structural stability of solutions to the riemann problem for a scalar conservation law the aim of this paper is to study the structural stability of solutions to the riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients it is proved that the riemann solution is possibly instable when one of the riemann initial data is at the vacuum furthermore we point out that the riemann solution is also possibly instable even when the riemann initial data stay far away from vacuum in order to deal with it we perturb the riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively it is also proved that the riemann solutions are unstable in some certain situations under the local small perturbations of the riemann initial data by letting the perturbed parameter e tend to zero in addition the interaction of the delta standing wave and the contact vacuum state is considered which appear in the riemann solutions

a note on set theoretic solutions of the yang baxter equation abstract this paper shows that every finite non degenerate involutive set theoretic solution x r of the yang baxter equation whose permutation group g x r has cardinality which is a cube free number is a multipermutation solution some properties of finite braces are also investigated it is also shown that if a is a left brace whose cardinality is an odd number and a b a b for all a b a then a is a two sided brace and hence a jacobson radical ring it is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level

global bifurcation of limit cycles in a family of polynomial systems in this paper we study the number of limit cycles in a family of polynomial systems using bifurcation methods and melnikov functions we obtain the maximal number of limit cycles in global bifurcation

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

the smallest upper bound on the number of zeros of abelian integrals abstract in this paper we develop a new method to estimate the smallest upper bound on the number of isolated zeros of abelian integrals and give an algebraic criterion for the case of abelian integrals along energy level ovals of potential systems as applications of our main result we first obtain a criterion guaranteeing that two abelian integrals have the chebyshev property with accuracy one and give an example of hyperelliptic abelian integrals then we get a criterion that any nontrivial linear combination of three abelian integrals has at most two isolated zeros using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non algebraic abelian integral along the oval y 2 1 2 1 x 2 ln x 1 2 h with 0 h which corresponds to the perturbation of a quadratic reversible system with two centers moreover we derive all possible configurations of limit cycles from the poincare bifurcation of this perturbation to our knowledge these problems can not be solved by other known approaches

multiplicity of fixed points and growth of e neighborhoods of orbits abstract we study the relationship between the multiplicity of a fixed point of a function g and the dependence on e of the length of e neighborhood of any orbit of g tending to the fixed point the relationship between these two notions was discovered in elezovic ubrinic and upanovic 2007 5 in the differentiable case and related to the box dimension of the orbit here we generalize these results to non differentiable cases introducing a new notion of critical minkowski order we study the space of functions having a development in a chebyshev scale and use multiplicity with respect to this space of functions with the new definition we recover the relationship between multiplicity of fixed points and the dependence on e of the length of e neighborhoods of orbits in non differentiable cases applications include in particular poincare maps near homoclinic loops and hyperbolic 2 cycles and abelian integrals this is a new approach to estimate the cyclicity by computing the length of the e neighborhood of one orbit of the poincare map for example numerically and by comparing it to the appropriate scale

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

alien limit cycles in lienard equations this paper aims at providing an example of a family of polynomial lienard equations exhibiting an alien limit cycle this limit cycle is perturbed from a 2 saddle cycle in the boundary of an annulus of periodic orbits given by a hamiltonian vector field the hamiltonian represents a truncated pendulum of degree 4 in comparison to a former polynomial example not only the equations are simpler but a lot of tedious calculations can be avoided making the example also interesting with respect to simplicity in treatment

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

counting real zeros of analytic functions satisfying linear ordinary differential equations abstract we suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients if the equation formula has no singular points in a small neighborhooduof a real segmentk all the coefficientsaj t have absolute value aonuanda0 t 1 then any solution of this equation may have no more than a zeros onk where u k is a geometric constant depending only onkandu if the principal coefficienta0 t is nonconstant but its modulus is at leasta 0 somewhere onk then the number of real zeros onkof any solution analytic inu does not exceed a a with some u k

