using a flopped version first used by cheltsov and rubinstein of the test configurations introduced by ross and thomas as an application we provide new obstructions for the existence of constant scalar curvature kahler metrics on polarized smooth del pezzo surfaces

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

infinitely many solutions for a floquet type bvp with superlinearity indefinite in sign abstract by the application of a technique developed by g j butler we find infinitely many solutions of the bvp formula where q 0 r is allowed to change sign g r r is superlinear and g x x 0 for all x 0 and r 2 r 2 is a continuous positively homogeneous and nondegenerate map at first we apply the main result to obtain solutions with a prescribed large number of zeros when is the rotation of a fixed angle second we find infinitely many subharmonic solutions of any order and again solutions with a prescribed large number of zeros for the periodic problem associated to the equation c q t g x 0 with q and g as above and c r

multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree abstract we study the periodic boundary value problem associated with the second order nonlinear differential equation u c u a t a t g u 0 where g u has superlinear growth at zero and at infinity a t is a periodic sign changing weight c r and 0 is a real parameter our model includes for c 0 the so called nonlinear hill s equation we prove the existence of 2 m 1 positive solutions when a t has m positive humps separated by m negative ones in a periodicity interval and is sufficiently large thus giving a complete solution to a problem raised by g j butler in 1976 the proof is based on mawhin s coincidence degree defined in open possibly unbounded sets and applies also to neumann boundary conditions our method also provides a topological approach to detect subharmonic solutions

pairs of positive periodic solutions of second order nonlinear equations with indefinite weight abstract we study the problem of the existence and multiplicity of positive periodic solutions to the scalar ode u a t g u 0 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight with negative mean value we first show the nonexistence of solutions for some classes of nonlinearities g x when is small then using critical point theory we prove the existence of at least two positive t periodic solutions for large some examples are also provided

fast adaptive summation of point forces in the two dimensional poisson equation direct summation of the velocity field introduced by point vortices tends to be time consuming since the velocity of each vortex is found as a sum over all other vortices the resulting number of numerical operations is proportional to the square of the number of vortices here a relatively simple procedure is outlined which significantly reduces the number of operations by replacing selected partial sums by asymptotic series the resulting number of operations appears to vary roughly in proportion to the number of unknowns corresponding to a fast solver

remark on krasnosielski s guiding functions and srzednicki s periodic segments abstract we show that the existence of the complete set of guiding functions guarantees the existence of periodic isolating segment that carries the same information concerning periodic solutions of the non autonomous periodic equations x f x t

convergence characteristics and computational cost of two algebraic kernels in vortex methods with a tree code algorithm we study the convergence characteristics of two algebraic kernels used in vortex calculations the rosenhead moore kernel which is a low order kernel and the winckelmans leonard kernel which is a high order kernel to facilitate the study a method of evaluating particle cluster interactions is introduced for the winckelmans leonard kernel the method is based on taylor series expansion in cartesian coordinates as initially proposed by lindsay and krasny j comput phys 172 2001 pp 879 907 for the rosenhead moore kernel a recurrence relation for the taylor coefficients of the winckelmans leonard kernel is derived by separating the kernel into two parts and an error estimate is obtained to ensure adaptive error control the recurrence relation is incorporated into a tree code to evaluate vorticity induced velocity next comparison of convergence is made while utilizing the tree code both algebraic kernels lead to convergence but the winckelmans leonard kernel exhibits a superior convergence rate the combined desingularization and discretization error from the winckelmans leonard kernel is an order of magnitude smaller than that from the rosenhead moore kernel at a typical resolution simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting in particular numerical simulations of the side by side collision of two identical vortex rings suggest that the three dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel we find that the winckelmans leonard kernel is able to perform the same task with a much smaller number of vortex elements than the rosenhead moore kernel greatly reducing the overall computational cost

hyperbolicity and averaging for the srzednicki wojcik equation for the srzednicki wojcik equation the planar nonautonomous ode parameterized by r z z 1 z 2exp i t z t c using averaging we show how the region of hyperbolicity grows with based on this we give bounds on the sizes of bounded orbits

bifurcation and stability properties of periodic solutions to two nonlinear spring mass systems abstract the behavior of a periodically forced linearly damped mass suspended by a linear spring is well known in this paper we study the nature of periodic solutions to two nonlinear spring mass equations our nonlinear terms are similar to earlier models of motion in suspension bridges we contrast the multiplicity bifurcation and stability of periodic solutions for a piecewise linear and smooth nonlinear restoring force we find that while many of the qualitative properties are the same for the two models the nature of the secondary bifurcations period doubling and quadrupling differs significantly

