degenerate diffusion and the stefan problem on etudie le probleme i u t u xx f x u dans 0 1 r u 0 t 0 u 1 t 0 dans r u x 0 u 0 x dans 0 1 avec certaines hypotheses sur f et u 0 on etudie le caractere des solutions quand s annule sur un intervalle

limit cycles of planar quadratic differential equations abstract since hilbert posed the problem of systematically counting and locating the limite cycles of polynomial systems on the plane in 1900 much effort has been expended in its investigation a large body of literature chiefly by chinese and soviet authors has addressed this question in the context of differential equations whose field is specified by quadratic polynomials this paper considers the class of quadratic differential equations which admit a unique equilibrium state and establish sufficient conditions for the existence and uniqueness of limit cycles the work is based upon insights and techniques developed in an earlier analysis of such systems motivated by questions from mathematical control theory

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

an exponential formula for polynomial vector fields ii lie series exponential substituition and rooted trees abstract grobner s lie series and the exponential formula provide different explicit formulas for the flow generated by a finite dimensional polynomial vector field the present paper gives 1 a generalization of the lie series in case of non commuting variables called exponential substituition 2 a structural understanding of the three formulas and their mutual relationships in terms of rooted trees and 3 as a byproduct new results on the enumeration coding and statistics of different kinds of rooted trees

stability of parallel triangular system solvers several parallel algorithms have been proposed for the solution of triangular systems the stability of four of them is analysed here a fan in algorithm a block elimination method a method based on a factorized power series expansion of the matrix inverse and a method based on a divide and conquer matrix inversion technique new forward error and residual bounds are derived including an improvement on the bounds of sameh and brent for the fan in algorithm a forward error bound is identified that holds not only for all the methods described here but for any triangular equation solver that does not rely on algebraic cancellation among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems

moduli of stability for germs of homogeneous vectorfields on r3 abstract in this paper we will show that there are an uncountable number of homogeneous quadratic vectorfields on r 3 so that no two of these vectorfields are topologically equivalent

stabilizability of second order bilinear systems this note states necessary and sufficient conditions for the existence of a linear state feedback controller such that a second order bilinear system has a globally asymptotically stable closed loop a suitable controller is constructed for each system which satisfies the conditions

uniqueness of limit cycles for polynomial first order differential equations abstract we study the uniqueness of limit cycles periodic solutions that are isolated in the set of periodic solutions in the scalar ode x k 1 m a k sin i k t cos j k t x n k in terms of i k j k n k our main result characterizes under some additional hypotheses the exponents i k j k n k such that for any choice of a 1 a m r the equation has at most one limit cycle the obtained results have direct application to rigid planar vector fields thus planar systems of the form x y x r x y y x y r x y where r x y k 1 m a k x i k y j k concretely when the set i k j k k 1 m has at least three elements or exactly one and another technical condition is satisfied we characterize the exponents i k j k such that the origin of the rigid system is a center for any choice of a 1 a m r and also when there are no limit cycles surrounding the origin for any choice of a 1 a m r

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

the accuracy of solutions to triangular systems triangular systems play a fundamental role in matrix computations it has been prominently stated in the literature but is perhaps not widely appreciated that solutions to triangular systems are usually computed to high accuracy higher than the traditional condition numbers for linear systems suggest this phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems results of wilkinson are unified and extended among the conclusions are that the conditioning of a triangular system depends on the right hand side as well as the coefficient matrix that use of pivoting in lu qr and cholesky factorisations can greatly improve the conditioning of a resulting triangular system and that a triangular matrix may be much more or less ill conditioned than its transpose

periodic solutions of polynomial non autonomous differential equations let n denote the set of n 1 tuples of c 1 periodic functions the point p p n p 1 p 0 n is identified with the differential equation we examine the way in which the total number of periodic solutions can vary as p traverses a path in n

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

stability of the partitioned inverse method for parallel solution of sparse triangular systems several authors have recently considered a parallel method for solving sparse triangular systems with many right hand sides the method employs a partition into sparse factors of the product form of the inverse of the coefficient matrix it is shown here that while the method can be unstable stability is guaranteed if a certain scalar that depends on the matrix and the partition is small and that this scalar is small when the matrix is well conditioned moreover when the partition is chosen so that the factors have the same sparsity structure as the coefficient matrix the backward error matrix can be taken to be sparse

optimal parallel solution of sparse triangular systems this paper considers the parallel solution of a sparse system lx b with triangular matrix l which is often a performance bottleneck in parallel computation when many systems with the same matrix are to be solved the parallel efficiency can be improved by representing the inverse of l as a product of a few sparse factors the factorization with the smallest number of factors is constructed subject to the requirement that no new nonzero elements are created applications are to iterative solvers with triangular preconditioners to structural analysis or to power systems applications experimental results on the connection machine show the method to be highly valuable

existence of at most 1 2 or 3 zeros of a melnikov function and limit cycles abstract we investigate the existence of at most one two or three limit cycles bifurcated from a periodic annulus of a hamiltonian system under a class of perturbations and obtain some sufficient conditions which ensure that the corresponding melnikov function has at most one two or three zeros in an open interval we also give applications to some systems which appear in codimension two bifurcations and to some lienard systems

the stefan problem with heating appearance and disappearance of a mushy region on considere un materiau qui est initialement dans l etat solide et qui sous l action de la chaleur commence la fusion on decrit la region dans laquelle le materiau n est ni un solide ni un liquide