fractal just infinite nil lie superalgebra of finite width abstract the grigorchuk and gupta sidki groups play fundamental role in modern group theory they are natural examples of self similar finitely generated periodic groups as their natural analogues there are constructions of nil lie p algebras over a field of characteristic 2 40 and arbitrary positive characteristic 52 in characteristic zero similar examples of lie algebras do not exist by a result of martinez and zelmanov 32 the second author constructed analogues of the grigorchuk and gupta sidki groups in the world of lie superalgebras of arbitrary characteristic the virtue of that construction is that the lie superalgebras have clear monomial bases 41 that lie superalgebras have slow polynomial growth and are graded by multidegree in the generators in particular a self similar lie superalgebra q is z 3 graded by multidegree in 3 generators its z 3 components lie inside an elliptic paraboloid in space the components are at most one dimensional thus the z 3 grading of q is fine an analogue of the periodicity is that homogeneous elements of the grading q q 0 q 1 are ad nilpotent in particular q is a nil finely graded lie superalgebra which shows that an extension of the mentioned result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid but computations with q are rather technical in this paper we construct a similar but simpler and smaller example namely we construct a 2 generated fractal lie superalgebra r over arbitrary field we find a clear monomial basis of r and unlike many examples studied before we find also a clear monomial basis of its associative hull a the latter has a quadratic growth the algebras r and a are z 2 graded by multidegree in the generators positions of their z 2 components are bounded by pairs of logarithmic curves on plane the z 2 components of r are at most one dimensional thus the z 2 grading of r is fine as an analogue of periodicity we establish that homogeneous elements of the grading r r 0 r 1 are ad nilpotent in case of n graded algebras a close analogue to being simple is being just infinite unlike previous examples of lie superalgebras we are able to prove that r is just infinite but not hereditary just infinite our example is close to the smallest possible example because r has a linear growth with a growth function r m 3 m as m moreover r is of finite width 4 char k 2 in case char k 2 we obtain a lie algebra of width 2 that is not thin thus we have got a more handy analogue of the grigorchuk and gupta sidki groups the constructed lie superalgebra r is of linear growth of finite width 4 and just infinite it also shows that an extension of the result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

on the finite cyclicity of open period annuli let be an open relatively compact period annulus of real analytic vector fieldx0 on an analytic surface we prove that the maximal number of limit cycles which bifurcate from under a given multiparameter analytic deformation x of x0 is finite provided that x0 is either a hamiltonian or generic darbouxian vector field 1 statement of the result let s be a real analytic surface without border compact or not and let x0 be a real analytic vector field on it an open period annulus of x0 is a union of period orbits of x0 which is bianalytic to the standard annulus s1 0 1 the image of each circle s1 u being a periodic orbit of x0 let x r 0 be an analytic family of analytic vector fields on s and let be an open period annulus of x0 the cyclicity cycl x of with respect to the deformation x is the maximal number of limit cycles of x which tend to as tends to zero see definition 2 clearly the vector field x0 has an analytic first integral f in the period annulus which has no critical points in what follows we suppose that the open period annulus is relatively compact i e its closure s is compact definition 1 we say that x0 is a hamiltonian vector field provided that it has a first integral with isolated critical points in a complex neighborhood of we say that x0 is a generic darbouxian vector field provided that all singular points of x0 in a neighborhood of are orbitally analytically equivalent to linear saddles x y with 0 remark 1 note that if x0 is a plane vector field with a first integral h as above then x0 h1 hy x hx y duke mathematical journal vol 152 no 1 c 2010 doi 10 1215 00127094 2010 005 received 3 july 2008 revision received 1 may 2009 2000 mathematics subject classification primary 34c07 secondary 34c10

bounding the number of zeros of certain abelian integrals abstract in this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n abelian integrals to have at most n k 1 zeros counted with multiplicities this condition involves the functions in the integrand of the abelian integrals and it can be checked in many cases in a purely algebraic way

on the number of zeros of abelian integrals a constructive solution of the infinitesimal hilbert sixteenth problem we prove that the number of limit cycles generated from nonsingular energy level ovals periodic trajectories in a small non conservative perturbation of a hamiltonian polynomial vector field on the plane is bounded by a double exponential of the degree of the fields this solves the long standing infinitesimal hilbert 16th problem

