marcinkiewicz zygmund inequalities and interpolation by spherical harmonics abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical harmonics in s d with respect to the l p norm moreover we prove that there are no complete interpolation families for p 2

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

parallelization of the lyapunov constants and cyclicity for centers of planar polynomial vector fields christopher in 2006 proved that under some assumptions the linear parts of the lyapunov constants with respect to the parameters give the cyclicity of an elementary center this paper is devote to establish a new approach namely parallelization to compute the linear parts of the lyapunov constants more concretely it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms to show the power of this approach we study the cyclicity of the holomorphic center iz z 2 z 3 z n under general polynomial perturbations of degree n for n 13 we also exhibit that from the point of view of computation among the hamiltonian time reversible and darboux centers the holomorphic center is the best candidate to obtain high cyclicity examples of any degree for n 4 5 13 we prove that the cyclicity of the holomorphic center is at least n 2 n 2 this result give the highest lower bound for m 6 m 7 m 13 among the existing results where m n is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n as a direct corollary we also obtain the highest lower bound for the hilbert numbers h 6 40 h 8 70 and h 10 108 because until now the best result was h 6 39 h 8 67 and h 10 100

polynomial interpolation and marcinkiewicz zygmund inequalities on the unit circle abstract the objective of this paper is to derive an intimate relationship among three important mathematical tools namely polynomial interpolation marcinkiewicz zygmund inequalities and a p weights in particular it is shown that minimum separation of sample points on the unit circle together with certain uniform a p weights generated by these sample points constitute a necessary and sufficient condition for the validity of the marcinkiewicz zygmund inequality evaluated at these points which in turn is equivalent to the jackson type estimate using the popov andreev module of continuity of polynomial interpolation again at these sample points

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

bifurcations of certain family of planar vector fields tangent to axes abstract the topological versality of the family v x x 1 x y y y 2 ax by cy 2 of vector fields on the plane is proven the asymptotic formulas of the bifurcational curves are given the proofs rely on an analysis of the limit cycles and the non abelian integrals along irrational curves

locally factorial generic zariski surfaces are factorial in this article we study normal afline sufaces in a defined by equations of the form zp g x y where the ground field k is assumed to be algebraically closed of characteristic p o if g is such that g g and g g g2 have no points in common then such surfaces are called zariski surf es a title given to them by p blass in bl if we also assume that zp g has the maximum possible number of singularities deg gl if p does not divide deg g or deg g 3 deg g 3 otherwise then we call such a surface a generic zaraski surface because zp g is of this type for a generic choice of g in this paper we develop a technique for calculating the divisor class group of zariski surfaces at a singular point we then employ this technique to show that locally factorial generic zariski surfaces are factorial

a quadratic system with two parallel straight line isoclines abstract in this paper a quadratic system with two parallel straight line isoclines is considered this system corresponds to the system of class ii in the classification of ye yanqian ye yanqian et al theory of limit cycles transl math monogr vol 66 american mathematical society providence ri 1986 using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles we prove that the maximum number of limit cycles in a quadratic system with two parallel straight line isoclines and two finite singular points is equal to two besides we obtain the same result in a different way applying the wintner perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in v a gaiko global bifurcation theory and hilbert s sixteenth problem kluwer boston 2003

singularity conditions on the class group of zariski surfaces let k be an algebraically closed field of characteristic p 0 and xg ak3 be a normal surface defined by an equation of the form zp g x y assume the number of singularities of xg is the maximum possible which is very often the case this paper defines an equivalence relation on the singularities of xg in terms of the hessian from which it derives a fundamental decomposition of the group of weil divisors of the surface from the decomposition various results are obtained relating the structure of the equivalence classes to that of the class group

the transverse dynamics of a holomorphic flow on considere un champ vectoriel polynomial homogene p z dans c n 1 les solutions du systeme d equations differentielles dz i dt p i z t c i 0 n feuillette c n 1 par des courbes holomorphes ayant des singularites le long du lieu commun p i z 0 on definit la notion de dynamique transverse du feuilletage

global qualitative analysis of a quartic ecological model abstract in this paper we complete the global qualitative analysis of a quartic ecological model in particular studying global bifurcations of singular points and limit cycles we prove that the corresponding dynamical system has at most two limit cycles

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

limit cycles of quadratic systems abstract in this paper the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving hilbert s sixteenth problem in this special case of polynomial systems is suggested using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is 3 1 v a gaiko global bifurcation theory and hilbert s sixteenth problem kluwer boston 2003 besides applying the wintner perko termination principle for multiple limit cycles to our canonical system we prove in a different way that a quadratic system has at most three limit cycles around a singular point focus and give another proof of the same conjecture

polynomial systems a lower bound for the hilbert numbers let h n be the maximum possible number of limit cycles of systems dot x p x y dot y q x y where p and q are polynomials of degree at most n we are concerned with the rate of growth of h n as n increases it is known that h n geq kn 2 for some constant k in this paper we show that h n grows at least as rapidly as n 2 log n

on local behavior of holomorphic functions along complex submanifolds of ℂ n in this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of n as a corollary we present multi dimensional generalizations of an important result of coman and poletsky on bernstein type inequalities on transcendental curves in 2

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

marcinkiewicz zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical polynomials in the weighted l p 0 p space with respect to a doubling weight moreover we show that these conditions are sharp

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

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

genericity conditions for finite cyclicity of elementary graphics abstract in this paper we explore the khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice the genericity conditions needed in that case form a proper subset of the usual methods moreover some of the conditions are non intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions hence we introduce an extension of the method which treats the usual functional pfaffian systems together with the admissible changes of coordinates in the functional equations

multiple limit cycle bifurcations of the fitzhugh nagumo neuronal model abstract in this paper we complete the global qualitative analysis of the well known fitzhugh nagumo neuronal model in particular studying global limit cycle bifurcations and applying the wintner perko termination principle for multiple limit cycles we prove that the corresponding dynamical system has at most two limit cycles

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

on lagrange interpolation at disturbed roots of unity let z nk e itnk 0 t n0 0 such that l n f f p 0 f 1 n whenever n p n 1 it must be emphasized that p necessarily depends on p in the sense that there exists a family z nk k 0 n with n 2 n 1 and such that l n f f 2 o f 1 n for all f a but sup l n f p f a f 1 diverges for sufficiently large values of p