remarks on chirka s proof of slodkowski s theorem by applying schauder s fixed point theorem to an integral operator chirka de veloped a remarkably simple proof for slodkowski s theorem on extending holomorphic motions in this paper we first present our understanding of a motive for designing the integral opera tor from which we also recapitulate the main work in chirka s proof when applying chirka s operator in different ways to obtain the extensions of holomorphic motions a question on the uniqueness of extensions arises after a brief summary on some known sufficient conditions for the uniqueness of extensions we raise two questions concerning the necessary conditions for the uniqueness of the extensions of the holomorphic motions of four points at the end we present an application when the extension is unique that is we show that there is a holomorphic map from the unit open disk d to the teichmuller space of symmetric circle homeomorphisms such that it can not be lifted to a holomorphic map from d to the complex banach space of vanishing beltrami coefficients

geometrical implications of certain infinite dimensional decompositions we investigate the connections between the global structure of a banach space i e the existence of certain finite and infinite dimensional de compositions and the geometrical properties of the closed convex bounded sub sets of such a space i e the existence of extremal and other topologically distin guished points the global structures of various supposedly pathological examples of banach spaces constructed by r c james turn out to be more universal than expected for instance james tree type resp james matrix type decompositions characterize banach spaces with the point of continuity property resp the radon nikodym property moreover the convex point of continuity property is stable under the formation of james infinitely branching tree type sums of infinite dimensional factors we also give several counterex amples to various questions relating some topological and geometrical concepts in banach spaces

types in the n adic tree and minimal analytic gaps abstract we develop some technical tools for the study of the record structure of the n adic tree and we give as an application the list of the nine minimal analytic gaps

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

interpolation banach lattices containing no isomorphic copies of c0 abstract this article is concerned with the question of when the banach lattices generated by the interpolation method of constants contain no isomorphic copies of c 0 for this we study the kothe dual spaces of banach lattices obtained by the methods of constants and means we find an explicit formula for the kothe dual of banach lattices generated by the means method with a mild restriction on the parameters we apply these results to describe a large class of banach lattices generated by the method of constants containing no subspaces isomorphic to c 0 moreover we present applications to the strict singularity of certain inclusion maps between banach sequence lattices

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

on the near differentiability property of banach spaces abstract let be a scalar measure of bounded variation on a compact metrizable abelian group g suppose that has the property that for any measure whose fourier stieltjes transform vanishes at the measure has radon nikod m derivative with respect to the haar measure on g then l pigno and s saeki showed that itself has radon nikod m derivative such property is not shared by vector measures in general we say that a banach space x has the near differentiability property if every x valued measure of bounded variation shares the above property we prove that banach spaces with the radon nikod m property have the near differentiability property while banach spaces with the near differentiability property enjoy the near radon nikod m property we also show that the banach spaces l 1 0 1 and l 1 h 0 1 have the near differentiability property lastly we show that banach spaces with the near differentiability property have type ii radon nikod m property whenever is a riesz subset of type 0 of g

the analytic complete continuity property abstract we introduce the notion of the analytic complete continuity property of banach spaces we give different characterizations of this property we show that this property is different from known related properties such as the complete continuity property and the analytic radon nikod m property

new examples of c 0 saturated banach spaces for every p 1 an isomorphically polyhedral banach space ep is constructed which has an unconditional basis and does not embed isomorphically into a c k space for any countable and compact metric space k moreover ep admits a quotient isomorphic to lp

modified mixed tsirelson spaces we study the modified and boundedly modified mixed tsirelson spacestm fkn n n 1 andtm s fkn n n 1 respectively defined by a subsequence fkn of the sequence of schreier families fn these are reflexive asymptotic l1spaces with an unconditional basis ei ihaving the property that every sequence xi ni 1of normalized disjointly supported vectors contained in ei i nis equivalent to the basis of ln1 we show that if lim 1 nn 1 then the spacet fn n n 1 and its modified variationstm fn n n 1 ortm s fn n n 1 are totally incomparable by proving thatc0is finitely disjointly representable in every block subspace oft fn n n 1 next we present an example of a boundedly modified mixed tsirelson spacexm 1 u tm 1 fkn n n 1 which is arbitrarily distortable finally we construct a variation of the spacexm 1 uwhich is hereditarily indecomposable