abelian integrals and limit cycles abstract the paper deals with generic perturbations from a hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles in this paper we show that near a 2 saddle cycle the number of limit cycles produced in unfoldings with one unbroken connection can exceed the number of zeros of the related abelian integral even if the latter represents a stable elementary catastrophe we however also show that in general finite codimension of the abelian integral leads to a finite upper bound on the local cyclicity in the treatment we introduce the notion of simple asymptotic scale deformation

limit cycle bifurcations near a 2 polycycle or double 2 polycycle of planar systems abstract in this paper we study the bifurcation problem of limit cycles for a near hamiltonian system near a 2 polycycle or double 2 polycycle we obtain new results on the number of limit cycles by using the method of stability changing of a homoclinic loop this method also provides a new way to find alien limit cycles especially we found some alien limit cycles of new type

efficient well balanced hydrostatic upwind schemes for shallow water equations the proposed work concerns the numerical approximations of the shallow water equations with varying topography the main objective is to introduce an easy and systematic technique to enforce the well balance property and to make the scheme able to deal with dry areas to access such an issue the derived numerical method is obtained by involving the free surface instead of the water height and this produces the scheme well balanced independently from the numerical flux function associated with the homogeneous problem as a consequence we obtain an easy well balanced scheme which preserves non negative water height when compared with the well known hydrostatic reconstruction the presented topography discretization does not involve any max function known to introduce some numerical errors as soon as the topography admits very strong variations or discontinuities a second order muscl accurate reconstruction is adopted the proposed hydrostatic upwind scheme is next extended for considering 2d simulations performed over unstructured meshes several 1d and 2d numerical experiments are performed to exhibit the relevance of the scheme

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

fractal nil graded lie superalgebras abstract the grigorchuk and gupta sidki groups play fundamental role in modern group theory they are natural examples of self similar finitely generated periodic groups the author constructed their analogue in case of restricted lie algebras of characteristic 2 31 shestakov and zelmanov extended this construction to an arbitrary positive characteristic 41 thus we have examples of self similar finitely generated restricted lie algebras with a nil p mapping in characteristic zero similar examples of lie algebras do not exist by a result of martinez and zelmanov 27 the goal of the present paper is to construct analogues of the grigorchuk and gupta sidki groups in the world of lie superalgebras of an arbitrary characteristic we construct two examples r q of finitely generated self similar lie superalgebras over a field k of an arbitrary characteristic and study their properties and properties of their associative hulls in case char k 2 these examples turn into restricted lie algebras the virtue of the present construction is that the lie superalgebras have clear monomial bases these lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators the lie superalgebra r is z 2 graded while q has a multidegree z 3 gradation and a z 2 gradation both algebras r and q have similar constructions computations for r are simpler but q enjoys some more specific interesting properties the z 3 components of q lie inside an elliptic paraboloid in space they are at most one dimensional thus the z 3 grading of q is fine in the z 2 gradation of q all components q n m n m z are infinite dimensional except for q 00 0 the z 2 gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras q q q in both examples ad a is nilpotent a being even or odd with respect to the z 2 grading as lie superalgebras this property is an analogue of the periodicity of the grigorchuk and gupta sidki groups in particular q is a nil finely graded lie superalgebra which shows that an extension of a theorem due to martinez and zelmanov 27 for the lie superalgebras of characteristic zero is not valid both lie superalgebras are self similar contain infinitely many copies of themselves let us also call them fractal due to this property

abelian integrals and limit cycles for a class of cubic polynomial vector fields of lotka volterra type with a rational first integral of degree two abstract in this paper we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of lotka volterra type having a rational first integral of degree 2 under polynomial perturbations of degree n the analysis is carried out by estimating the number of zeros of the corresponding abelian integrals moreover using chebyshev criterion we show that the sharp upper bound for the number of zeros of the abelian integrals defined on each period annulus is 3 for n 3 the simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered all configurations u v with 0 u v 3 u v 5 are realizable

