design of state detectors for nonlinear systems using symmetries and semi invariants two techniques are proposed for the design of asymptotic state detectors for nonlinear systems both in the continuous time and discrete time cases the first technique allows one to design linear state detectors for nonlinear systems belonging to a given class which is studied and characterized in the paper the second technique extends the ideas of the first by using semi invariants to compute invariant sets and study their stability

inverse problems for multiple invariant curves planar polynomial vector fields which admit invariant algebraic curves darboux integrating factors or darboux first integrals are of special interest in the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors under generic assumptions regarding the multiple invariant algebraic curves involved in particular we prove in this generic scenario that the existence of a darboux integrating factor implies darboux integrability furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these

on a lyapunov equation for polynomial continuous time systems the primary goal of this paper is to describe the classes of all polynomial vector fields such that the associated dynamical system admits some polynomial lyapunov function v x by fixing or constraining the time derivative of v x in different ways several different cases are studied hints are given about how to use the results for stability analysis and for non polynomial systems

dynamics of cubic siegel polynomials we study the one dimensional parameter space of cubic polynomials in the complex plane which have a fixed siegel disk of rotation number where is a given irrational number of brjuno type the main result of this work is that when is of bounded type the boundary of the siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial we also show that these boundaries vary continuously as one moves in the parameter space this is most nontrivial near the set of cubics with both critical points on the boundary of their siegel disk we prove that this locus is a jordan curve in the parameter space most of the techniques and results can be generalized to polynomials of higher degrees

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

siegel discs herman rings and the arnold family we show that the rotation number of an analytically linearizable element of the arnold family fa b x x a bsin 27rx mod 1 a b c ir o b 1 27r satisfies the brjuno condition conversely for every brjuno rotation number there exists an analytically linearizable element of the arnold family along the way we prove the necessity of the brjuno condition for linearizability of p x d z az 1 z d d and e z azez a e27ia at 0 we also investigate the complex arnold family and classify its possible fatou components finally we show that the siegel discs of pa d and e are quasidiscs with a critical point on the boundary if the rotation number is of constant type

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

all bounded type siegel disks of rational maps are quasi disks we prove that every bounded type siegel disk of a rational map must be a quasi disk with at least one critical point on its boundary this verifies douady sullivan s conjecture in the case of bounded type rotation numbers

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

lie symmetries for the orbital linearization of smooth planar vector fields around singular points abstract this work is a generalization of the method proposed in i a garcia s maza linearization of analytic isochronous centers from a given commutator j math anal appl 339 1 2008 740 745 of linearization of analytic isochronous centers from a given commutator in this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on c 2 around an elementary singular point i e a singular point with associated eigenvalues c satisfying 0 or a nilpotent singular point from a given infinitesimal generator of a lie symmetry

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

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

vandermonde matrices on integer nodes this paper deals with vandermonde matrices widetilde v whose nodes are the first n integer numbers we give an analytic factorization of such matrices and explicit formulas for the entries of their inverses and explore their computational issues we also give asymptotic estimates of the frobenius norm of both widetilde v and its inverse

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

on siegel disks of a class of entire maps let f c c be an entire map of the form f z p z exp q z where p and q are polynomials of arbitrary degrees we allow the case q 0 building upon a method pioneered by m shishikura we show that if f has a siegel disk of bounded type rotation number centered at the origin then the boundary of this siegel disk is a quasicircle containing at least one critical point of f this unifies and generalizes several previously known results

linearization of analytic isochronous centers from a given commutator in this paper we propose an algorithmic way to get the change of variables that linearizes an analytic isochronous center from a given commutator moreover we use this method in order to obtain the linearization of some isochronous centers of the existent literature

linearizability and integrability of vector fields via commutation abstract in this paper we consider complex smooth and analytic vector fields x in a neighborhood of a nondegenerate singular point it is proved the equivalence between linearizability and commutation i e the existence of a commuting vector field y such that the lie brackets x y 0 for complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point it is also proved the equivalence between integrability and the existence of a smooth vector field y such that y is a normalizer of x i e x y x

on the structure of the small quantum cohomology rings of projective hypersurfaces we give an explicit procedure which computes for degree d 3 the correlation functions of topological sigma model a model on a projective fano hypersurface x as homogeneous polynomials of degree d in the correlation functions of degree 1 number of lines we extend this formalism to the case of calabi yau hypersurfaces and explain how the polynomial property is preserved our key tool is the construction of universal recursive formulas which express the structure constants of the quantum cohomology ring of x as weighted homogeneous polynomial functions of the constants of the fano hypersurface with the same degree and dimension one more we propose some conjectures about the existence and the form of the recursive laws for the structure constants of rational curves of arbitrary degree our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation assuming the validity of the conjectures we find the recursive laws for rational curves of degree four

