bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

eigenschemes and the jordan canonical form abstract we study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix the two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the jordan canonical form of the matrix we also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space this interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint

on the number of limit cycles of a class of polynomial differential systems we study the number of limit cycles of polynomial differential systems of the form where g1 f1 g2 and f2 are polynomials of a given degree note that when g1 x f1 x 0 we obtain the generalized pol

oblique boundary value problems for augmented hessian equations ii abstract in this paper we continue our investigations into the global theory of oblique boundary value problems for augmented hessian equations we construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented hessian is only assumed regular this enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case finally we give some applications to existence theorems which embrace standard hessian equations as special cases

regularity for an obstacle problem of hessian equations on riemannian manifolds abstract in this paper we study the regularity for solutions to an obstacle problem of hessian type fully nonlinear equations on riemannian manifolds as an application the existence of a c 1 1 solution is proved

flow of nonconvex hypersurfaces into spheres the flow of surfaces by functions of their principal curvatures has been intensively studied it started with the work of brakke 1 who used the formalism of geometric measure theory a more classical approach had been chosen by huisken who looked at the so called inward flow in 5 the outward flow of surfaces this term will be explained in the sequel is scaling invariant and therefore more natural than the inward flow producing more general results so far huisken 6 and urbas 8 proved that the outward flow of convex surfaces by suitable functions of their principal curvatures converges to spheres the convexity assumption is essential in their work in this paper we would like to present a different method for proving the convergence of star shaped surfaces into spheres via the outward flow let be a symmetric positive function homogeneous of degree one being defined on an open cone of rn with vertex in the origin which contains the positive diagonal i e all tuples of the form

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

torsion free aluffi algebras abstract a pair of ideals j i r has been called aluffi torsion free if the aluffi algebra of i j is isomorphic to the corresponding rees algebra we give necessary and sufficient conditions for the aluffi torsion free property in terms of the first syzygy module of the form ideal j in the associated graded ring of i for two pairs of ideals j 1 j 2 i such that j 1 j 2 i 2 we prove that if one pair is aluffi torsion free the other one is so if and only if the first syzygy modules of j 1 and j 2 have the same form ideals we introduce the notion of strongly aluffi torsion free ideals and present some results on these ideals

restricted independence in displacement function for better estimation of cyclicity abstract since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center focus equilibrium in this paper we consider a restricted independence to a parametric curve which gives a method not only to increase the lower bound for the cyclicity of the center focus equilibrium but also to be available when those focal values are not independent we apply this method to a nondegenerate cubic center focus variety and prove that the cyclicity reaches its an upper bound

hypersurfaces of prescribed mean curvature in lorentzian manifolds we give a new existence proof for closed hypersurfaces of prescribed mean curvature in lorentzian manifolds

small amplitude limit cycles of some lienard type systems abstract as we know the lienard system and its generalized forms are classical and important models of nonlinear oscillators and have been widely studied by mathematicians and scientists the main problem considered by most people is the number of limit cycles in this paper we investigate two kinds of lienard systems and obtain the maximal number i e the least upper bound of limit cycles appearing in hopf bifurcations by applying some known bifurcation theorems with technical analysis

the obstacle problem for hessian equations on riemannian manifolds abstract in this paper we consider the obstacle problem for a class of fully nonlinear equations on riemannian manifolds the c 1 1 regularity of the greatest viscosity solutions is established

global c2 estimates for convex solutions of curvature equations we establish a c2 a priori estimate for convex hypersurfaces whose principal curvatures 1 n satisfy k x f x x the weingarten curvature equation we also obtain such an estimate for admissible 2 convex hypersurfaces in the case k 2 our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations 2015 wiley periodicals inc

immersed hypersurfaces with constant weingarten curvature in a recent paper korevaar 1 5 used the alexandrov reflection principle to show that closed embedded hypersurfaces in n 1 lhn 1 or the upper hemisphere of s 1 are umbilic spheres provided a certain function f of the principal curvatures t t 22 is constant he only had to assume that f is positive on the positive cone 212i 0 vi and that f is elliptic on the component f of 21 f 0 which contains here f is said to be elliptic if 0f d21 0 for all i 1 inn this generalization of earlier sphere theorems see 9 and 10 for references cannot be extended to hypersurface immersions in view of recent counterexamples 1 11 however assuming additional curvature conditions walter derived in 10 global results for hypersurface immersions in a space n l c of constant curvature c which have a constant higher mean curvature function h here h is the r th symmetric function of the principal curvatures it was shown that such hypersurfaces are of constant mean curvature h1 provided they have nonnegative sectional curvature and non negative principal curvatures as a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified here we show that it is not necessary to assume all principal curvatures to be non negative moreover we extend waiter s result to general symmetric functions f f 2 let m be a smooth connected and compact manifold without boundary then we have the following result