robustness of muscl schemes for 2d unstructured meshes we consider second order accuracy muscl schemes to approximate the solutions of hyperbolic system of conservation laws in the context of the 2d unstructured grids we propose a limitation procedure on the gradient reconstruction to enforce several stability properties we establish that the muscl scheme preserves the invariant domains and satisfy a set of entropy inequalities a conservation assumption is not useful in the present work to define the piecewise linear approximations and the proposed limitation can be understood as a systematic correction of the standard gradient reconstruction procedure the numerical method is applied to the compressible euler equations the gradient reconstruction is performed using the characteristic variables several numerical tests exhibit stability and robustness of the scheme

efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes this paper presents efficient second order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows for compressible unsteady flows a time dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables as a result a compact scheme on the unstructured meshes is developed for incompressible steady flows a continuous second order gas kinetic bgk type scheme is presented for which the time dependent gas distribution function with a continuous particle velocity is used on unstructured meshes the efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells therefore even with the stencil of a first order scheme a high resolution method is constructed numerical examples are presented which are compared with the benchmark solutions and the experimental measurements

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

new family of abelian integrals satisfying chebyshev property abstract this paper is motivated by the chebyshev property of a family of abelian integrals along non algebraic level ovals a new criterion is established to study the chebyshev property of these abelian integrals as its applications we give three examples to show that the ordered sets defined by abelian integrals are ect systems moreover the number of the abelian integrals can be an arbitrary positive integer

redundant picard fuchs system for abelian integrals abstract we derive an explicit system of picard fuchs differential equations satisfied by abelian integrals of monomial forms and majorize its coefficients a peculiar feature of this construction is that the system admitting such explicit majorants appears only in dimension approximately two times greater than the standard picard fuchs system the result is used to obtain a partial solution to the tangential hilbert 16th problem we establish upper bounds for the number of zeros of arbitrary abelian integrals on a positive distance from the critical locus under the additional assumption that the critical values of the hamiltonian are distant from each other after a proper normalization we were able to majorize the number of all real and complex zeros in the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non homogeneity of uni and bivariate complex polynomials are studied

modeling dissipation equation in supersonic turbulent mixing layers with high density gradients in this work the ability of a ke model to compute mixing layers with significant density gradients is discussed it is shown that as in boundary layers deficiencies appear for compressible flows even if the ke model is improved by compressibility corrections a modification of the sige coefficient is proposed to compute efficiently compressible mixing layers with high density variations this new formulation leads to a separate modelling of the subsonic and supersonic effects the results are presented on mixing layer computations with density ratios between 1 7 velocity density co gradients and 7 velocity density cross gradients and the convective mach number evolves from 0 3 to 1 they show a clear improvement of the capability of the ke model to accurately compute high speed mixing layers

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

a relaxation method for two phase flow models with hydrodynamic closure law this paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two phase flows in pipelines the pdes are closed by two highly nonlinear algebraic relations namely a pressure law and a hydrodynamic one the severe nonlinearities encoded in these laws make the classical approximate riemann solvers virtually intractable at a reasonable cost of evaluation we propose a strategy for relaxing solely these two nonlinearities the relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate such a property not only makes it trivial to solve the riemann problem but also enables us to enforce some further stability requirements in addition to those coming from a chapman enskog analysis the new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions extensive numerical evidences are provided

the cyclicity of period annuli for a class of cubic hamiltonian systems with nilpotent singular points abstract this paper deals with the limit cycles of a class of cubic hamiltonian systems under polynomial perturbations we suppose that the corresponding hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x axis and y axis and also the origin is a nilpotent singular point hence the hamiltonian h x y can be written as h x y x 2 a x 4 b x 2 y 2 c y 4 a b c r 3 c 0 for the above h x y the corresponding vector field is x 2 y b x 2 2 c y 2 y 2 x 1 2 a x 2 b y 2 we first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits for the following perturbed system x 2 y b x 2 2 c y 2 e f x y y 2 x 1 2 a x 2 b y 2 e g x y where 0 e 1 f x y and g x y are polynomials in x y with degree n we give an estimation of the number of isolated zeros of the corresponding abelian integral the number of limit cycles follows from poincare pontryagin theorem