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

isolating chains and chaotic dynamics in planar nonautonomous odes abstract we present a topological method for detecting complicated dynamics in nonautonomous periodic ode s the method is based on the notion of isolating chain as an application we show the chaotic behaviour of processes generated by the equations in the complex plane z z k e i t z where k 3 is an even number and is close to 0

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

superlinear indefinite equations on the real line and chaotic dynamics abstract in this paper we are concerned with a differential equation of the form x c x q t g x 0 t a b where a b q has infinitely many zeros in a b and g is superlinear we prove the existence of solutions with prescribed nodal properties in the intervals of negativity and positivity of q when c 0 and q is periodic we show that the equation under consideration exhibits chaotic like dynamics

on periodic solutions inside isolating chains abstract we develop a geometric approach to problems concerning the existence of t periodic solutions of a non autonomous time t periodic ordinary differential equation we consider isolating segments subsets of the extended phase space of the equation which in some ways resemble isolating blocks from the theory of isolated invariant sets the union of several contiguous isolating segments is called an isolating chain isolating segments determine some homomorphisms in reduced singular homologies the main theorem asserts that the lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the poincare map of the equation in the set of initial values of t periodic solutions contained inside the chain we give some applications of the theorem to planar polynomial equations in particular we prove that the equation z z 5 sin 2 t z has four nonzero periodic solutions provided 0 336

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

superlinear indefinite elliptic problems and pohoz aev type identities abstract we prove the existence of a nontrivial solution for a nonlinear elliptic problem u u a x g u with dirichlet boundary condition on a bounded domain where g is superlinear both at zero and at infinity a x changes sign and 0

example of a suspension bridge ode model exhibiting chaotic dynamics a topological approach abstract using an elementary phase plane analysis combined with some recent results on topological horseshoes and fixed points for planar maps we prove the existence of infinitely many periodic solutions as well as the presence of chaotic dynamics for a simple second order nonlinear ordinary differential equation arising in the study of lazer mckenna suspension bridges model

a multipole expansion technique in solving boundary integral equations abstract the boundary integral equation method reduces the spatial dimension of an elliptic problem by converting the original n dimensional partial differential equation to an n 1 dimensional boundary integral equation defined on the boundary of the domain of the original problem at the same time the discretisation of the problem is also remarkably simplified the price of these advantages however is that the structure as well as the algebraic properties of the resulting boundary element matrices are somewhat unpleasant since they are neither sparse nor self adjoint in general even if the original problem is self adjoint consequently the computational cost of the boundary integral equation method seems to be unnecessarily high to make the method more economic from a computational point of view we present a numerical technique based on the multipole expansion method which reduces the computational cost of the appearing matrix vector multiplications i e the evaluations of the discretized boundary integrals by a remarkable amount the method is applicable also to the reconstruction problem when the inner solution is to be reconstructed from the boundary solutions

a fast adaptive vortex method in three dimensions the method of local corrections mlc developed by anderson for two spatial dimensions is a particle particle particle mesh method in which the calculation of the velocity field induced by a collection of vortices is split into two parts i a finite difference velocity field calculation using a fast poisson solver the results of which are used to represent the velocity field induced by vortices far from the evaluation point and ii an n body calculation to compute the velocity field at a vortex induced by nearby vortices we present a fast vortex method for incompressible flow in three dimensions based on the extension of the mlc algorithm from two to three spatial dimensions and the use of adaptive mesh refinement in the finite difference calculation of the mlc calculations with a vortex ring in three dimensions show that the break even point between the mlc with amr and the direct method is at n 3000 on a cray y mp for n 64 000 mlc with amr can be 12 times faster than the direct method results from calculations of two colliding inviscid vortex rings demonstrate the increased resolution which can be obtained using fast methods

the method of holomorphic foliations in planar periodic systems the case of riccati equations abstract we develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients this method relies on analysis of geometry of leaves of certain holomorphic foliations in cp1 cp1 we concentrate our attention on the case of riccati equations we determine the dynamics of the non autonomous equations using some local invariants of holomorphic vector fields defining the foliations in particular new examples of riccati equations without periodic solutions are presented

a geometric criterion for positive topological entropy we prove that a diffeomorphism possessing a homoclinic point with a topological crossing possibly with infinite order contact has positive topological entropy along with an analogous statement for heteroclinic points we apply these results to study area preserving perturbations of area preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections in the heteroclinic case the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection in the homoclinic case the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection these results significantly simplify the application of the poincare arnold melnikov sotomayor method the results apply even when the contraction and expansion at the fixed point is subexponential