extension operators for real analytic functions on compact subvarieties of let x be a compact coherent real analytic subvariety of r it is shown that a continuous linear operator which extends real analytic functions on x to real analytic functions on r exists if and only if x is of type pl which means that in every point of x the local complexification satisfies hormander s local phragmenlindelof condition this is in particular true if x is a manifold in the present paper we study the question under which condition on a compact coherent subvariety x rd there is a continuous linear operator which extends the real analytic functions on x to the whole of rd or equivalently under which condition on x the ideal of x in the algebra a rd is complemented we solve the problem completely and it turns out that the characterizing condition is well known and appears in various other connections an extension operator exists if and only if in every point of x the local complexification of x satisfies hormander s local phragmen lindelof condition this condition is well studied see for that section 1 below there is a significant difference to the corresponding problem for complex subvarieties of cd while our condition is purely local and in fact is a condition on the type of singularities this aspect plays no role in the complex case if cd there it is true for algebraic varieties zahariuta 27 djakov mityagin 7 and more general if and only if the variety is of strong liouville type 23 24 with 29 moreover it is true for strictly pseudoconvex with c2 boundary and a complex manifold transversal at mityagin henkin 20 theorem 4 2 here the difficulties one has to overcome at the boundary resemble those we have to overcome at the singularities a proof of this result in the spirit of

on the dependence of analytic solutions of partial differential equations on the right hand side given a nonzero polynomial p z a m a0z0 on cn martineau proved in the 1960s that for each convex domain g of cn the partial differential operator p d f eai m aa0f c acting on the frechet space a g of all analytic functions on g is surective in the present paper it is investigated whether solutions f of the equation p d f g can be chosen as f r g with a continuous linear operator r a g a g for bounded g we give a necessary and sufficient condition for the existence of such an r

parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences we show that a linear partial differential operator with constant coefficients p d is surjective on the space of e valued ultra distributions over an arbitrary convex set if e is a nuclear frechet space with property dn in particular this holds if e is isomorphic to the space of tempered distributions s or to the space of germs of holomorphic functions over a one point set h 0 this result has an interpretation in terms of solving the scalar equation p d u f such that the solution u depends on parameter whenever the right hand side f also depends on the parameter in the same way a suitable analogue for surjective convolution operators over rd is obtained as well to get the above results we develop a splitting theory for short exact sequences of the form 0 x y z 0 where z is a frechet schwartz space and x y are pls spaces like the spaces of distributions or real analytic functions or their subspaces in particular an extension of the dn splitting theorem of vogt and wagner is obtained

approximation of analytic sets by nash tangents of higher order the aim of this paper is to show that for every locally analytic subset x of cm and every a in x there exist a neighborhood v of a in cm and a sequence x nu of nash subsets of v converging to x cap v such that x and x satisfy a certain condition for tangency of order next it is shown that this condition implies that for sufficiently large the multiplicities of x and x at a are equal

characterization of splitting for frechet hilbert spaces via interpolation based on the methods from interpolation theory we give a characterization of pairs e f of frechet hilbert spaces so that for each frechet hilbert space g each short exact sequence 0 rightarrow f rightarrow g rightarrow e rightarrow 0 splits this characterization essentially depends on a key condition s of an interpolation nature an equivalent description of s in terms of appropriate families of interpolation functions for fre spaces is presented we also define and study general variants of interpolation type of some well known linear topological invariants as an application we obtain an extension of the well known dn splitting theorem of vogt