bifurcation theory for finitely smooth planar autonomous differential systems abstract in this paper we establish bifurcation theory of limit cycles for planar c k smooth autonomous differential systems with k n the key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of hopf bifurcation at a fine focus or a center and of poincare bifurcation in a period annulus we especially study the smoothness of the first order melnikov function in degenerate hopf bifurcation at an elementary center as we know the smoothness problem was solved for analytic and c differential systems but it was not tackled for finitely smooth differential systems here we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case

runge kutta discontinuous galerkin methods for compressible two medium flow simulations one dimensional case the runge kutta discontinuous galerkin rkdg method for solving hyperbolic conservation laws is a high order finite element method which utilizes the useful features from high resolution finite volume schemes such as the exact or approximate riemann solvers tvd runge kutta time discretizations and limiters in this paper we investigate using the rkdg finite element method for compressible two medium flow simulation with conservative treatment of the moving material interfaces numerical results for both gas gas and gas water flows in one dimension are provided to demonstrate the characteristic behavior of this approach

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

how to preserve the mass fractions positivity when computing compressible multi component flows abstract we are interested in the numerical investigation of the compressible flow of a gaseous mixture considering a hyperbolic system including the euler equations for the mixture and a mass conservation equation for each species we propose a new approximation scheme for the convective term of the species equations this approximation relies on some properties of the exact solution of the riemann problem for the multi component system and applies when an upwind godunov type scheme is used for the euler equations its main interest lies in the fact that it preserves the positivity and monotonicity of the mass fractions of all species

a conservative fractional step method to solve non isentropic euler equations abstract a conservative fractional step method to solve the non isentropic euler set of equations on unstructured meshes is presented herein the present two step method which is based on approximate riemann solvers is compared with roe s scheme the actual numerical rate of convergence is examined the c f l like conditions which must be fulfilled to obtain discrete positive values of density and pressure variables are then discussed it is also shown that the extension to the multi component case enables to fulfil the maximum principle for the mass fraction

on the conservative and accurate cfd approximations for moving meshes and moving boundaries we are concerned with the conservative approximations of compressible flows in the context of moving mesh or interface tracking this paper makes a review of some numerical approaches based on the geometrical conservation law gcl property for computation with moving mesh or moving boundary those methods are reformulated and some comparisons are made we also propose a methodology to approximate accurately the mesh point trajectories and obtain accurate geometrical parameters in 2d and 3d satisfying the gcl property for time accurate schemes higher order geometrical parameters also act as a stabilization of the approximation the numerical effects of the higher order trajectories are illustrated for a 2d moving airfoil the efficiency of a mesh relaxation based on the modified linear elasticity equations is demonstrated for 3d computations of flow behavior in the combustion chamber of a four valves piston engine

hopf takens bifurcations and centres the paper deals with hopf takens bifurcations both in generic families and in families containing centres attention goes to the relation between normal forms lyapunov coefficients and the bautin ideal in case the bautin ideal has only one generator we pay attention to the first non zero melnikov function

computational methods for nucleosynthesis and nuclear energy generation this review concentrates on the two principle methods used to evolve nuclear abundances within astrophysical simulations evolution via rate equations and via equilibria because in general the rate equations in nucleosynthetic applications form an extraordinarily stiff system implicit methods have proven mandatory leading to the need to solve moderately sized matrix equations efforts to improve the performance of such rate equation methods are focused on efficient solution of these matrix equations by making best use of the sparseness of these matrices recent work to produce hybrid schemes which use local equilibria to reduce the computational cost of the rate equations is also discussed such schemes offer significant improvements in the speed of reaction networks and are accurate under circumstances where calculations with complete equilibrium fail