tutte polynomial of scale free networks the tutte polynomial of a graph or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics the computation of this invariant for a graph is np hard in general in this paper we focus on two iteratively growing scale free networks which are ubiquitous in real life systems based on their self similar structures we mainly obtain recursive formulas for the tutte polynomials of two scale free networks lattices one is fractal and large world while the other is non fractal but possess the small world property furthermore we give some exact analytical expressions of the tutte polynomial for several special points at x y plane such as the number of spanning trees the number of acyclic orientations etc

disjointly homogeneous banach lattices and compact products of operators the notion of disjointly homogeneous banach lattice is introduced in these spaces every two disjoint sequences share equivalent subsequences it is proved that on this class of banach lattices the product of a regular am compact and a regular disjointly strictly singular operators is always a compact operator

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

on the growth constant for square lattice self avoiding walks the growth constant for two dimensional self avoiding walks on the honeycomb lattice was conjectured by nienhuis in 1982 and since that time the corresponding results for the square and triangular lattices have been sought for the square lattice a possible conjecture was advanced by one of us ajg more than 20 years ago based on the six significant digit estimate available at the time this estimate has improved by a further six digits over the intervening decades and the conjectured value continued to agree with the increasingly precise estimates we discuss the three most successful methods for estimating the growth constant including the most recently developed topological transfer matrix method due to another of us jlj we show this to be the most computationally efficient of the three methods and by parallelising the algorithm we have estimated the growth constant significantly more precisely incidentally ruling out the conjecture which fails in the 12th digit our new estimate of the growth constant is

applications of the theory of semi embeddings to banach space theory let x and y be banach spaces and t x y an injective bounded linear operator t is called a semi embedding if t maps the closed unit ball of x to a closed subset of y this concept was introduced by lotz peck and porta proc edinburgh math soc 22 1979 233 240 it is proved that if x semi embeds in y and x is separable then x has the radon nikodym property provided y does it is shown that if l1 semi embeds in y then y fails the schur property and contains a subspace isomorphic to l1 as a consequence of the proof it is shown that if x is a subspace of l1 either l1 embeds in x or l1 embeds in l1x the simpler result that l1 does not semi embed in c0 is treated separately this result is used to deduce the classic result of menchoff that there exists a singular probability measure on the circle with fourier coefficients vanishing at infinity some generalizations of the notion of semi embedding are given and several complements and open questions are discussed

decompositions of banach lattices into direct sums we consider the problem of decomposing a banach lattice z as a direct sum z x y where x and y are complemented subspaces satisfying a condition of incomparability e g every operator from y to x is strictly singular we treat both the atomic and nonatomic cases in particular we answer a question of wojtaszczyk by showing that l1 ffll2 has unique structure as a nonatomic banach lattice one of the most important problems in the theory of banach lattices which is still open is whether any complemented subspace of a banach lattice must be linearly isomorphic to a banach lattice the main difficulty seems to lie in the fact that most of the criteria for a banach space to be isomorphic to a lattice do not really distinguish between lattices and their complemented subspaces we do not actually treat this question in the present paper but rather consider the situation z x d3 y where z is a banach lattice and x and y two complemented subspaces which are assumed to satisfy different conditions that make them distinct in some or another sense this line of research was initiated by p wojtaszczyk 128 and also by i s edelstein and p wojtaszczyk 3 who proved that if z has a normalized unconditional basis zn n l i e it is a separable atomic lattice so that every linear operator from y into x is compact then zn n l splits into two disjoint parts which are respectively equivalent to bases of x and y in particular both x and y have unconditional bases the proof of this result is based on a fundamental theorem from 28 and 3 which is mentioned below as theorem a we give here a different proof which does not make use of theorem a but instead is based on a simple change of signs result from 2 which is described below as theorem b we also consider the case when the compactness assumption above is replaced by the total incomparability of x and y for which we prove a similar result provided x and y have unconditional bases unfortunately the most interesting case when every operator from y into x is assumed to be strictly singular which was raised as an open problem in 28 remains unsolved we conclude the section devoted to the atomic case with a simple theorem on block bases of a space with unconditional basis zn 1 whose span is complemented such a block basis splits into two disjoint parts the first equivalent to a subsequence of zn n received by the editors september 5 1986 1980 mathematics subject classification 1985 revision primary 46b30 46b15 secondary 46a40 this research was supported by grant no 84 00210 from the united states israel binational science foundation bsf jerusalem israel the research of the first author was also supported in part by nsf grant dms 8500938 the research of the second author was also supported in part by nsf grant dms 8601401 r 1987 american mathematical society ooo2 9947 87 1 00 25 per page

