orbitally universal centers abstract in this paper we define when a polynomial differential system is orbitally universal and we show the relevance of this notion in the classical center problem i e in the problem of distinguishing between a focus and a center

analytic integrability and characterization of centers for generalized nilpotent singular points a method which provides necessary conditions for obtaining a local analytic integral in a neighborhood of a generalized nilpotent singular point is developed this method is applied to several classes of generalized nilpotent systems we obtain sufficient conditions for each case finding an inverse integrating factor or a local first integral defined at the origin we study which ones of the found cases possess a center at the origin

rational extensions of the trigonometric darboux poschl teller potential based on para jacobi polynomials the possibility for the jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by calogero and yi who termed them para jacobi polynomials such polynomials are used here to build seed functions of a darboux backlund transformation for the trigonometric darboux poschl teller potential as a result one step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter are constructed for each n value the eigenstates of these extended potentials are associated with a novel family of dependent polynomials which are orthogonal on 1 1

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

universal characters integrable chains and the painleve equations abstract the universal character is a generalization of the schur polynomial attached to a pair of partitions see adv math 74 1989 57 we prove that the universal character solves the darboux chain the n periodic closing of the chain is equivalent to the painleve equation of type a n 1 1 consequently we obtain an expression of rational solutions of the painleve equations in terms of the universal characters

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

the center focus problem and reversibility in this paper we discuss the relation between time reversibility and the center focus problem we show the following for any analytic planar system x if j1x the one jet of x is conjugated to y x then x is analytically time reversible if and only if it is a center if j1x is conjugated to y 0 then some sufficient and necessary conditions for x to have a center are given in particular the reversibility of certain types of polynomial vector fields is studied

the analytic and formal normal form for the nilpotent singularity abstract we study orbital normal forms for analytic planar vector fields with nilpotent singularity we show that the takens normal form is analytic in the case of generalized cusp we present the complete formal orbital normal form it contains functional moduli we interprete the coefficients of these moduli in terms of the hidden holonomy group

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

analytic integrability around a nilpotent singularity abstract in this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions

classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree abstract in this paper we classify the centers localized at the origin of coordinates and their isochronicity for the polynomial differential systems in r 2 of degree d that in complex notation z x i y can be written as z i z a z d n 1 2 z d n 1 2 b z d n 1 2 z d n 1 2 c z d 1 2 z d 1 2 d z d 2 j n 1 2 z d 2 j n 1 2 where j is either 0 or 1 if j 0 then d 5 is an odd integer and n is an even integer satisfying 2 n d 1 2 if j 1 then d 3 is an integer and n is an integer with converse parity with d and satisfying 0 n d 1 3 where denotes the integer part function furthermore r and a b c d c note that if d 3 and j 0 we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in k e malkin 1964 17 n i vulpe and k s sibirskii 1988 25 j llibre and c valls 2009 15 and if d 2 j 1 and c 0 we are also obtaining as a particular case the quadratic polynomial differential systems studied in n n bautin 1952 2 h zoladek 1994 26 so the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations

multi indexed jacobi polynomials and maya diagrams multi indexed jacobi polynomials are defined by the wronskian of four types of eigenfunctions of the poschl teller hamiltonian we give a correspondence between multi indexed jacobi polynomials and pairs of maya diagrams and we show that any multi indexed jacobi polynomial is essentially equal to some multi indexed jacobi polynomial of two types of eigenfunction as an application we show a wronskian type formula of some special eigenstates of the deformed poschl teller hamiltonian

corrigendum to the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems j differential equations 227 2006 406 426 y f1 x y em1 x y ex f2 x y em2 x y where m1 x f1 f1 y and m2 x f1 f1 x f1 here f1 x y is an analytic function starting with terms of degree 2 in x and y and it appears in the inverse change of variables 8 i e u x f1 x y v y f2 x y we remark that this correction only has implications in the proof of proposition 6 now this proof becomes simpler because we have only an arbitrary function f1 x y in the perturbation of the nilpotent system

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

universal characters and an extension of the kp hierarchy the universal character is a polynomial attached to a pair of partitions and is a generalization of the schur polynomial in this paper we define vertex operators which play roles of raising operators for the universal character by means of the vertex operators we obtain a series of non linear partial differential equations of infinite order called the uc hierarchy we regard it as an extension of the kp hierarchy we investigate also solutions of the uc hierarchy the totality of the space of solutions forms a direct product of two infinite dimensional grassmann manifolds and its infinitesimal transformations are described in terms of the lie algebra

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

quasi homogeneous normal forms in this paper we extend the concepts of the normal form theory for vector fields that are expanded in quasi homogeneous components of a fixed type these expansions have been used by some authors in the analysis of the determinacy of a given singularity also the use of reparametrizations in the time are considered namely beyond the use of c conjugation to determine normal forms we present a method useful to determine how much a vector field can be simplified by using c equivalence the results obtained are applied in the case of the bogdanov takens singularity firstly using c conjugation and later showing the improvements provided by the c equivalence

extended krein adler theorem for the translationally shape invariant potentials considering successive extensions of primary translationally shape invariant potentials we enlarge the krein adler theorem to mixed chains of state adding and state deleting darboux backlund transformations it allows us to establish novel bi linear wronskian and determinantal identities for classical orthogonal polynomials

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

exceptional orthogonal polynomials and generalized schur polynomials we show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized schur polynomials introduced by sergeev and veselov

the geometry of consonant belief functions simplicial complexes of necessity measures in this paper we extend the geometric approach to the theory of evidence in order to include the class of necessity measures represented on a finite domain of frame by consonant belief functions b f s the correspondence between chains of subsets and convex sets of b f s is studied and its properties analyzed eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex in particular we focus on the set of outer consonant approximations of a belief function showing that for each maximal chain of subsets these approximations form a polytope the maximal such approximation with respect to the weak inclusion relation between b f s is one of the vertices of this polytope and is generated by a permutation of the elements of the frame

equivalences of the multi indexed orthogonal polynomials multi indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion multi indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions for multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types two different index sets may give equivalent multi indexed orthogonal polynomials we clarify these equivalences multi indexed orthogonal polynomials with both type i and ii indices are proportional to those of type i indices only or type ii indices only with shifted parameters

analytic integrability for some degenerate planar vector fields abstract in this paper we study the analytic integrability of degenerate vector fields of the form y 3 2 a x 3 y x 5 3 a x 2 y 2 around the origin for these vector fields it is proved that integrability does not imply formal orbital equivalence to the hamiltonian leading part moreover it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible

the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems abstract in this work we study the centers of planar analytic vector fields which are limit of linear type centers it is proved that all the nilpotent centers are limit of linear type centers and consequently the poincare liapunov method to find linear type centers can be also used to find the nilpotent centers moreover we show that the degenerate centers which are limit of linear type centers are also detectable with the poincare liapunov method

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

zeros of wronskians of hermite polynomials and young diagrams abstract for a certain class of partitions a simple qualitative relation is observed between the shape of the young diagram and the pattern of zeros of the wronskian of the corresponding hermite polynomials in the case of the two term wronskian w h n h n k we give an explicit formula for the asymptotic shape of the zero set as n some empirical asymptotic formulas are given for the zero sets of three term and four term wronskians

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences