biorthogonal systems and the infinite companion matrix abstract a nonsymmetric analogue of a gram matrix is used to represent the infinite companion matrix introduced by the author in 4 as a projection operator in h2 the method consists in constructing for a given set 0 fn 1 of linearly independent vectors another set g0 gn 1 which lies in a given space and satisfies fi gk ik the infinite companion matrix may then be interpreted as an interpolation operator connections with lyapunov type equations are also explained

loewner and bezout matrices abstract the notion of intertwining matrices introduced recently by the authors is used to study loewner matrices as in the case of hankel and bezout matrices compatible with a polynomial f of degree not necessarily equal to the size of the matrices the theory requires a careful handling of polynomials with a root at infinity the authors establish first some technical results essentially of a projective character concerning divisibility of such polynomials these are then used to represent products of the form b1b 12b3b 14 b 12rb2r 1 where the bj are bezoutian matrices corresponding to a polynomial f again in the form of a bezoutian matrix corresponding to the same f this result can be deduced from the barnett formula if the degree of f equals the size of the matrices but requires the intertwining theory if this assumption is dropped among other results a strengthening and a conceptual proof of a recent result of z vavrin on inverses of loewner matrices is given in the last section the authors investigate analogies of barnett type formulae for bezoutian matrices b f g where the degree of f is allowed to be smaller than the size of the matrix the barnett formula does not extend immediately but it can be shown that an analogous formula holds asymptotically in the general case

the dynamics of maps tangent to the identity and with nonvanishing index let f be a germ of a holomorphic self map of c 2 at the origin o tangent to the identity and with o as a nondicritical isolated fixed point a parabolic curve for f is a holomorphic f invariant curve with o on the boundary attracted by o under the action of f it has been shown by m abate 2001 that if the characteristic direction v p t o c 2 has residual index not belonging to q then there exist parabolic curves for f tangent to v in this paper we prove using a different method that the conclusion still holds just assuming that the residual index is not vanishing at least when f is regular along v

analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

an extremal problem for operators let a be a banach algebra f a compact set in the complex plane and h a function holomorphic in some neighborhood of the set f thus h a is meaningful for each element a e a whose spectrum a is contained in f and it is possible to evaluate the norm h a problem compute the supremum of the norms h a as a ranges over all elements of a with spectrum contained in f and whose norm does not exceed one that is compute sup h a a e a a f a 1 this problem was first formulated and treated by the author in the particular case where a is the algebra of all linear operators on a finite dimensional hilbert space and f is the disc z z r for a given positive number r 1 the paper discusses motivation connections with complex function theory convergence of iterative processes critical exponents and the infinite companion matrix

bezout hankel and loewner matrices abstract relations between the classes of bezout hankel and loewner matrices and of their inverses are investigated all these classes appear as matrix realizations of one operator in suitable bases different bases in the space of all polynomials of degree not exceeding n 1 are considered these bases correspond to different expansions of the polynomial w z p x p z w x z x we introduce a matrix k corresponding to this polynomial this matrix together with its limit forms may be used to represent all the classes of matrices mentioned in particular we obtain a new interpretation of the variant of generalized loewner matrices introduced recently by the second named author

an algebraic method to solve the minimal partial realization problem for scalar sequences abstract we give an algebraic method to construct all minimal partial realizations of a finite sequence of scalar numbers we use the minimal number of parameters to characterize all these minimal partial realizations three examples are worked out to illustrate the method

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

residual indices of holomorphic maps relative to singular curves of fixed points on surfaces let m be a two dimensional complex manifold and let f m to m be a holomorphic map that fixes pointwise a possibly singular compact reduced and globally irreducible curve c subset m we give a notion of degeneracy of f at a point of c it turns out that f is non degenerate at one point if and only if it is non degenerate at every point of c when f is non degenerate on c we define a residual index for f at each point of c then we prove that the sum of the indices is equal to the self intersection number of c

bezoutians and projections abstract with each polynomial p of degree n whose roots lie inside the unit disc we may associate the n dimensional space of all solutions of the recurrence relation whose coefficients are those of p considered as a subspace of 1 2 the main result consists in establishing a close relation between the bezoutian of two such polynomials of the same degree and the projection operator onto one of the corresponding spaces along the complement of the other the note forms a loose continuation of the author s investigations of the infinite companion matrix the generating function of the infinite companion matrix of a polynomial p appears thus as a particular case the corresponding bezoutian is that of the pair p and z n

