necessary and sufficient conditions for oscillation of second order neutral differential equations consider the second order nonlinear neutral differential equation with delays e d2 dt2 y t py t q t f y t 0 for t 0 where q t f x are continuous functions q t 0 yf y 0 if y 0 and 0 0 0 when f y satisfies either the superlinear or sublinear conditions which include the special case f y y y 1 of 1 and 0 1 respectively we give necessary and sufficient conditions for the oscillation of all continuable solutions of e when p 0 in e these results reduce to the well known theorems of atkinson and belohorec in the special case when f y y y 1 1

axisymmetric shape of gas bubbles in a uniform flow this paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method bifurcation and turning points are located by applying a special type of hermite pade approximation

extreme points and support points of subordination families this paper surveys the main results and open questions in the theory of extreme points and support points of subordination families of analytic functions an effort has been made to make the list of references as complete as possible

a reliable method for computing m pade approximants on arbitrary staircases abstract the m pade approximation problem is defined which contains as a special case the hermite pade approximation as well as the newton pade approximation different well known methods for computing m pade approximations are studied they are based on a normal solution table it is shown that the concept of one of them computing bases of sets induced by interpolation conditions can be generalized to the singular case a characterization of special bases leads to a representation of the m pade solution set like in beckermann this journal 1990 a simple efficient and reliable method for computing such bases by so called transfer matrices is derived finally examples of hermite pade and newton pade approximation are given the connection to other well known reliable algorithms for rational interpolation is discussed

on extreme points of subordination families with a convex majorant abstract let z c z b 0 denote the set of functions analytic in and satisfying z 0 0 suppose f is analytic and univalent in and maps it onto a convex domain d other than a strip a wedge or a half plane let s f fo b 0 let es f denote the set of extreme points of s f d j hallenbeck and t h macgregor asked in 3 whether the set es f ever lies strictly between the minimal fo b 0 lim r 1 re i 1 a e and the maxima fc is an extreme point of b 0 sets we prove that this always happens when the boundary of d contains a line segment let d f be any convex domain and let f be subordinate to f we show that if f is an extreme point of s f then the closure of the boundary values f of f f lim r 1 f re i a e has a nonempty intersection with the set of extreme points of the boundary of d we also completely determine extreme points of s f when the boundary of d consists of finitely many arcs wih positive curvature

an oscillation criterion for second order sublinear differential equations abstract an oscillation criterion is given for the second order sublinear differential equation y a t f y 0 where a t is not assumed to be non negative and f y is non decreasing in y and yf y 0 for y 0 f y also satisfies a sublinear condition which includes the special case f y y y sgn y 0 this result extends earlier oscillation theorems of philos and the first author

on the theory and computation of nonperfect pade hermite approximants abstract for a vector of k 1 power series we introduce two new types of rational approximations the weak pade hermite form and the weak pade hermite fraction a recurrence relation is then presented which computes pade hermite forms along with their weak counterparts along a sequence of perfect points in the pade hermite table the recurrence relation results in a fast algorithm for calculating a pade hermite approximant of any given type when the vector of power series is perfect the algorithm is shown to calculate a pade hermite form of type n0 nk in o kn 2 operations where n n0 nk this complexity is the same as that of other fast algorithms the new algorithm also succeeds in the nonperfect case usually with only a moderate increase in cost

on a system of classical polynomials of simultaneous orthogonality abstract we introduce a system of classical polynomials of simultaneous orthogonality study the differential equation of third order recurrence relation and precise the ratio asymptotic and zeros distribution of polynomials

the structure of the singular solution table of the m pade approximation problem abstract a general interpolation problem is defined which contains as special cases the m pade approximation problem as well as classical interpolation problems e g newton pade approximation to study the structure of the solution table a new concept the minimal solution is introduced statements on the local and global structure of the solution table are obtained as a special case the well known block structure of the pade table is derived finally a characterization of the solution set is given

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

a sublinear oscillation theorem abstract an oscillation criterion is given for the second order nonlinear differential equation y a t y 0 where a t is continuous but is not assumed to be non negative for all large values of t and y is non decreasing in y and satisfies y y 0 for y 0 this result generalizes a recent extension of belohorec s theorem due to kwong and wong and also incorporates an earlier result of coles

oscillation of sublinear second order differential equations with integrable coefficients abstract an oscillation criterion is given for the second order sublinear differential equation y a t f y 0 where a t c 0 and f y c is nondecreasing with yf y 0 for y 0 and also satisfies a sublinear condition which covers the special case f y y sgn y 0 the coefficient a t is allowed to be negative for large values of t and its integral over the nonnegative reals is finite this result extends an earlier oscillation theorem of kwong and wong for the special case

oscillation criteria for second order sublinear differential equation abstract the basic purpose of this paper is to present a new oscillation criteria for second order sublinear differential equation a t x t x t q t t 0 t t 0 0 where a a c 1 t 0 is positive function q c t 0 has no restriction on its sign c 1 r are such that x 0 x x 0 for x 0