the stability of second order quadratic differential equations this paper investigates the stability properties of second order systems dot x f x where f x contains either quadratic terms system 1 or linear and quadratic terms system 2 in x the principal contributions are summarized in two theorems which give necessary and sufficient conditions for stability and asymptotic stability in the large of systems 1 and 2 respectively

computing center conditions for vector fields with constant angular speed we investigate the planar analytic systems which have a center focus equilibrium at the origin and whose angular speed is constant the conditions for the origin to be a center in fact an isochronous center are obtained concretely we find conditions for the existence of a cw commutator of the field we cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families y x h1 hm x y h1 hm t and y x h2 h2n x y h2 h2n t with hi homogeneous polynomial in x y of degree i in these cases the maximum number of limit cycles which can bifurcate from a fine focus is determined

partitioning a chordal graph into transitive subgraphs for parallel sparse triangular solution abstract a recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form the number of general communication steps required by this approach is proportional to the number of factors in the factorization the triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations and thereby the complexity of the parallel algorithm can be minimized algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work let f l lt denote the symmetric filled matrix corresponding to a cholesky factor l and let gf denote the adjacency graph of f we consider the problem of minimizing the number of factors over all permutations which preserve the structure of gf the graph model of this problem is to partition the vertices gf into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship the solution to this chordal graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step further the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph this solution relies on several results concerning transitive perfect elimination orderings introduced in this paper we describe a partitioning algorithm with o v e time and space complexity

periodic solutions of a quartic nonautonomous equation on considere des equations de la forme z z 4 t z 3 t z 2 t z on donne un ensemble qui a au moins huit solutions periodiques les coefficients sont des polynomes quadratiques en cost et sint

a clique tree algorithm for partitioning a chordal graph into transitive subgraphs a partitioning problem on chordal graphs that arises in the solution of sparse triangular systems of equations on parallel computers is considered roughly the problem is to partition a chordal graph g into the fewest transitively orientable subgraphs over all perfect elimination orderings of g subject to a certain precedence relationship on its vertices in earlier work a greedy scheme that solved the problem by eliminating a largest subset of vertices at each step was described and an algorithm implementing the scheme in time and space linear in the number of edges of the graph was provided here a more efficient greedy scheme obtained by representing the chordal graph in terms of its maximal cliques which eliminates a subset of the leaf cliques of the current graph at each step is described several new results about minimal vertex separators in chordal graphs and in particular the concept of a em critical separator of a leaf clique are employed to prove that the new scheme solves the partitioning problem we provide an algorithm implementing the scheme in time and space linear in the size of the clique tree

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

uniform in time convergence of numerical methods for non linear degenerate parabolic equations gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations conforming and non conforming finite element mixed finite element and finite volume methods we show here that this framework can be applied to a family of degenerate non linear parabolic equations which contain in particular the richards stefan s and leray lions models and we prove a uniform in time strong in space convergence result for the gradient scheme approximations of these equations in order to establish this convergence we develop several discrete compactness tools for numerical approximations of parabolic models including a discontinuous ascoli arzela theorem and a uniform in time weak in space discrete aubin simon theorem the model s degeneracies which occur both in the time and space derivatives also requires us to develop a discrete compensated compactness result

non autonomous equations related to polynomial two dimensional systems periodic solutions of certain one dimensional non autonomous differential equations are investigated equation 1 4 the independent variable is complex the motivation which is explained in the introductory section is the connection with certain polynomial two dimensional systems several classes of coefficients are considered in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients in particular we need to know when there is a full neighbourhood of periodic solutions we give a number of sufficient conditions and investigate the implications for the corresponding two dimensional systems

limit cycles of a class of polynomial vector fields in the plane on montre que pour 0 suffisamment petit le systeme quadratique dx dt x y x x y dy dt x y xy a une source spirale hyperbolique unique a l origine et un cycle limite unique stable entourant l origine

limit cycles of polynomial systems with homogeneous non linearities abstract we study the class of vector fields of the form x p1 x y pn x y y q1 x y qn x y where pk and qk are real homogeneous polynomials of degree k n 2 and the origin is a non degenerate critical point of index 1 for this class of vector fields we give four theorems about their limit cycles

an exponential formula for polynomial vector fields abstract the well known matrix exponentiation for linear first order systems of ode is generalised to polynomial vector fields the substitution process of our exponential formula is closely related to the usual iteration of polynomial mappings

solving nonlinear equations from higher order derivations in linear stages for all polynomials a r of degree less than or equal to d and all higher order derivations e e s e 1 and f f s f 1 where 1 s and 1 i m of length s less or equal to r the reason for wanting such derivations e1 em is simple given several derivations f1 fm it is useful to have derivations e1 em which are good local approximations to the fi and which are easy to compute with notice that since the right hand sides of equations are known while the left hand sides involve the unspecified coefficients of the polynomial functions the equations are equivalent to a system of nonlinear algebraic equations involving the coefficients it is the purpose of this paper to give an algorithm which solves such a system when a solution exists in fact as the title suggests this algorithm will work by solving a sequence of r linear systems

the stefan problem with mushy regions continuity of the interface on considere un probleme de stefan a une dimension avec un terme source sous l hypothese que le profil initial est monotone on obtient la continuite des frontieres libres le solide le melange et la region liquide