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

germs of holomorphic vector fields in c3 without a separatrix oblatum 28 xii 1988 26 iii 1990 10 x 1990 13 vi 1991 30 ix 1991 supported by universite de toulouse conacyt cnrs and caicyt 87 336 fulltext preview

characterizations of bezout and hankel bezout matrices abstract a characterization of bezout matrices is presented as matrices the entries of which satisfy a certain system of homogeneous quadratic equations also the class of matrices which are both hankel and bezout is characterized and formulae for the inversion of such matrices are given

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

the infinite companion matrix abstract a systematic account is given of the properties and applications of the notion of infinite companion matrix of a polynomial p it turns out to be a particular case of a companion matrix corresponding to a pair of polynomials p w of degree n if w is taken to be zn connections with bezoutians projection operators reproducing kernels and dilation theory are explained

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

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

more on loewner matrices abstract the study of loewner matrices is continued in a direction close to the recent work by fiedler ptak and vav in all results are stated for the general case of multiple interpolation nodes i e for generalized loewner matrices both the loewner bezoutian connection of antoulas and anderson and the loewner hankel connection of vav in in the general case which are re proved in a simple way provide a good opportunity for finding various properties of loewner matrices some of them being generalizations of results published by other authors for simple nodes only there are also results which are new especially the barnett like formulas and an intertwining relation characteristic for loewner matrices as well as the loewner matrix hankel vector connection in order to handle loewner matrices further on the authors introduce and deal with a class of so called r matrices it turns out that some known results concerning bezoutians hankel matrices and their products can in a natural way be lifted to the level of r matrices and of the products of loewner and r matrices

invariant curves and topological invariants for real plane analytic vector fields abstract let z be a germ of a singular real analytic vector field at o r 2 we give conditions on the multiplicity and the milnor number of z which imply that the foliation defined by z has a characteristic orbit or an analytic invariant curve with the hypothesis that z is a real generalized curve then it is proved that for a non dicritical real generalized curve the multiplicity mod 2 is invariant under bilipshitz homeomorphisms preserving foliations

extendable cohomologies for complex analytic varieties we introduce a cohomology called extendable cohomology for abstract complex singular varieties based on suitable differential forms beside a study of the general properties of such a cohomology we show that given a complex vector bundle one can compute its topological chern classes using the extendable chern classes defined via a chern weil type theory we also prove that the localizations of the extendable chern classes represent the localizations of the respective topological chern classes thus obtaining an abstract residue theorem for compact singular complex analytic varieties as an application of our theory we prove a camacho sad type index theorem for holomorphic foliations of singular complex varieties

a new aspect of hankel matrices via krylov matrix abstract an investigation is made of the krylov matrices associated to the companion matrix c of a monic polynomial p x of degree n in this paper we propose a new approach to the study of hankel matrices as krylov matrices of c t firstly we focus on algebraic structures of krylov matrices of c over a field f we then discuss an equivalent condition for the hankel matrix to be a krylov matrix of c t by the notion of compatibility as a result we derive new determinant formulas for such hankel matrices in terms of eigenvalues and eigenvectors of c respectively

hankel matrices generated by the markov parameters of rational functions abstract finite hankel matrices s i j are considered for which s i are the markov parameters of a given rational function p a the problems of inversion and kernel description are reduced to the problem of solving some polynomial congruences in terms of p and a an application to hankel matrices of the form y t c i j a x where c a is a companion matrix is included

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

bubble interaction model for hydrodynamic unstable mixing the analytic model for the evolution of single and multiple bubbles in rayleigh taylor mixing is presented for the system of arbitrary density ratio the model is the extension of zufiria s potential theory which is based on the velocity potential with point sources we present solutions for a single bubble at various stages from the model and show that the solutions for the bubble velocity and curvature are in good agreement with numerical results we demonstrate the evolution of multiple bubbles for finite density contrast and investigate dynamics of bubble competition whereby leading bubbles grow in size at the expense of neighbors the model shows that the growth coefficient ensuremath alpha for the scaling law of the bubble front depends on the atwood number and increases logarithmically with the initial perturbation amplitude it is also found that the aspect ratio of the bubble size to the bubble height exhibits a self similar behavior in the bubble competition process and its values are insensitive to the atwood number the predictions of the model for the similarity parameters are in accordance with experimental and numerical results