some characterizations of the analytic radon nikodým property in banach spaces the analytic radon nikod m property for a complex banach space x is characterized in several ways analogous to the common characterizations of the radon nikod m property for example the banach space x has the analytic radon nikod m property if and only if any operator from l1 to x that factors through l1h01 is representable the characterizations are used to study complemented subspaces of l1h01 and to define the analytic radon nikod m property for real banach spaces

non asplund banach spaces and operators let w and z be banach spaces such that z is separable and let r w z r w z be a continuous linear operator we study consequences of the adjoint operator r r having non separable range from our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between banach spaces we also obtain an operator theoretic characterisation of separable banach spaces with non separable dual

operators on c 0 1 preserving copies of asymptotic l1 spaces given separable banach spaces x y z and a bounded linear operator t x y then t is said to preserve a copy of z provided that there exists a closed linear subspace e of x isomorphic to z and such that the restriction of t to e is an into isomorphism it is proved that every operator on c 0 1 which preserves a copy of an asymptotic l1 space also preserves a copy of c 0 1

the three space problem for l 1 we construct a subspace x of l1 such that x is an 1 sum of spaces isomorphic to 11 but such that l1 x does not contain a copy of l1 we also construct two banach spaces e1 e2 that do not contain a copy of li but such that e1 x e2 contains a copy of l1 moreover the projections this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms the three space problem for l 29 of li on each factor are one to one and the images of the unit ball of l1 are closed these examples settle questions of j lindenstrauss p pelczynski j bourgain and h p rosenthal equipe d analyse tour 46 u a au c n r s n0 754 universitt paris vi 4 place jussieu 75230 paris cedex 05 department of mathematics ohio state university 231 w 18th avenue columbus ohio 43210 this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms

operators on c k spaces preserving copies of schreier spaces it is proved that an operator t c k x k compact metrizable x a separable banach space for which the e szlenk index of t bx is greater than or equal to 1 is an isomorphism on a subspace of c k isomorphic to x the schreier space of order as a corollary one obtains that a complemented subspace of c k with szlenk index equal to 1 contains a subspace isomorphic to x

a banach space block finitely universal for monotone bases a reflexive banach space x with a basis ei is constructed having the property that every monotone basis is block finitely representable in each block basis of x

a density version of the halpern lauchli theorem abstract we prove a density version of the halpern lauchli theorem this settles in the affirmative a conjecture of r laver specifically let us say that a tree t is homogeneous if t has a unique root and there exists an integer b 2 such that every t t has exactly b immediate successors we show that for every d 1 and every tuple t 1 t d of homogeneous trees if d is a subset of the level product of t 1 t d satisfying lim sup n d t 1 n t d n t 1 n t d n 0 then there exist strong subtrees s 1 s d of t 1 t d having a common level set such that the level product of s 1 s d is a subset of d