darboux linearization and isochronous centers with a rational first integral abstract in this paper we study isochronous centers of polynomial systems it is known that a center is isochronous if and only if it is linearizable we introduce the notion of darboux linearizability of a center and give an effective criterion for verifying darboux linearizability if a center is darboux linearizable the method produces a linearizing change of coordinates most of the known polynomial isochronous centers are darboux linearizable moreover using this criterion we find a new two parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family we also determine all hamiltonian cubic systems which are darboux linearizable in the second part of this work we restrict to the study of isochronous centers having a rational first integral we prove that under certain conditions the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center

a matrix approach to polynomials we present a matrix formalism to study univariate polynomials the structure of this formalism is beautiful enough to be worth seeing on its own yet we give another motivation to this by presenting three new theorems and applying the formalism to give new proofs of some known results

are there critical points on the boundaries of singular domains generalizing a result of e ghys we prove a general theorem that implies that if a rational functionf of the riemann sphere of degree 2 leaves invariant a singular domainc a disk or a ring on which the rotation number off satisfies a diophantine condition provided that on bar c f is injective then each boundary component ofc contains critical point off the injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off z z n a withne n 2 andae we also show that the singular disks associated to periodic elliptic points off z e az that satisfy a diophantine condition are unbounded in in the end of the paper we give a survey of the theory of iteration of entire functions of

orbital linearization around equilibria of vector fields in rn and lie symmetries abstract in this paper we study some relationships between analytic vector fields x in r n which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set n x of its locally analytic normalizers analytic infinitesimal generators of lie point orbitally symmetries in that neighborhood characterization and necessary and sufficient conditions for the orbital linearization of x via the existence of special elements in n x are given

quadratic like dynamics of cubic polynomials a small perturbation of a quadratic polynomial f with a non repelling fixed point gives a polynomial g with an attracting fixed point and a jordan curve julia set on which g acts like angle doubling however there are cubic polynomials with a non repelling fixed point for which no perturbation results into a polynomial with jordan curve julia set motivated by the study of the closure of the cubic principal hyperbolic domain we describe such polynomials in terms of their quadratic like restrictions

bounded type siegel disks of a family of sine families abstract we prove that for any positive integer n there is a polynomial p z with degree n such that the entire function g z p sin z has a bounded type siegel disk bounded by a quasi circle in the plane which passes through at least one critical point of g in addition let 0 1 be an irrational number of bounded type we prove that for any integer k 0 the boundary of the siegel disk of g z 0 sin z e 2 i 1 s 2 k d s centered at the origin is a quasi circle passing through exactly two critical points 2 and 2 with multiplicity k 1 respectively

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

generalized pascal functional matrix and its applications in this paper we introduce the generalized pascal functional matrix and show that the existing variations of pascal matrices are special cases of this generalization we study some algebraic properties of such generalized pascal functional matrices in addition we demonstrate a direct application of these properties by deriving several novel combinatorial identities and a nontraditional approach for lu decompositions of some well known matrices such as symmetric pascal matrices

matrix approach to polynomials 2 abstract we continue our study of the structure initiated in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 our main emphasis is exploring further structure into the formalism introduced in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 this formalism reveals beautiful interplay between certain elementary operators and provides tools for example for checking handling and generalizing combinatorial identities as we show in examples in addition to that we discover a group structure among vandermonde matrices a fascinating diophantine equation and new proofs and generalizations of recently found related results

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

julia sets in parameter spaces given a complex number of modulus 1 we show that the bifurcation locus of the one parameter family fb z z bz2 z3 b contains quasi conformal copies of the quadratic julia set j z z2 as a corollary we show that when the julia set j z z2 is not locally connected for example when z z z2 has a cremer point at 0 the bifurcation locus is not locally connected to our knowledge this is the first example of complex analytic parameter space of dimension 1 with connected but non locally connected bifurcation locus we also show that the set of complex numbers of modulus 1 for which at least one of the parameter rays has a non trivial accumulation set contains a dense g subset of s1

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

on the dynamics of a family of generated renormalization transformations abstract we study the family of renormalization transformations of the generalized d dimensional diamond hierarchical potts model in statistical mechanic and prove that their julia sets and non escaping loci are always connected where d 2 in particular we prove that their julia sets can never be a sierpinski carpet if the parameter is real we show that the julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models moreover the asymptotic formula of the hausdorff dimension of the julia set is calculated as the parameter tends to infinity

pascal k eliminated functional matrix and its property abstract in this paper we shall first define the pascal k eliminated functional matrix for two variables over an arbitrary field f 1 8 then using the previous results we obtain several interesting combinatorial identities we also investigate the relationship between these matrices and cesaro matrices 9 finally we give an affirmative answer to an open problem raised in 1 for minimal polynomial of pascal k eliminated functional matrix over the field z p

continuity of capacity of a holomorphic motion abstract we study the behavior of various set functions under holomorphic motions we show that under such deformations logarithmic capacity varies continuously while analytic capacity may not