bezoutians and intertwining matrices abstract using the notion of an intertwining matrix introduced in a previous paper the authors show that the class of all n by n bezoutians compatible with a nonzero polynomial of degree n may be characterized by an intertwining relation as well as by four other conditions formulated in terms of the structure of these matrices

lyapunov bezout and hankel abstract given a polynomial f of degree n we denote by c its companion matrix and by s the truncated shift operator of order n we consider lyapunov type equations of the form x sxc w and x cxs w we derive some properties of these equations which make it possible to characterize bezoutian matrices as solutions of the first equation with suitable right hand sides w similarly for hankel and the second equation and to write down explicit expressions for these solutions this yields explicit factorization formulae for polynomials in c for the schur cohn matrix and for matrices satisfying certain intertwining relations as well as for bezoutian matrices

lyapunov equations and gram matrices if p is a polynomial with all roots inside the unit disc and c its companion matrix then the lyapunov equation x c xc p has a unique solution for every positive semidefinite matrix p we characterize sets of vectors x0 xn 1 and y0 yn 1 such that x g x0 xn 1 g y0 yn 1 1 geometrical connections between such bases and contractions with one dimensional defect spaces are established

quasidirect decompositions of hankel and toeplitz matrices decompositions over an algebraically closed field of a hankel matrix into a sum of hankel matrices the sum of the ranks of which is equal to the rank of the original matrix are completely described similar results hold for toeplitz matrices

zero location by hermitian forms the singular case abstract the schur cohn criterion for the number of zeros of a polynomial inside and outside the unit disc fails if the polynomial has a pair of conjugate zeros or a zero on the unit circle the corresponding quadratic form is singular recently the authors have shown 5 that the classical schur cohn criterion may be deduced from a simple algebraic identity this yields not only a very simple proof but also a substantial generalization the method produces a whole family of quadratic forms which may be used for testing the zeros in the present paper the same algebraic identity is used to show that singularity of these quadratic forms is always due to the presence of pairs of conjugate zeros or zeros on the unit circle there is a method for ascertaining zero distribution in the singular case by differentiation we give a derivation of this test on the basis of our matrix theoritic treatment the second section deals with the same problem in the case of a general circle or half plane

note on products of bezoutians and hankel matrices abstract a quantitative theorem due to fiedler and ptak on products of bezoutians and their inverses is extended to a more general case using as the main tool an alternative to fiedler s assertion on the quasidirect decomposition of hankel matrices

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

meromorphic vector fields with single valued solutions on complex surfaces abstract we study ordinary differential equations in the complex domain given by meromorphic vector fields on compact complex kahler surfaces we prove that if such an equation has a maximal single valued solution in particular an entire one whose image is not contained in a proper analytic subset then up to a bimeromorphic transformation either the vector field is holomorphic or it preserves a fibration

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

complete algebraic vector fields on affine surfaces part ii in this work we study the singularities at infinity of algebraic vector fields in dimension 2 these singularities will be classified under a mild assumption the general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in part ii our main results are local and therefore can be carried over more general surfaces whereas we deal with complete vector fields the results also apply to complete ones thanks to a theorem of forstneric 7

explicit expressions for bezoutians abstract the following commutator identity is proved u s v s v 1 s u 1 s here s is the n by n matrix of the truncated shift operator s i i 1 i 0 1 n 1 and u v are two polynomials of degree not exceeding n the reciprocal polynomial f 1 of a polynomial f of degree n is defined by f 1 z z n f 1 z the commutator identity is closely related to some properties of the bezoutian matrix of a pair of polynomials it is used to obtain the bezoutian matrix in the form of a simple expression in terms of s and s to demonstrate the advantage of this expression we show how it can be used to obtain simple proofs of some interesting corollaries

hankel matrices and the infinite companion abstract a formula of barnett type relating the bezoutian b f g to the hankel matrix h g f is extended to rectangular bezoutians the proof is based on an interesting relation between the family of all hankel matrices corresponding to the markov parameters of g f and the infinite companion matrix corresponding to f