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

numerical evaluation of an energy relaxation method for inviscid real fluids we consider finite volume methods for the euler equations of a real fluid with a general pressure law we present some numerical tests concerning an energy relaxation method proposed by coquel and perthame siam j numer anal 35 1998 pp 2223 2249 we use some particular equations of state for which classical methods can be used this allows to compare the results obtained with the classical methods and the energy relaxation method

linear estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians abstract an upper bound b n 7n 5 is derived for the number of zeros of abelian integrals i h h g x y dy f x y dx on the open interval where h is an oval lying on the algebraic curve h x y 1 2 y2 u x h deg u x 4 and is the maximal interval of existence of h f x y g x y are polynomials of x and y and n max deg f x y deg g x y

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop ii abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x for h 1 6 is three and for h 0 is four counted with multiplicity for all parameters and this implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for e 0 outside an eye figure loop is less than or equal to three

multicomponent gas flow computations by a discontinuous galerkin scheme using l2 projection of perfect gas eos a new formulation of multicomponent gas flow computation suited to a discontinuous galerkin discretization is here presented and discussed the original key feature is the use of l 2 projection form of the perfect gas equation of state that allows all thermodynamic variables to span the same functional space this choice greatly mitigates problems encountered by the front capturing schemes in computing discontinuous flow field retaining at the same time their conservation properties at the discrete level and ease of use this new approach combined with an original residual based artificial dissipation technique shows itself capable through a series of tests illustrated in the paper to both control the spurious oscillations of flow variables occurring in high order accurate computations and reduce them increasing the degree of the polynomial representation of the solution this result is of great importance in computing reacting gaseous flows where the local accuracy of temperature and species mass fractions is crucial to the correct evaluation of the chemical source terms contained in the equations even if the presence of the physical diffusivities somewhat brings relief to these problems the present work can therefore also be considered among many others already presented in the literature as the authors first step toward the construction of a new discontinuous galerkin scheme for reacting gas mixture flows

the number of limit cycles for a family of polynomial systems in this paper the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods with the help of a computer algebra system e g maple 7 0 we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems 2 1 and 2 2 is 5n 5 1 1 n 2 for c 0 and n for c 0

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

zeroes of complete elliptic integrals for 1 2 resonance abstract we give an estimate for the number of zeroes of the elliptic integral i h a x y dx b x y dy along the ovals h x y h h x y y 2 2 x 2 x 4 in terms of the degrees of the symmetric polynomials a and b

unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops abstract in this paper we present a complete study of quadratic 3 parameter unfoldings of some integrable system belonging to the classqr3 and having two centers and two unbounded heteroclinic loops we restrict to unfoldings that are transverse toqr3 obtain a versal bifurcation diagram and all global phase portraits including the precise number and configuration of the limit cycles it is proved that 3 is the maximal number of limit cycles surrounding a single focus and only the 1 1 configuration can occur in case of simultaneous nests of limit cycles essentially the proof relies on a careful analysis of a related non conservative abelian integral

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

cyclicity of several planar graphics and ensembles through three singular points without generic conditions abstract this paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle node p 0 and two hyperbolic saddles p 1 and p 2 for the non generic cases of r 1 0 1 r 2 0 1 and r 1 0 1 r 2 0 1 where r 1 0 and r 2 0 are the hyperbolicity ratio of the saddles p 1 and p 2 respectively for the case of r 1 0 1 r 2 0 1 we suppose that the connection from p 0 to p 2 and the connection from p 0 to p 1 keep unbroken we prove that these graphics and ensembles are of finite cyclicity respectively moreover the cyclicity is linearly dependent on the order of the neutral saddle p 1 if p 2 is contractive and r 2 0 q we also show that the nearer r 2 0 is close to 1 the more the limit cycles are bifurcated for the case of r 1 0 1 r 2 0 1 we obtain that these graphics and ensembles are of finite cyclicity respectively if p 1 is of finite order and the hp connection from p 0 to p 2 keeps unbroken

on the number of zeros of abelian integrals for a kind of quartic hamiltonians an explicit upper bound b n is derived for the number of zeros of abelian integrals i h email protected c hg x y dx f x y dy on the open interval 0 1 4 where c h is an oval lying on the algebraic curve h x y x 2 y 2 x 4 ax 2y 2 y 4 with a 2 f x y and g x y are polynomials in x and y of degrees not exceeding n assume i h not vanish identically then b n 3n 14 12n 34 23

