equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

manifold subbotin spline the subbotin quadratic spline is considered and is extended to the spline on spline case which can be employed to approximate the derivatives y r r 1 2 3

on the weights of certain quadratures for the numerical evaluation of cauchy principal value integrals and their derivatives we consider interpolatory quadrature rules of the type begin gathered frac d k d lambda k int limits 1 1 1 x alpha 1 x beta frac f x x lambda dx sum limits i 1 n w ni k lambda f x ni r n k f hfill k 0 1 alpha beta 0 hfill end gathered where the nodes x ni are the zeros of the jacobi polynomialp n x in particular we prove that sum limits i 1 n left w ni k lambda right o n k log n uniformly in 1 e 1 e furthermore we show that whenk 1 f x c p 1 1 p 1 andf p x h 1 1 0 1 we have r n 1 f o n p mu 1 delta 0 small as we like uniformly in 1 e 1 e

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

spline collocation for singular integro differential equations over 0 1 this paper analyses the convergence of spline collocation methods for singular integro differential equations over the interval 0 1 as trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms we give asymptotic error estimates and some numerical results

a comparison of thee 3 and not a knot cubic splines abstract the algorithms and schemes of the cubic spline interpolation with two end conditions which do not require thederivative information at the end points are of great practical importance and have been included in several general purpose software libraries the cubic spline with not a knot end conditions is one of them and it is implemented in the widely used imsl and nag software libraries in this paper we compare thee 3 cubic spline with the not a knot cubic spline and we will show that thee 3 cubic spline is more accurate than the second one and also it has superconvergence properties which the not a knot cubic spline does not have these superconvergence properties of thee 3 cubic spline can be used in different fields for better approximation

the use of spline on spline for the approximation of cauchy principal value integrals in this paper the spline on spline interpolatory polynomials are used to approximate cauchy principal value integrals and their derivatives this technique yields better results than using the derivatives of the traditional single spline functions which has been considered by orsi improved order of convergence for these approximations are discussed

symmetry of large solutions of nonlinear elliptic equations in a ball let g be a locally lipschitz continuous real valued function which satisfies the keller osserman condition and is convex at infinity then any large solution of u g u 0 in a ball is radially symmetric

cubic and quintic spline on spline interpolation this paper is concerned with the cubic spline on spline procedure considered by dolezal and tewarson 3 we show that when the knots are equally spaced with spacing h then the order of accuracy of the spline on spline approximations can be better by one power of h than that predicted by the results of 3 we also indicate how the cubic spline on spline process can be extended to quintic splines

on the convergence of some cubic spline interpolation schemes cubic spline interpolation schemes which require no derivative information at the end points are of great practical importance and have been included in several general purpose software libraries in this paper optimal order error estimates are developed for three popular schemes of this derivative free type the approximation of c 1 a b backslash c 2 a b functions by any such derivative free method that reproduces cubics necessarily displays some dependence on the local mesh ratio however for the spline interpolants studied here this dependence is restricted to the first and last subintervals

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

pulling back fixed points if x y are g spaces spaces on which a compact lie group g acts and f x y is a g map then the image of a fixed point in x is fixed in i1 when does the inverse image of a fixed point contain a fixed point in particular if x y are g manifolds and f has degree prime to gi order of g in unpublished work of the author1 this is shown for elementary abelian p groups with x y more general spaces n near manifolds in this paper we show that this is true for abelian p groups g acting smoothly on x with p odd or with p 2 with an additional complex hypothesis we give a counterexample for nonabelian g the method is to prove a generalization of a theorem of bredon 1973 for 7lip actions which says that maps of degree prime to p induce injections on the mod p cohomology of the fixed set we generalize this theorem to abelian p groups where the action on x is smooth and p 4 2 or other hypothesis this seems to be new even in the case of elementary abelian p groups of rank 1 we actually prove a stronger theorem for a generalization of the notion of degree which may be non zero for maps between manifolds of different dimension the proof involves complex linear equivariant k theory in an essential way which requires smoothness and p 2 or a complexity hypothesis on x in order to produce the appropriate complex linear bundles among the corollaries we show that an abelian p group p odd cannot act smoothly on a closed manifold with exactly one isolated fixed point generalizing an old result of atiyah and bott 1964 for g z p this was found independently by ewing and stong 1986 by a different argument i am indebted to ib madsen for his comments in particular for discussions on equivariant k theory thanks are due also to michael davis for carefully reading the manuscript and pointing out several difficulties i also wish to thank the referee for his comments