local levi flat hypersurfaces invariants by a codimension one holomorphic foliation in this paper we study codimension one holomorphic foliations leaving invariant real analytic hypersurfaces in particular we prove that a germ of real analytic levi flat hypersurface with sufficiently small singular set is given by the zeroes of the imaginary part of a holomorphic function

vandermonde factorization and canonical representations of block hankel matrices abstract we study to which extent well known facts concerning vandermonde factorization or canonical representation of scalar hankel matrices transfer to block hankel matrices with p q blocks it is shown that nonsingular block hankel matrices can be factored like in the scalar case into nonconfluent vandermonde matrices and that the theorem on full rank factorization of arbitrary hankel matrices transfers in a weak version to the 2 2 block case but not to larger block sizes in general the minimal rank of a vandermonde factorization both with finite nodes and affine is described in terms of the hankel matrix the main tools are realization partial realization and moebius transformations

holomorphic dynamics near germs of singular curves let m be a two dimensional complex manifold p m and open image in new window a germ of holomorphic foliation of m at p letopen image in new window be a germ of an irreducible possibly singular curve at p in m which is a separatrix for open image in new window we prove that if the camacho sad suwa index ind open image in new window then there exists another separatrix for open image in new window at p a similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points

an equation of lyapunov type abstract a new method of finding explicit solutions of lyapunov equations is described based on a lemma on one dimensional perturbations of invertible operators if y satisfies the equation y cyc bb for an appropriate vector b then x y 1 satisfies x c xc aa for a given vector a a concrete example with a 1 0 0 t is given

bezoutian representation via vandermonde matrices abstract the result on benoutian representation via vandermonde matrices is provided with a proof which is based on polynomial models

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

invariant curves and analytic integrability of a planar vector field abstract we give an expression of the irreducible invariant curves at the singular point for analytically integrable systems we provide an expression of its primitive first integral this fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

sturm and sylvester algorithms revisited via tridiagonal determinantal representations first we show that sturm algorithm and sylvester algorithm which compute the number of real roots of a given univariate polynomial lead to two dual tridiagonal determinantal representations of the polynomial next we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

extending the notions of companion and infinite companion to matrix polynomials the notion of infinite companion matrix is extended to the case of matrix polynomials including polynomials with singular leading coefficient for row reduced polynomials a finite companion is introduced as the compression of the shift matrix the methods are based on ideas of dilation theory connections with systems theory are indicated applications to the problem of linearization of matrix polynomials solution of systems of difference and differential equations and new factorization formulae for infinite block hankel matrices having finite rank are shown as a consequence any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n deg det d

a fast method to diagonalize a hankel matrix abstract we consider a vandermonde factorization of a hankel matrix and propose a new approach to compute the full decomposition in o n 2 operations the method is based on the use of a variant of the lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the hankel matrix by adapting existing methods to solve for these eigenvalues and then for the coefficients one arrives at a method to compute the entire decomposition in o n 2 operations the method is illustrated with a simple numerical example

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type in this note a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type is introduced the operator representation relative to a pair of dual bases and the generalized barnett type factorization formula are derived an intertwining relation and a bezoutian reduction to a block diagonal form by congruence via a generalized vandermonde matrix are presented fujiwara hermite and routh hurwitz criteria in terms of this generalized bezout matrix are obtained

recursive solution of lowner vandermonde systems of equations ii abstract an o n 2 algorithm for the solution of a linear system the n n coefficient matrix of which consists of a vandermonde and a lowner part is presented in the first part of the algorithm the corresponding vandermonde system is solved in the second part the last rows of the vandermonde matrix are successively replaced by rows of the lowner structure the algorithm is convenient if the vandermonde part is dominant the main tool is the connection of such systems with rational interpolation problems

explicit expressions for the evolution of single mode rayleigh taylor and richtmyer meshkov instabilities at arbitrary atwood numbers we present explicit analytic expressions for the evolution of the bubble amplitude in rayleigh taylor rt and richtmyer meshkov rm instabilities these expressions are valid from the linear to the nonlinear regime and for arbitrary atwood number a our method is to convert from the linear to the nonlinear solution at a specific value ensuremath eta of the amplitude for which explicit analytic expressions have been given previously for a 1 k o mikaelian phys rev lett 80 508 1998 by analyzing a recent extension of layzer s theory to arbitrary a v n goncharov phys rev lett 88 134502 2002 we find a simple transformation that generalizes our solutions to arbitrary a we compare this model with another explicit model attributed to fermi and with numerical simulations fermi s model agrees with numerical simulations for the rt case but its extension to the rm case disagrees with simulations the model proposed here agrees with hydrocode calculations for both rt and rm instabilities