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

on a nonlinear delay population model we investigate uniform permanence of positive solutions of nonlinear delay equations in several examples we give explicit estimates for lower and upper limit of solutions we give conditions which imply that all solutions are asymptotically equivalent the nonlinear delay differential equation x t r t g t x t h x t t 0 is considered sufficient conditions are established for the uniform permanence of the positive solutions of the equation in several particular cases explicit formulas are given for the upper and lower limit of the solutions in some special cases we give conditions which imply that all solutions have the same asymptotic behavior in particular when they converge to a periodic or constant steady state

on the matrix equation xa ax xp abstract we study the matrix equation xa ax x p in m n k for 1 p n it is shown that every matrix solution x is nilpotent and that the generalized eigenspaces of a are x invariant for a being a full jordan block we describe how to compute all matrix solutions combinatorial formulas for a m x l x l a m and ax l are given the case p 2 is a special case of the algebraic riccati equation

the coherence of complemented ideals in the space of real analytic functions we characterize when an ideal of the algebra a r d of real analytic functions on r d which is determined by the germ at r d of a complex analytic set v is complemented under the assumption that either v is homogeneous or v r d is compact the characterization is given in terms of properties of the real singularities of v in particular for an arbitrary complex analytic variety v complementedness of the corresponding ideal in a r d implies that the real part of v is coherent we also describe the closed ideals of a r d as sections of coherent sheaves

inheritance of surjectivity for partial differential operators on spaces of real analytic functions abstract for an open set r n let a be the space of real analytic functions on improving our previous results we prove a new quantitative characterization of the linear partial differential operators p d which are surjective on a this implies that p d is surjective on a r n if p d is surjective on a for some further inheritance properties for the surjectivity of p d on a are also obtained

explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative volterra type operators abstract in this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative volterra type operators in a recent paper of us a variation of constants formula has been given for such systems and this is particularly suited to handling the studied problems first we investigate the boundedness of the solutions then some estimates are given for the upper and lower limits of the solutions by using this we study the existence of the limit of the solutions and obtain a formula for the limit the applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities synchronisation and lotka volterra type delay functional differential systems

phragmen lindelof principles on algebraic varieties from several results in recent years starting with h6rmander s characterization of the constant coefficient partial differential equations p d u f that have a real analytic solution u for every real analytic function f it has become clear that certain properties of the partial differential operator p d are equivalent to estimates of phragmen lindelof type for plurisubharmonic functions on the algebraic variety

the geometry of analytic varieties satisfying the local phragmen lindelof condition and a geometric characterization of the partial differential operators that are surjective on ℝ the local phragmen lindelof condition for analytic subvarieties of c n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators as hormander has shown here necessary geometric conditions for this phragmen lindelof condition are derived they are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in c 3 the latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on r 4

the algebraic surfaces on which the classical phragmen lindelof theorem holds let v be an algebraic variety in open image in new window we say that v satisfies the strong phragmen lindelof property spl or that the classical phragmen lindelof theorem holds on v if the following is true there exists a positive constant a such that each plurisubharmonic function u on v which is bounded above by z o z on v and by 0 on the real points in v already is bounded by a im z for algebraic varieties v of pure dimension k we derive necessary conditions on v to satisfy spl and we characterize the curves and surfaces in open image in new window which satisfy spl several examples illustrate how these results can be applied

boundedness and persistence of delay differential equations with mixed nonlinearity for a nonlinear equation with several variable delays x t k 1 m f k t x h 1 t x h l t g t x t where the functions fk increase in some variables and decrease in the others we obtain conditions when a positive solution exists on 0 as well as explore boundedness and persistence of solutions finally we present sufficient conditions when a solution is unbounded examples include the mackey glass equation with non monotone feedback and two variable delays its solutions can be neither persistent nor bounded unlike the well studied case when these two delays coincide