support points of families of analytic functions described by subordination we determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination the families we study include the functions with a positive real part the typically real functions and the functions which are subordinate to a given majorant if the majorant f is univalent then each support point has the form f 0 where p is a finite blaschke product and o 0 o this completely characterizes the set of support points when f is convex the set of support points is found for some specific majorants including f z 1 z l z p where p 1 let k and st denote the set of normalized convex and starlike mappings respectively we find the support points of the families k and st defined by the property of being subordinate to some member of k or st respectively introduction let a z e c i z i 1 and let e denote the set of functions analytic in a then 6 is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of la by a continuous linear functional on e we mean a complex valued functional defined on 6f that is linear and continuous in other words if j is such a functional then j af bg aj f bj g whenever a and b belong to c and f and g belong to b also if fn e 6b n 1 2 and fn f then j fn j f each continuous linear funcn tional j on ef is given by a sequence bn n 0 1 which satisfies limn n i n i 1 and

vector orthogonal polynomials abstract the following well known framework will be applied to hermite pade approximants first the hermite pade type approximants will be defined from the error of approximation some supplementary conditions will be fixed in order to obtain hermite pade forms vector orthogonal polynomials will be deduced from these conditions their existence and uniqueness are studied in the normal case these results are ensured then some properties of vector orthogonal polynomials will be proved

support points of families of starlike mappings abstract the support points are obtained for the class of those analytic functions starlike of order m fold symmetric that are real on 1 1 we also determine the set of support points for the m fold symmetric analytic functions which are starlike of order

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

integral averages and oscillation criteria of second order nonlinear differential equations the purpose of this paper is to present some new sufficient criteria for the oscillation of all solutions of the second order nonlinear differential equation a t j x x q t f x 0 t t 0 0 where a c 1 t 0 is a positive function qc t 0 has no restriction on its sign j c r c 1 r are such that 0 and j 0 email protected 0 for 0 these criteria are obtained by using a general class of the parameter functions in the averaging techniques and represent extension as well as improvement of known oscillation criteria of several authors for the generalized emden fowler equation

oscillation criteria for second order nonlinear perturbed differential equations abstract some criteria are obtained which guarantee that all solutions of the second order nonlinear perturbed differential equations r t y t p t y t q t y t h t y t y t are oscillatory

approximants de pade hermite 1ere partie theorie we present in this first paper a generalization of pade approximants which gives us as particular cases shafer s and baker sd log approximants first we define these approximants following an old idea of hermite then we prove some fundamental properties for their constructions

hermite pade approximation basic notions and theorems abstract our purpose is to give a brief exposition of basic notions and facts on hermite pade approximation theory a hermite pade form hpf is defined here in the same way as a pade approximant is as a consequence in some cases hpf does not exist even in the modified version hpf need not be unique therefore it was worthwhile to choose some hpf as the optimal one all criteria of existence unicity etc are phrased in a unified manner they base on few quantities depending on the power series under consideration and are usable in the theory rather than in the numerical practice

vector orthogonal polynomials we give the relation between hermite pade approximants and vector orthogonal polynomials an algorithm for calculating vector orthogonal polynomials near the diagonal is described an exact multiple integral formula for vector orthogonal polynomials is proved an example is given showing how the recurrence relations of pade may be used to calculate hermite pade approximants of degrees m mu mu with mu increasing

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

integral averages and oscillation of second order nonlinear differential equations new oscillation criteria are given for the second order sublinear differential equation left a left t right psi left x left t right right x left t right right q left t right f left x left t right right 0 t geqslant t 0 0 where a c 1 t 0 is a nonnegative function f c with x 0 xf x x 0 for x 0 f have continuous derivative on 0 with f x x03c8 x 0 for x 0 and q c t 0 has no restriction on its sign this oscillation criteria involve integral averages of the coefficients q and a and extend known oscillation criteria for the equation x t q t x t 0

b splines and hermite pade approximants to the exponential function this paper is the continuation of a work initiated in p sablonniere an algorithm for the computation of hermite pade approximations to the exponential function divided differences and hermite pade forms numer algorithms 33 2003 443 452 about the computation of hermite pade forms hpf and associated hermite pade approximants hpa to the exponential function we present an alternative algorithm for their computation based on the representation of hpf in terms of integral remainders with b splines as peano kernels using the good properties of discrete b splines this algorithm gives rise to a great variety of representations of hpf of higher orders in terms of hpf of lower orders and in particular of classical pade forms we give some examples illustrating this algorithm in particular another way of constructing quadratic hpf already described by different authors finally we briefly study a family of cubic hpf

classes of analytic functions subordinate to convex functions and extreme points in this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping f when the image of the unit disk under f is a convex domain other than a half plane a strip or an infinite wedge

oscillation of sublinear differential equations of second order on etudie les solutions d equations differentielles qui existent sur des intervalles de la forme t