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

small amplitude limit cycles of certain lienard systems the paper is concerned with the number of limit cycles of systems of the form y f x g x where f and g are polynomials for several classes of such systems the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in f and g is obtained in terms of the degree of f and g

absolute cyclicity lyapunov quantities and center conditions abstract in this paper we consider analytic vector fields x 0 having a non degenerate center point e we estimate the maximum number of small amplitude limit cycles i e limit cycles that arise after small perturbations of x 0 from e when the perturbation x is fixed this number is referred to as the cyclicity of x at e for near 0 in this paper we study the so called absolute cyclicity i e an upper bound for the cyclicity of any perturbation x for which the set defined by the center conditions is a fixed linear variety it is known that the zero set of the lyapunov quantities correspond to the center conditions caubergh and dumortier 2004 6 if the ideal generated by the lyapunov quantities is regular then the absolute cyclicity is the dimension of this so called lyapunov ideal minus 1 here we study the absolute cyclicity in case that the lyapunov ideal is not regular

the resolution of the ideal of 2 2 minors of a 2 nmatrix of linear forms abstract let k be an algebraically closed field and let s k x 1 x m let m be a 2 n matrix of linear forms of s and let i 2 m denote the ideal generated by the determinants of the 2 2 minors of m we study in this paper the minimal finite free resolution of s i 2 m as an s module m corresponds to a certain 2 dimensional vector space l of m n matrices that is to a matrix pencil the kronecker weierstrass theory of such matrix pencils provides a normal form for l and we characterize the resolution of s i 2 m in terms of this normal form in particular if the general element of l is injective we explicitly construct the minimal resolution of s i 2 m by repeated application of the horseshoe lemma for any m we express the regularity of s i 2 m as a function of the invariants of l

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

hopf bifurcation for two types of lienard systems abstract in this paper we study the maximal number of limit cycles in hopf bifurcations for two types of lienard systems and obtain an upper bound of the number in some cases the upper bound is the least called the hopf cyclicity

small amplitude limit cycles in lienard systems via multiplicity abstract a recent paper of christopher and lloyd reduces the computation of the order of degeneracy of a weak focus for a polynomial lienard system x y f x y g x to the computation of the multiplicity of a polynomial map in this paper we first take advantage of that approach to obtain new lower and upper bounds for the maximum order of degeneracy of the origin in terms of the degrees of f and g later on we implement an algorithm to compute this maximum order for concrete values of these degrees as far as we know we enlarge the set of degrees for which this maximum order was known finally we extend christopher and lloyd s result to analytic degenerate or not lienard equations

intersection theory of linear embeddings we study intersection theoretic properties of subschemes defined by ideal sheaves of linear type in particular their behavior with respect to blowups and segre classes

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

surfaces of prescribed mean curvature vector in semi riemannian manifolds abstract solving a second order quasilinear initial value problem we prove existence and uniqueness of both space like and time like immersions of prescribed mean curvature vector in general semi riemannian manifolds such surfaces arise for example as critical points of generalized area functionals as well as self similar solutions of the mean curvature flow

conformal metrics with prescribed curvature functions on manifolds with boundary we study the dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on riemannian manifolds with boundary as a consequence we prove the existence of a conformal metric given its value on the boundary as a prescribed metric conformal to the induced background metric with a prescribed curvature function of the schouten tensor

center problem for several differential equations via cherkas method cherkas method characterizes centers for analytic lienard differential equations we extend his method to degenerate lienard differential equations and we apply this extension to solve the center problem for several families of polynomial differential equations in particular we give all centers for some differential equations given by a vector field which is the sum of two quasi homogeneous ones

the aluffi algebra of the jacobian of points in projective space torsion freeness abstract the algebra in the title has been introduced by p aluffi let j i be ideals in the commutative ring r the embedded aluffi algebra of i on r j is an intermediate graded algebra between the symmetric algebra and rees algebra of the ideal i j over r j a pair of ideals has been dubbed an aluffi torsion free pair if the surjective map of the aluffi algebra of i j onto the rees algebra of i j is injective in this paper we focus on the situation where j is the ideal of points in general linear position in projective space and i is its jacobian ideal

mean curvature flow of spacelike graphs we prove the mean curvature flow of a spacelike graph in 1 2 g1 g2 of a map f 1 2 from a closed riemannian manifold 1 g1 with ricci1 0 to a complete riemannian manifold 2 g2 with bounded curvature tensor and derivatives and with sectional curvatures satisfying k2 k1 remains a spacelike graph exists for all time and converges to a slice at infinity we also show with no need of the assumption k2 k1 that if k1 0 or if ricci1 0 and k2 c c 0 constant any map f 1 2 is trivially homotopic provided f g2 0 and in case k2 0 this largely extends some known results for ki constant and 2 compact obtained using the riemannian structure of 1 2 and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo riemannian setting then in the riemannian one

periodic orbits in 3 dimensional systems and application to a perturbed volterra system abstract we give a theorem about the existence of periodic orbits for a 3 dimensional system then use it to study a perturbation of a 3 dimensional volterra system and prove the existence and uniqueness of its periodic orbits on a 2 dimensional invariant manifold

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

mean curvature flow of compact spacelike submanifolds in higher codimension we prove the longtime existence for mean curvature flow of a smooth n dimensional spacelike submanifold of an n m dimensional manifold whose metric satisfies the timelike curvature condition

aluffi torsion free ideals abstract a special class of algebras which are intermediate between the symmetric and the rees algebras of an ideal was introduced by p aluffi in 2004 to define characteristic cycle of a hypersurface parallel to conormal cycle in intersection theory these algebras are recently investigated by a nasrollah nejad and a simis who named them aluffi algebras for a pair of ideals j i of a commutative ring r the aluffi algebra of i j is called aluffi torsion free if it is isomorphic to the rees algebra of i j in this paper ideals generated by 2 minors of a 2 n matrix of linear forms and also edge ideals of graphs are considered and some conditions are presented which are equivalent to aluffi torsion free property of them

transverse fully nonlinear equations on sasakian manifolds and applications abstract we prove a priori estimates for a class of transverse fully nonlinear equations on sasakian manifolds and give some geometric applications such as the transversion calabi yau theorem for transverse balanced and strongly gauduchon metrics we also explain that similar results hold on compact oriented taut transverse hermitian foliated manifold of complex co dimension n and give some geometric applications such as the transverse calabi yau theorems for transverse hermitian and strongly gauduchon metrics