the discrete lyapunov equation in controllable canonical form a new method of treating lyapunov equations is proposed if d and d are two self adjoint one dimensional matrices related in a certain way then x c ast xc d if and only if x 1 cx 1 c ast d as an application a generalization of a recent result is given if f is the vector f 0 0 0 1 t then the solution of x cxc ast ff ast is shown to be inverse of the schur cohn matrix

effects of viscosity on the growth of rayleigh taylor instability using zufiria s potential flow model with a point source in the velocity potential we investigate the effects of viscosity on the growth of the bubble in rayleigh taylor instability the nonlinear asymptotic solutions are obtained analytically for the velocity and curvature of the bubble we find that the viscosity decreases the velocity but does not affect the curvature of the bubble consistent with sohn s results based on layzer s potential flow model without a point source in the velocity potential furthermore the dependences of the bubble velocity on the reynolds number and the atwood number are studied and we find that both models give the same results at high reynolds numbers and low atwood numbers but smaller velocities are found in zufiria s model at low reynolds numbers and high atwood numbers finally evolution of the velocity with time is studied and the results based on zufiria s model are found to be in better agreement with recent experimental data than that based on layzer s model

kinetic theoretical approach to turbulence in variable density incompressible statistically inhomogeneous fluids the kinetic theory is developed for the mass mixing of two incompressible immiscible fluids due to rayleigh taylor instability as an example for turbulence in variable density statistically inhomogeneous incompressible fluids an expression is derived for the fine grain force in terms of the mass density and velocity fields this expression enables the conversion of the navier stokes equation into an exact explicit conservation equation in phase space the equation is a generalization to the variable density case of the lundgren equation t s lundgren phys fluids 10 969 1967 the conserved quantity is the fine grain density velocity distribution fgdvd the fine grain mass density and fluid velocity fields are the two lowest moments of the fgdvd the joint density velocity probability density function dvpdf is the ensemble average of the fgdvd using detailed numerical solutions of the navier stokes equation it is found that the correlation between the acceleration and the fgdvd is weak this result identifies a small parameter which enables the derivation by controlled approximations of closed equations for the dvpdfs the lowest order yields the mean field approximation it is shown by a numerical solution of the closed kinetic equation in the mean field approximation that it properly describes the time evolution of the system for periods shorter than the relaxation time closure schemes beyond the mean field are discussed

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

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

on invariance of plurigenera for foliations on surfaces let x be a smooth algebraic surface a foliation f on x is roughly speaking a subline bundle t f of the tangent bundle of x the dual of t f is called the canonical bundle of the foliation k f in the last few years birational methods have been successfully used in order to study foliations more precisely geometric properties of the foliation are translated into properties of the canonical bundle of the foliation one of the most important invariants describing the properties of a line bundle l is its kodaira dimension kappa l which measures the growth of the global sections of l and its tensor powers the kodaira dimension of a foliation f is defined as the kodaira dimension of its canonical bundle kappa k f in their fundamental works brunella and mcquillan give a classfication of foliations on surfaces on the model of enriques kodaira classification of surfaces the next step is the study of the behaviour of families of foliations brunella proves that for a family of foliations x t f t of dimension one on surfaces satisfying certain hypotheses of regularity the kodaira dimension of the foliation does not depend on t by analogy with siu s invariance of plurigenera it is natural to ask whether for a family of foliations x t f t the dimensions of global sections of the canonical bundle and its powers depend on t in this talk we will discuss to which extent an invariance of plurigenera for foliations is true and under which hypotheses on the family of foliations it holds

multiplication of diagonal transforms of loewner matrices abstract a new formula for multiplication of matrices arising as appropriate diagonal transformations of loewner matrices is derived in the case of symmetric loewner matrices it is shown that special subclasses form multiplicative groups some extensions of known results about polynomials associated with hankel and loewner matrices are given also a previously found formula of the author describing the inverses of loewner matrices is shown to be a consequence of the present results