on closed invariant sets in local dynamics abstract we investigate the dynamical behaviour of a holomorphic map on an f invariant subset c of u where f u c k we study two cases when u is an open connected and polynomially convex subset of c k and c u closed in u and when u has a p s h barrier at each of its points and c is not relatively compact in u in the second part of the paper we prove a birkhoff s type theorem for holomorphic maps in several complex variables i e given an injective holomorphic map f defined in a neighborhood of u with u star shaped and f u a runge domain we prove the existence of a unique forward invariant maximal compact and connected subset of u which touches u

on line bin packing of items of random sizes ii this paper describes the construction of an on line algorithm with the following property there exist universal constants k alpha such that given any probability measure mu on 0 1 and a sequence x 1 ldots x n ldots of items independently and identically distributed according to mu the algorithm packs x 1 ldots x n into at most t n x 1 ldots x n kn 1 2 log n 3 4 unit size bins where t n x 1 ldots x n denotes the minimum number of bins needed to pack x 1 ldots x n and does this with probability greater than or equal to 1 k exp alpha log n 3 2 in contrast with the authors previous work on this problem the algorithm is now independent of mu

an interpolation theorem for proper holomorphic embeddings given a stein manifold x of dimension n 1 a discrete sequence a j subset x and a discrete sequence b j subset mathbb c m where m ge n left frac 3n 2 right 1 there exists a proper holomorphic embedding f colon x hookrightarrow mathbb c m satisfying f a j b j for every j 1 2

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

extension operators and automorphisms abstract many of the commonly studied extension operators that take a normalized locally univalent function f defined in the open unit disk d c to a normalized locally biholomorphic mapping of the unit ball b c n n 2 are of the form f 0 f where 0 f z f z 1 f z 1 z z b z z 2 z n is the roper suffridge extension of f and is a first coordinate shear an automorphism of c n of the form w w 1 g w w where g c n 1 c is holomorphic and contains no terms of degree 0 or 1 in its taylor expansion at the origin other studies have considered the roper suffridge extension operator with the multiplier f z 1 replaced by a different expression recently it has been shown that the set of restrictions of automorphisms of c n to b satisfying some geometric property such as convexity or starlikeness is dense in the full respective family of geometric mappings when n 2 we study large families of biholomorphic mappings on b generated by automorphisms of c n abstract generalizations of the roper suffridge extension operator unitary rotations and families of locally univalent functions on d in many cases families arising from distinct automorphisms will be seen to be disjoint despite the fact that they contain the restrictions of the respective automorphisms to b and the previously noted density result of particular interest given our consideration of more general automorphisms related to extension operators we show that the compositions of the roper suffridge extension operator with a particular class of first coordinate overshears automorphisms of c n of the form w h w w 1 g w w for g h c n 1 c holomorphic and h never 0 extend certain bounded starlike mappings of d to starlike mappings of b we conclude with several open questions inspired by this work

universal entire functions for affine endomorphisms of c n abstract in this paper the affine endomorphisms of c n which support compositionally universal entire functions are completely characterized

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

loewner chains and nonlinear resolvents of the caratheodory family on the unit ball in cn abstract in this paper we study various properties of nonlinear resolvents of holomorphic mappings in the caratheodory family m b n where b n is the euclidean unit ball in c n first we prove certain characterizations of inverse loewner chains f z t e 0 t a d z on b n 0 where a 0 c is a locally lebesgue integrable function such that a t 0 for a e t 0 and which satisfies a natural assumption next we prove that if f m b n then the nonlinear resolvent family j r r 0 is an inverse loewner chain on b n and the associated herglotz vector field is of divergence type where j r j r f i n r f 1 r 0 this result is a generalization to higher dimensions of a recent result due to elin shoikhet and sugawa we prove that 1 r j r can be embedded as the first element of a normal loewner chain and the family 1 r j r f 0 r f m b n is a compact subset of s 0 b n also we deduce that the shearing of 1 r j r r 0 associated with the family m b 2 is quasi convex of type a and also starlike of order 4 5 we give a sufficient condition for 1 r j r to be quasiconformal on b n and to be extended to a quasiconformal homeomorphism of c n onto itself finally sharp coefficient bounds for the nonlinear resolvent families of certain subsets of m b n and also examples of support points of the compact families generated by nonlinear resolvent mappings on b 2 will be obtained

free dense subgroups of holomorphic automorphisms we show the existence of free dense subgroups generated by two elements in the holomorphic shear and overshear group of complex euclidean space and extend this result to the group of holomorphic automorphisms of stein manifolds with the density property provided there exists a generalized translation the conjugation operator associated to this generalized translation is hypercyclic on the topological space of holomorphic automorphisms