on the burnside problem for periodic groups narain gupta 1 and said sidki 2 1 university of manitoba department of mathematics winnipeg manitoba r3t 2n2 canada 2 university of brasilia department of mathematics brasilia d f brazil introduction the generalized burnside problem refers to the question are finitely generated periodic groups finite this was answered in the negative by golod 1 who proved that for each prime p there exists a finitely generated infinite p group golod s construction in not however direct and is based on his celebrated work with safarevi recently grigor uk 2 has given a direct and elegant construction of an infinite 2 group which is generated by three elements of order 2 in this paper we give for each odd prime p a direct construction of an infinite p group on two generators each of order p our group is a subgroup of the automorphism group of a regular tree of degree p and as might be expected it is residually finite and has infinite exponent preliminaries let p be an odd prime and let t 0 be the infinite regular tree of degree p with vertex 0 so that through each vertex u of t 0 there are p regular subtrees

a runge kutta discontinuous galerkin approach to solve reactive flows a runge kutta discontinuous galerkin method to solve the hyperbolic part of reactive navier stokes equations written in conservation form is presented complex thermodynamics laws are taken into account particular care has been taken to solve the stiff gaseous interfaces correctly with no restrictive hypothesis 1d and 2d test cases are presented

computation of unsteady viscous flows around moving bodies using the k e turbulence model on unstructured dynamic grids abstract we consider the numerical solution on unstructured dynamic meshes of the averaged navier stokes equations equipped with the k e turbulence model and a wall function we discuss discretization issues pertaining to conservation laws moving grids and numerical dissipation we also present a robust spring analogy method for constructing dynamic meshes we validate our implementation of this two equation turbulence model and justify its usage for a class of vortex shedding problems by correlating our computational results with experimental data obtained for a flow past a square cylinder we also apply our solution methodology to the two dimensional aerodynamic stability analysis of the tacoma narrows bridge and report numerical results that are in good agreement with observed data

limit cycle bifurcations by perturbing a class of integrable systems with a polycycle abstract in this paper we deal with the problem of limit cycle bifurcation near a 2 polycycle or 3 polycycle for a class of integrable systems by using the first order melnikov function we first get the formal expansion of the melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion based on the coefficients we obtain a lower bound for the maximal number of limit cycles near the polycycle as an application of our main results we consider quadratic integrable polynomial systems obtaining at least two limit cycles

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

multiple limit cycles of some strongly nonlinear lienard van der pol oscillator in this paper we investigate limit cycles of some lienard van der pol oscillator the system has a double homoclinic loop and two cuspidal loops if the damping effect has vanished by the related melnikov function theory and bifurcation theories the limit cycles near the singular circle and center with their distribution are found the number of limit cycles obtained also reveals that some recent results on the lower bounds of the maximal number of limit cycles bifurcated from this kind of systems can be improved

a class of reversible quadratic systems with piecewise polynomial perturbations abstract this paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n all possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques then we present the algebraic structure of its corresponding first order melnikov function which can be used to study the limit cycle bifurcation problem in this direction we use it to investigate the hopf bifurcation of the perturbed systems and obtain an upper bound for the hopf cyclicity meanwhile we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians with two centers abstract in this paper we give the upper bound of the number of zeros of abelian integral i h h p x y d x q x y d y where h is the closed orbit defined by h x y x 2 x 4 y 4 h 0 h is the maximal open interval on which the ovals h exist p x y and q x y are real polynomials in x and y of degree at most n

alien limit cycles in abel equations abstract the aim of this paper is to study the existence of limit cycles for a family of generalized abel equations x a t x m b t x n m n 2 under certain assumptions it is proved that there exists a non trivial limit cycle this limit cycle has the characteristic that it arises from neither a hopf bifurcation nor a perturbation of periodic orbits in a period annulus around the centre at the origin