efficient parallel rooting of complex polynomials on the unit circle an efficient method is presented for rooting complex polynomial equations of any order where the root space is restricted to the unit circle the method restates the evaluation of polynomials as a recursive algorithm involving only additions by means of the bilinear transformation the straight line uniformly spaced recursive polynomials evaluation method of a h nuttall 1987 is extended to the unit circle where the root positions are determined by thresholding general coefficient transformations are provided along with a comparison to the horner method

approximation of biholomorphic mappings by automorphisms of cn in this paper we prove several results on approximation of biholomorphic mappings between domains in c n 2 by holomorphic automorphisms of c recall that the group of holomorphic automorphisms of c denoted aut c consists of those holomorphic mappings c c that have a holomorphic inverse 4 1 c c with the topology of uniform convergence on compact sets aut c is a topological group while the group aut c 1 of automorphisms of the complex plane consists only of the linear mappings az b a b e c a 4 0 the group aut c is very large and complicated when n 2 let us choose a linear coordinate system on c and write the coordinates as z z w where z z1 zn i c c n 1 and w z c then autc n contains mappings of the form

carleman approximation by conformal minimal immersions and directed holomorphic curves abstract let r be an open riemann surface in this paper we prove that every continuous function m r n n 3 defined on a divergent jordan arc m r can be approximated in the carleman sense by conformal minimal immersions thus providing a new generalisation of carleman s theorem in fact we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property furthermore the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map as a consequence we give an approximate solution to a plateau problem for divergent jordan curves in the euclidean spaces

strongly pseudoconvex domains as subvarieties of complex manifolds in this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex stein domains our sufficient condition for the existence of such subvarieties in a complex manifold x is expressed in terms of the morse indices and the number of positive levi eigenvalues of an exhaustion function on x examples show that our conditions cannot be weakened in general we obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with griffiths positive normal bundle in the projective case these generalize classical theorems of remmert bishop and narasimhan concerning proper holomorphic maps and embeddings to bbb c n bbb p n backslash bbb p n 1

number theoretic fast algorithms for bilinear and other generalized transformations fast algorithms based on the mersenne and fermat number theoretic transforms are used to perform the bilinear transformation of a continuous transfer function to a discrete equivalent the computations are carried out in finite precision arithmetic require no multiplications and can be implemented in parallel using very simple processors although the bilinear transform is presently emphasized similar algorithms are easily derived for any transformation from the s plane to the z plane involving the ratio of two polynomials with integer coefficients

closed holomorphic 1 forms without zeros on stein manifolds we prove that a stein mainfold admits a closed holomorphic 1 form without zeros in every class of the first cohomology group we also prove an approximation result for closed holomorphic 1 forms without zeros defined in a neighborhood of a compact subset

technical communique the schur stability via the hurwitz stability analysis using a biquadratic transformation in this note it is shown that instead of a bilinear transformation a biquadratic transformation can be used to determine the schur stability of a given discrete time polynomial by determining the hurwitz stability of the corresponding continuous time polynomial however using a biquadratic transformation the analysis is not limited by any restriction on the root positions several numerical examples are provided to illustrate the procedure

integrable embeddings and foliations a k submanifold l of an open n manifold m is called it weakly integrable wi resp it strongly integrable si if there exists a submersion phi m rightarrow bbb e n k such that l subset phi 1 0 resp l phi 1 0 in this work we study the following problem first stated in a particular case by costa et al it invent math 1988 which submanifolds l of an open manifold m are wi or si for general m we explicitly solve the case k n 1 and provide necessary and sufficient conditions for submanifolds to be wi and si in higher codimension as particular cases we recover the theorem of bouma and hector it indagationes math 1983 asserting that any open orientable surface is si in bbb e 3 and watanabe s and miyoshi s theorems it topology 1993 and 1995 claiming that any link is wi in an open 3 manifold in the case m bbb e n we fully characterize wi and si submanifolds we provide examples of 3 and 7 manifolds which are not wi and we show that a theorem by miyoshi it topology 1995 which states that any link in bbb e 3 is si does not hold in general the right analogue to miyoshi s theorem is also proved implying in particular the surprising result that no knot in bbb e 3 is si our results applied to the theory of foliations of euclidean spaces give rise to some striking corollaries using some topological invariants we classify all the submanifolds of bbb e n which can be realized as proper leaves of foliations we prove that bbb s 3 can be realized as a leaf of a foliation of bbb e n n geq 7 but not in bbb e 5 or bbb e 6 which partially answers a question by vogt it math ann 1993 we construct open 3 manifolds which cannot be leaves of a foliation of any compact 4 manifold but are proper leaves in bbb e 4 the theory of wi and si submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role phillips gromov h principle hirsh smale theory complete intersections seifert manifolds the theory of immersions and embeddings obstruction theory etc