embedding holomorphic discs through discrete sets the most interesting case is n 2 for n 3 this was proved by a different method in 16 for f c n and in 9 for all convex domains o c c n in the case when f2 c n 3 one can in addition prescribe a discrete set c c and require that f j z j for all j 16 for n 2 the methods of 9 and 16 only give proper holomorphic immersions of the disc through a given discrete set in 12 the main problem of course is that in dimension two one cannot remove self intersections o f complex curves by small deformations we do not know whether our theorem holds for non runge pseudoconvex domains in cn our methods do not seem to extend to this case in this direction it was proved in 6 that for every finite subset z in an arbitrary connected pseudoconvex domain f2 c c n n 1 there exist proper holomorphic mappings f d d o f the disc into 12 such that z c f d it is likely that a refinement of the construction in 6 gives proper holomorphic immersions f d f2 whose image contains a given discrete subset of 12 an example

criteria for the density property of complex manifolds in this paper we suggest new effective criteria for the density property this enables us to give a trivial proof of the original anders en lempert result and to establish almost free of charge the algebraic density property for all linear algebraic groups whose connected components are different from tori or c as another application of this approach we tackle the question asked among others by f forstneri v c about the density of algebraic vector fields on euclidean space vanishing on a codimension 2 subvariety

perfect packing theorems and the average case behavior of optimal and online bin packing we consider the one dimensional bin packing problem under the discrete uniform distributions u j k 1 leq j leq k 1 in which the bin capacity is k and item sizes are chosen uniformly from the set 1 2 ldots j note that for 0 u j k leq 1 this is a discrete version of the previously studied continuous uniform distribution u 0 u where the bin capacity is 1 and item sizes are chosen uniformly from the interval 0 u we show that the average case performance of heuristics can differ substantially between the two types of distributions in particular there is an online algorithm that has constant expected wasted space under u j k for any j k with 1 leq j k 1 whereas no online algorithm can have o n 1 2 expected waste under u 0 u for any 0 u leq 1 our u j k result is an application of a general theorem of courcoubetis and weber that covers all discrete distributions under each such distribution the optimal expected waste for a random list of n items must be either theta n theta n 1 2 or o 1 depending on whether certain perfect packings exist the perfect packing theorem needed for the u j k distributions is an intriguing result of independent combinatorial interest and its proof is a cornerstone of the paper we also survey other recent results comparing the behavior of heuristics under discrete and continuous uniform distributions

on the sum of squares algorithm for bin packing in this article we present a theoretical analysis of the online sum of squares algorithm ss for bin packing along with several new variants ss is applicable to any instance of bin packing in which the bin capacity b and item sizes s a are integral or can be scaled to be so and runs in time o nb it performs remarkably well from an average case point of view for any discrete distribution in which the optimal expected waste is sublinear ss also has sublinear expected waste for any discrete distribution where the optimal expected waste is bounded ss has expected waste at most o log n we also discuss several interesting variants on ss including a randomized o nb log b time online algorithm ss whose expected behavior is essentially optimal for all discrete distributions algorithm ss depends on a new linear programming based pseudopolynomial time algorithm for solving the np hard problem of determining given a discrete distribution f just what is the growth rate for the optimal expected waste

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

polynomial completion of symplectic jets and surfaces containing involutive lines motivated by work of dragt and abell on accelerator physics we study the completion of symplectic jets by polynomial maps of low degrees we use andersen lempert theory to prove that symplectic completions always exist and we prove the degree bound conjectured by dragt and abell in the physically relevant cases however we disprove the degree bound for 3 jets in dimension 4 this follows from the fact that if sigma is the disjoint union of r 7 involutive lines in mathbb p 3 then sigma is contained in a degree d 4 hypersurface i e the restriction morphism iota h 0 mathbb p 3 mathcal o mathbb p 3 4 rightarrow h 0 sigma mathcal o sigma 4 has a nontrivial kernel todd we give two new proofs of this fact and finally we show that if r d ne 7 4 then the map iota has maximal rank

algebraic density property of danilov gizatullin surfaces a danilov gizatullin surface is an affine surface v which is the complement of an ample section s for the ruling of a hirzebruch surface the remarkable theorem of danilov and gizatullin states that the isomorphism class of v depends only on the self intersection number s s in this paper we apply the theorem of danilov gizatullin to prove that the lie algebra generated by the complete algebraic vector fields on v coincides with the set of all algebraic vector fields of v

a unified approach to parallel space decomposition methods we consider relaxed additive and multiplicative iterative space decomposition methods for the minimization of sufficiently smooth functionals without constraints we develop a general framework which unites existing approaches from both parallel optimization and finite elements specifically this work unifies earlier research on the parallel variable distribution method in minimization space decomposition methods for convex functionals algebraic schwarz methods for linear systems and splitting methods for linear least squares we develop a general convergence theory within this framework which provides several new results as well as including known convergence results