remez type inequality on sets with cusps abstract a remez type inequality is proved for a large family of sets with cusps in r n including compact fat and semialgebraic or subanalytic sets

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

solution of discrete scaled systems via general discrete orthogonal polynomials abstract general discrete orthogonal polynomials are introduced to analyse and approximate the solution of a class of discrete scaled systems using the general discrete shift transformation matrix together with the general discrete scale matrix the discrete scaled system can be reduced to a set of simultaneous linear algebraic equations the coefficient vectors of general discrete orthogonal polynomials can be determined simply by the derived algorithm examples are included to show the applicability of the general discrete orthogonal polynomial approximations

optimal control using discrete laguerre polynomials this paper presents discrete laguerre polynomials which approximate the optimal control problems of digital systems the discrete variational principle in combination with the idea of a penalty function is used to construct the modified discrete euler lagrange equations the discrete laguerre series is then applied to simplify the modified equations into a set of linear algebraic ones for the approximate solutions of state and control variables an example is given and the results obtained are very accurate and attractive

walsh stretch matrices and functional differential equations this correspondence presents a new operational matrix for stretch via walsh functions and discusses some interesting properties of this matrix and its role in the solution of certain functional differential equations the solutions of differential equations are obtained in a series of walsh functions the results are piecewise constant with minimal mean square error the single term approximation combined with the technique of extension 8 reduces the computational effort to a minimum

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

l espace symetrique de drinfeld et correspondance de langlands locale ii we study the geometry and the cohomology of the tamely ramified cover of drinfeld s p adic symmetric space over a p adic field k for this tame level we prove in a purely local way most of a conjecture of harris on the form of the ell adic cohomologies as well as the torsion freeness of the integral cohomology in this paper we also compute the ell adic cohomology of coxeter deligne lusztig variety associated to mathrm gl d and some results of independent interest on the coefficient systems over the bruhat tits building associated to mathrm gl d k have been established

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

algebraic families of analytic functions i the aim of this paper is to consider analytic functions arising in connection with algebraic differential equations from the following point of view taylor coefficients of such functions usually turn out to be polynomials or rational functions in the parameters of the problem following pioneering work of bautin 1 we relate analytic properties of the functions considered with algebraic properties of various ideals generated by their taylor coefficients bautin s approach has been successfully applied to the study of the poincare mapping of the planar polynomial systems in 4 9 27 and to the study of the period function of polynomial hamiltonian systems in 5 6 12 13 however the study of the poincare mapping of the polynomial differential equations closely related to the second part of hilbert 16th problem is probably the most difficult among a wide variety of problems to which bautin s approach can be applied in particular counting zeroes of solutions of algebraic differential equations is an important and mostly open problem to our best knowledge the first application of bautin s method to this problem was given in 26 in the present paper we systematically apply this method in the following two situations to algebraic functions and to solutions of the first order polynomial differential equations higher order linear equations are also considered in section 5 article no de963250

the center conditions for a lienard system we obtain necessary and sufficient coefficient conditions for a center at the origin for a lienard system with nonlinearities of degree six the necessity of the conditions is derived from the first seven focus quantities and their sufficiency is proved by cherkas s method

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

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

the cyclicity of a cubic system with nonradical bautin ideal abstract we present a method for investigating the cyclicity of an elementary focus or center of a polynomial system of differential equations by means of complexification of the system and application of algorithms of computational algebra showing an approach to treating the case that the bautin ideal b of focus quantities is not a radical ideal more precisely when the ideal b k is not radical where b k is the ideal generated by the shortest initial string of focus quantities that like the bautin ideal determines the center variety we illustrate the method with a family of cubic systems

small amplitude limit cycles and the distribution of zeros of families of analytic functions we estimate the expected number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter we obtain this inequality by methods of pluripotential theory this inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

transcendence measures and algebraic growth of entire functions in this paper we obtain estimates for certain transcendence measures of an entire function f using these estimates we prove bernstein doubling and markov inequalities for a polynomial p z w in 2 along the graph of f these inequalities provide in turn estimates for the number of zeros of the function p z f z in the disk of radius r in terms of the degree of p and of r

estimating the number of limit cycles in polynomials systems abstract we describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field we apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system

critical period bifurcations of a cubic system we study bifurcations of the period function of a linear centre perturbed by third degree homogeneous polynomials the approach is based on making use of algorithms of computational algebra