comments on s plane to z plane mapping using a simultaneous equation algorithm based on the bilinear transformation by f d groutage et al the commenter directions attention to a conveniently implementable scheme for implementing the bilinear transformation used in the above mentioned paper ibid vol ac 32 no 7 p 635 7 july 1987 which is based on the use of the q sub n matrix he suggests that the overall advantages in the use of q sub n matrix cannot be matched by the authors approach

s plane to z plane mapping using a simultaneous equation algorithm based on the bilinear transformation this note presents an algorithm for transforming a continuous transfer function to an approximate discrete equivalent by means of the bilinear transformation

a foliation of the ball by complete holomorphic discs we show that the open unit ball mathbb b n of mathbb c n n 1 admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs

the lipschitz metric for real valued continuous functions abstract for a continuous real function f defined on a metric space x let f denote its minimal lipschitz constant if f is lipschitz and put f otherwise we study the extended real valued metric on the continuous real functions defined by d f g max f x 0 g x 0 f g when x a b this metric provides new insight into a classical result regarding the derivative of a limit of a sequence of real valued functions defined on the interval

extending holomorphic sections from complex subvarieties in this paper we treat the classical problem of extending holomorphic mappings and sections from closed complex subvarieties of a complexmanifold our main results theorems 1 1 and 1 4 extend those of grauert gr2 and cartan car results of this type are commonly referred to as the okagrauert principle on stein manifolds our methods are similar to those developed in the papers of gromov gro henkin and leiterer hl2 and the authors fp1 fp2 the main addition here is the interpolation of a given holomorphic section on a complex subvariety of a stein manifold our globalization scheme follows very closely the one developed in fp2 to state our first result we recall the notion of a dominating spray introduced by m gromov gro sect 0 5 given a holomorphic vector bundle p e y over a complex manifold y we denote by 0y ey p 1 y the zero element in the fiberey and we observe thatey is ac linear subspace of the tangent space t0ye definition 1 a spray on a complex manifold y is a holomorphic vector bundle p e y together with a holomorphic map s e y such that for each y y s 0y y and the derivative ds t0ye tyy maps ey surjectively onto tyy

strong uniform continuity abstract let b be an ideal of subsets of a metric space x d this paper considers a strengthening of the notion of uniform continuity of a function restricted to members of b which reduces to ordinary continuity when b consists of the finite subsets of x and agrees with uniform continuity on members of b when b is either the power set of x or the family of compact subsets of x the paper also presents new function space topologies that are well suited to this strengthening as a consequence of the general theory we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions

an fft algorithm based fast algorithm for arbitrary polynomial transformation a fast algorithm for the arbitrary polynomial transformation is described this algorithm is based on the fast fourier transform fft algorithm and reduces the computational complexity of a recently proposed recursive algorithm by an order of magnitude

comments on polytopes of polynomials with zeros in a prescribed set in a recent paper by m fu and b r barmish ibid vol 34 p 544 6 may 1989 the applicability of the edge theorem was significantly extended mainly through relaxing a simple connectedness assumption the class of subsets of the complex plane to which the main result applies is referred to as those subsets whose complements are pathwise connected on the riemann sphere the authors give a definition of pathwise connected on the riemann sphere that differs markedly from the literal meaning the commenter shows by example that although the result given by the authors is correct it does not apply to all subsets whose complements are pathwise connected on the riemann sphere in the usual sense the commenter introduces a result which strengthens the main result of the above paper as a special case of this result it is shown that for the case of real polynomials the applicability of the edge theorem does extend to subsets symmetric with respect to the real axis whose complements are pathwise connected on the riemann sphere in the usual sense

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

proper j holomorphic discs in stein domains of dimension 2 we prove the existence of global bishop discs in a strictly pseudoconvex stein domain in an almost complex manifold of complex dimension 2

density property for hypersurfaces uv p bar x we study hypersurfaces of mathbb c n 2 bar x u v given by equations of form uv p bar x where the zero locus of a polynomial p is smooth reduced the main result says that the lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the lie algebra of all algebraic vector fields consequences of this result for some conjectures of affine algebraic geometry and for the oka grauert gromov principle are discussed