symmetry results of large solutions for semilinear cooperative elliptic systems abstract in this paper we consider symmetry of large solutions of the semilinear cooperative elliptic system u i g i u 1 u m in b r u i on b r where u i c 2 b r g i r m r a c function b r is the open ball of center 0 and radius r 0 in r n n 3 m n i 1 2 m after introducing the weakly coupled fully coupled hypotheses and some assumptions for g we use the moving plane method to prove any large solution of the semilinear cooperative elliptic system is radially symmetric and radially increasing in order to use the moving plane method for our problem a strong maximum principle of the linear cooperative elliptic system is constructed furthermore we also get the symmetry result of large solutions of the semilinear cooperative elliptic system in an annulus

effects of a degeneracy in the competition model part ii perturbation and dynamical behaviour this is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of in part i we mainly discussed the existence of two kinds of steady state solutions of this system namely the classical steady states and the generalized steady states here we use these solutions to determine the dynamics of the model we do this with the help of the perturbed model where b x is replaced by b x e which itself is a classical competition model this approach also reveals the interesting relationship between the steady state solutions both classical and generalized of the above system and that of the perturbed system

the use of piecewise quadratic polynomials for the solution of singular integral equations of cauchy type abstract a method for the numerical solution of singular integral equations of cauchy type is developed the unknown function is expressed as a product of a weight function and a continuous function t the continuous function t is approximated by piecewise quadratic polynomials and the singular integral equation is reduced to a linear algebraic system numerical examples are given and comparisons are made with the widely used gauss type methods

classification of oriented equivariant spherical fibrations classifying spaces for oriented equivariant spherical fibrations are constructed and the notion of an equivariant sf fibration is introduced it is shown that equivariant sf fibrations are naturally oriented in ro g graded equivariant singular cohomology introduction this paper is intended as a sequel to w1 and much of the material there will be assumed here with reference the purpose of this note is to complete the classification theory for equivariant fibrations by constructing explicit classifying spaces for various categories of oriented equivariant spherical fibrations equivariant spherical fibrations with linear fibrewise action are peculiar in that two distinct fibers may look like the spheres s v and s w of unrelated representations even if the base space in question is connected thus given a generalized ro g graded equivariant cohomology theory eg there is no natural grading in which one might expect to find an orientation class in order to define the problem away one notes that e s0 is not concentrated in dimension zero and proceeds according to the ideas of wirthmuller w7 let e be an ro g graded cohomology theory with products in the sense of wirthmuller let a e ro g and let p x b be a g spherical fibration with thom complex tx then an element p e e tx is an e orientation of p in dimension a if for each fiber s vv in x the restriction of i to s v which lies in eah sv with hx the isotropy subgroup of p sv is a generator of this free eh s 0 module it may easily be seen that if a is an actual representation v the existence of such a class implies that eg s w e s0 for any fiber sw over a stationary point in b thus the existence of an orientation implies in general that the cohomology theory e fails to distinguish among representations of a given dimension on the other hand when b has connected fixed point sets all the fibers in x must look like s v for some fixed g module v in a sense to be made precise below we then refer to p as a v dimensional fibration and look for orientations in dimension v e ro g this leads to a more restricted notion of orientation and the mathematics is correspondingly less cumbersome there is also a natural notion of an equivariant sf fibration in this context and we show in 3 that such fibrations received by the editors april 17 1979 and in revised form march 17 1981 ams mos subject classifications 1970 primary 54h15

overconvergence properties of quintic interpolatory splines let q be a quintic spline with equi spaced knots on a b interpolating a given function y at the knots the parameters which determine q are used to construct a piecewise defined polynomial p of degree six it is shown that p can be used to give at any point of a b better orders of approximation to y and its derivatives than those obtained from q it is also shown that the superconvergence properties of the derivatives of q at specific points of a b are all simple consequences of the properties of p