power series space representations of nuclear frechet spaces let e be a nuclear graded frechet space such that the norms satisfy inequalities 2 ik ckjl lk li ik l for all k let f beagradedfrechet space such that the dual extended real valued norms satisfy inequalities ii2 dkil lii lk l for all k and let a be a tame resp linearly tame linear map from f to e then there exists a tame resp linearly tame factorization of a through a power series space a 2 a in the case of a tame quotient map e ib tamely equivalent to a power series space of infinite type this applies in particular to the range of a tame resp linearly tame projection in a power series space a2o a in this case one does not need nuclearity it also applies to the tame spaces in the sense of the various implicit function theorems if they are nuclear they are tamely equivalent to power series spaces ao a the question whether certain frechet spaces have isomorphic representations as kothe sequence spaces has become more and more important on one hand by examples of mityagin zobin 28 djakov mityagin 2 and others it has become clear that in general even for the nuclear case a representation is not possible on the other hand the theory of operators in frechet spaces has developed tools which are more or less connected to such representations an important open problem is for instance whether complemented subspaces of nuclear k6the spaces always have a basis i e whether they are again isomorphic to k6the spaces see pelczyfiski 15 this is not even clear for the complemented subspaces of the space s or more generally of a power series space a a of infinite type except for special cases see 4 5 10 23 27 moreover in this case it is of interest to determine the continuity estimates of isomorphisms in particular to determine whether there are tame or linearly tame isomorphisms see 6 in the case of finite type power series spaces any complemented subspace is again a finite type power series space see mityagin 12 and mityagin henkin 13 the consequence of this for questions on the linearization of the basic constructions of complex analysis are presented in mityagin henkin 13 extensions and generalizations of the method of mityagin henkin have also been given in dubinsky 3 and krone 8 9 received by the editors march 20 1987 1980 mathematics subject classification 1 98 5 revision primary 46a45 46a 1 2 58c 1 5 1990 american mathematical society 0002 9947 90 1 00 25 per page

a note on the matrix equation xa ax xp abstract another more elementary proof is given of proposition 2 3 in a recent paper by dietrich burde this proposition says that n n matrices a and x over an algebraically closed field k of characteristic zero satisfying the equality in the title of this paper can be simultaneously triangularized the proposed proof is based on the shemesh criterion for two matrices to have a common eigenvector

a note on permanence of nonautonomous cooperative scalar population models with delays for a large family of nonautonomous scalar delayed differential equations used in population dynamics some criteria for permanence are given as well as explicit upper and lower bounds for the asymptotic behavior of solutions the method described here is based on comparative results with auxiliary monotone systems in particular it applies to a nonautonomous scalar model proposed as an alternative to the usual delayed logistic equation

algebraic varieties on which the classical phragmen lindelof estimates hold for plurisubharmonic functions algebraic varieties v are investigated on which the natural analogue of the classical phragmen lindelof principle for plurisubharmonic functions holds for a homogeneous polynomial p in three variables it is shown that its graph has this property if and only if p has real coefficients no elliptic factors is locally hyperbolic in all real characteristics and the localizations in these characteristics are square free the last condition is shown to be necessary in any dimension

convex univalent functions and continuous linear right inverses abstract for surjective infinite order differential operators on the space of holomorphic functions on a convex domain g other than the entire complex plane we characterize when they admit a continuous linear right inverse the characterization is given by a smoothness condition on g which is phrased in terms of the boundary behavior of the riemann mapping of the unit disc onto g

on a delay population model with a quadratic nonlinearity without positive steady state a population model described by a nonlinear delay differential equation with a quadratic nonlinearity x t k 1 m k t x h k t t x 2 t t 0 is considered where m 1 is an integer functions k 0 0 are continuous functions h k 0 r are continuous such that t h k t t const 0 and for any t 0 the inequality h j t t holds for at least one index j 1 m although this equation does not have a positive steady state a new method not based on the existence of a positive steady state is developed and used to investigate the permanence global attractivity conditions and nonoscillation properties

optimal gevrey classes for the existence of solution operators for linear partial differential operators in three variables abstract for a constant coefficient linear partial differential operator acting on all infinitely differentiable functions or ultradifferentiable functions of beurling type on euclidean 3 space the existence of a continuous linear solution operator is investigated it is shown that there is an optimal weight in the sense that a solution operator exists for a weight if and only if o provided that such an operator exists for at least one weight furthermore the optimal class is either a gevrey class of rational exponent or the class of all infinitely differentiable functions