computational aspects of the bilinear transformation based algorithm for s plane to z plane mapping the author analyzes the computational complexity of an algorithm by f d groutage et al ibid vol ac 32 no 7 p 635 7 july 1987 for performing the transformation of a continuous transfer function to a discrete equivalent by a bilinear transformation groutage et al defend their method by noting that their technique is not limited to the bilinear transformation rather it can be extended to any higher order integration rule simpson runge kutta etc or to any higher order expansion of the ln function in general using the method s can be any appropriate mapping function s f z

real valued lipschitz functions and metric properties of functions abstract the purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real valued lipschitz function the composition has this property the key tools we use are the efremovic lemma 21 and a theorem of garrido and jaramillo 22 that says that a function h between metric spaces is lipschitz if and only if whenever it is followed by a lipschitz real valued function in a composition the composition is lipschitz we also present a streamlined proof of the garrido jaramillo result itself but one that still relies on their natural continuous linear operator from the lipschitz space for the target space to the lipschitz space for the domain separately we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for lipschitz type functions as special cases

oka s principle for holomorphic sections of elliptic bundles 2 the h principle for locally trivial fibrations 2 1 the h principle over small subsets 2 2 the h principle over totally real submanifolds 2 3 local extension of the h principle 2 4 localizable extensions 2 5 the h principle for cm bundles 2 6 manifolds with totally real souls 2 7 totally real extensions 2 8 nicely localizable extensions 2 9 the basic h principle for z x 2 10 homomorphisms and holomorphic maps of rank r

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk we give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains specifically we will give the second term in the expansion of the metrics this depends on an improved localisation result and estimates in the one variable case finally we will give a new characterisation of the unit disk in mathbb c in terms of the asymptotic behaviour of quotients of invariant metrics

oka s principle for holomorphic submersions with sprays the oka principle is a fundamental principle of complex analysis which says that on stein manifolds closed complex submanifolds of affine spaces analytic problems of cohomological or even homotopical nature have only topological obstructions the story began in 1939 when k oka proved that on a domain of holomorphy a secondcousin problem is solvable by holomorphic functions if it is solvable by continuous functions oka oka s result has the following equivalent formulation ife x is a holomorphicc bundle over adomainof holomorphy then every continuous section of e is homotopic to a holomorphic section in 1957 h grauert gra extended oka s result to sections of holomorphic fiber bundles with complex homogeneous fibers over stein manifolds see also the papers car fr hl in 1989 m gromov gr1 announced a major generalization of grauert s theorem replacing complex lie groups and homogeneous spaces by a larger class of complex manifolds with a spray a spray on a complex manifold y is a holomorphic vector bundle p e y with a holomorphic map s e y the spray map which is the identity on the zero section and whose restriction to each fibre of e is a submersion at zero such sprays exist on complex homogeneous spaces on complements of affine algebraic subvarieties of codimension at least two and more generally on manifolds whose holomorphic tangent bundle is spanned by finitely many c complete holomorphic vector fields the notion of spray extends in a natural way to submersions the following is theorem 4 5 in gr1 if h z x is a holomorphic submersion onto a stein manifold x such that every point x x has an open neighborhood u x such that z u h 1 u u admits a fiber dominating spray then the inclusion of the

recursive identification algorithms for continuous systems using an adaptive procedure abstract recursive identification algorithms for continuous systems from sampled input and output data are discussed the continuous system is identified through an approximated discrete time estimation model with continuous system parameters an approximated discrete time model of the continuous system under study is first obtained by bilinear transformation then using the estimated denominator of the transfer function of the discrete time model to construct adaptive iir filters introduced to avoid direct approximations of differentiations from sampled data an approximated discrete time estimation model with continuous system parameters is derived the discrete time estimation model is composed of filtered sampled system input and output signals with filtered inputs and delayed filtered outputs as instrumental variables some kinds of recursive instrumental variable identification algorithms are proposed to obtain consistent estimates in the presence of noise the proposed identification algorithms ha

the oka grauert principle without induction over the base dimension we give a new proof of grauert s theorem on oka s principle gra1 gra2 gra3 see also c in the case of smooth stein manifolds which does not use induction over the base dimension instead we use induction over the levels of a strictly plurisubharmonic exhausting function grauert s bump method the present paper is an edited version of our preprint h l2 from 1986 which is difficult to find and of bad printing quality we did not publish the paper in a journal at that time because we planned to write a book containing it but the book has not been written until now on the other hand in the meantime some interest to this proof appeared see e g the work of gromov gro therefore we think a publication of our proof could be useful even with a delay of 10 years this proof gives also some oka principle for arbitrary pseudoconvex manifolds see theorem 1 3 below for vector bundles vector bundles without further mention will refer to topological vector bundles with fibre cr this oka principle contains the following