new examples of c0 saturated banach spaces ii abstract for every banach space z with a shrinking unconditional basis satisfying an upper p estimate for some p 1 an isomorphically polyhedral banach space is constructed which has an unconditional basis and admits a quotient isomorphic to z it follows that reflexive banach spaces with an unconditional basis and non trivial type tsirelson s original space and c 0 l p for p 1 are isomorphic to quotients of isomorphically polyhedral banach spaces with unconditional bases

ramsey families of subtrees of the dyadic tree we show that for every rooted finitely branching pruned tree t of height w there exists a family f which consists of order isomorphic to t subtrees of the dyadic tree c 0 1 n with the following properties i the family f is a g subset of 2 c ii every perfect subtree of c contains a member of f iii if k is an analytic subset of f then for every perfect subtree s of c there exists a perfect subtree s of s such that the set a f a c s either is contained in or is disjoint from k

the unconditional basic sequence problem we construct a banach space that does not contain any infinite un conditional basic sequence and investigate further properties of this space for example it has no subspace that can be written as a topological direct sum of two infinite dimensional spaces this property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity in par ticular it is either strictly singular or fredholm with index zero this implies that the space is not isomorphic to any proper subspace trinity college cambridge cb2 1tq england current address department of mathematics university college london gower street lon don wc1e 6bt england e mail address wtg l o phx cam ac uk universit paris vii u f r de mathtmatiques 2 place jussieu 75251 paris cedex 05 france current address equipe d analyse et mathematiques appliquees universite de mamne la vallee 93160 noisy le grand france e mail address maurey logiquejussieu fr this content downloaded from 157 55 39 105 on fri 07 oct 2016 04 30 48 utc all use subject to http about jstor org terms

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

duality with real space renormalization and its application to bond percolation we obtain the exact solution of the bond percolation thresholds with inhomogeneous probabilities on the square lattice our method is based on the duality analysis with real space renormalization which is a profound technique invented in the spin glass theory our formulation is a more straightforward way than that of a very recent study on the same problem r m ziff et al j phys a math theor 45 494005 2012 the resultant generic formulas from our derivation can give several estimations for the bond percolation thresholds on other lattices rather than the square lattice

braiding of the attractor and the failure of iterative algorithms le mouvement de l attracteur dans une famille stable d applications rationnelles determine un groupe de tresses sur la sphere de riemann on montre que le groupe correspondant des classes d application est soit reductible soit fini soit fixe l un des attracteurs ceci conduit a un critere topologique d existence des bifurcations dans les algorithmes iteratifs de recherche de racines bases sur des applications rationnels

on the structure of non weakly compact operators on banach lattices in this paper we study extensions of theorems of hagler and stegall on lp spaces 11 and of rosenthal 24 25 and pelczynski 21 on c k spaces to more general banach spaces with some lattice structure which do not contain complemented copies of i r we show that i fx is a separable banach space such that co does not embed into x and t is a bounded linear operator from x into some banach space y such that t y is not separable then i if x has local unconditional structure h la gordon and lewis 7

banach lattice versions of strict singularity abstract we explore the relation between lattice versions of strict singularity for operators from a banach lattice to a banach space in particular we study when the class of disjointly strictly singular operators those not invertible on the span of any disjoint sequence coincides with that of lattice strictly singular operators i e those not invertible on any infinite dimensional sublattice new results are given which help to clarify the existing relation between these two classes

frontiers in complex dynamics rational maps on the riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems first rational maps are complexanalytic so a broad spectrum of techniques can contribute to their study quasiconformal mappings potential theory algebraic geometry etc the rational maps of a given degree form a finite dimensional manifold so exploration of this parameter space is especially tractable finally some of the conjectures once proposed for smooth dynamical systems and now known to be false seem to have a definite chance of holding in the arena of rational maps in this article we survey a small constellation of such conjectures centering around the density of hyperbolic rational maps those which are dynamically the best behaved we discuss some of the evidence and logic underlying these conjectures and sketch recent progress towards their resolution

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

continuity of julia set and its hausdorff dimension of yang lee zeros abstract we show the continuity of the julia set and its hausdorff dimension about a family of rational maps concerning 2 dimensional diamond hierarchical potts models about anti ferromagnetic coupling in statistical mechanics

transfer matrix computation of generalized critical polynomials in percolation percolation thresholds have recently been studied by means of a graph polynomial pb p henceforth referred to as the critical polynomial that may be defined on any periodic lattice the polynomial depends on a finite subgraph b called the basis and the way in which the basis is tiled to form the lattice the unique root of pb p in 0 1 either gives the exact percolation threshold for the lattice or provides an approximation that becomes more accurate with appropriately increasing size of b initially pb p was defined by a contraction deletion identity similar to that satisfied by the tutte polynomial here we give an alternative probabilistic definition of pb p which allows for much more efficient computations by using the transfer matrix than was previously possible with contraction deletion we present bond percolation polynomials for the 4 82 kagome and 3 122 lattices for bases of up to respectively 96 162 and 243 edges much larger than the previous limit of 36 edges using contraction deletion we discuss in detail the role of the symmetries and the embedding of b for the largest bases we obtain the thresholds pc 4 82 0 676 803 329 pc kagome 0 524 404 998 pc 3 122 0 740 420 798 comparable to the best simulation results we also show that the alternative definition of pb p can be applied to study site percolation problems this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

proximity to l1and distortion in asymptotic l1spaces abstract for an asymptotic l 1 space x with a basis x i certain asymptotic l 1 constants x are defined for 1 x measures the equivalence between all normalized block bases y i k i 1 of x i which are s admissible with respect to x i s is the th schreier class of sets and the unit vector basis of l k 1 this leads to the concept of the delta spectrum of x x which reflects the behavior of stabilized limits of x the analogues of these constants under all renormings of x are also defined and studied we investigate x both in general and for spaces of bounded distortion we also prove several results on distorting the classical tsirelson s space t and its relatives

on connectivity of julia sets of yang lee zeros we prove that the julia sets of the yang lee zeros of the potts model on the diamond hierarchical lattice are connected sets on the complex plane

radon nikodým properties associated with subsets of countable discrete abelian groups with any subset of a countable discrete abelian we associate with it three banach space properties these properties are radon nikodym type properties the relationship between these properties is investigated the results are applied to give new characterizations of riesz subsets and rosenthal subsets of countable discrete abelian groups

gδ embeddings in hilbert space abstract it is shown that a separable banach space x has the point of weak to norm continuity property resp the radon nikodym property if and only if there exists a compact g embedding resp an h embedding from x into l2 this solves several questions of j bourgain and h p rosenthal j funct anal 52 1983 it is also shown that every non relatively compact sequence in a banach space with property pc has a difference subsequence which is a boundedly complete basic sequence this solves a question of pelczynski and extends some results of w b johnson and h p rosenthal studia math 43 1972 77 92 various related questions asked in the above bourgain rosenthal reference and by g a edgar and r f wheeler pac j math 115 1984 and n ghoussoub and h p rosenthal math ann 264 1983 321 332 are also settled

borel extractions of converging sequences in compact sets of borel functions abstract it is well known by a classical result of bourgain fremlin talagrand that if k is a pointwise compact set of borel functions on a polish space then given any cluster point f of a sequence f n n in k one can extract a subsequence f n k k converging to f in the present work we prove that this extraction can be achieved in a borel way this will prove in particular that the notion of analytic subspace of a separable rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence

percolation critical polynomial as a graph invariant every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial with the root in 0 1 providing the threshold recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice the polynomial depends on the lattice and on its decomposition into identical finite subgraphs but once these are specified the polynomial is essentially unique on lattices for which the exact percolation threshold is unknown the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size in this paper i show how the critical polynomial can be viewed as a graph invariant like the tutte polynomial in particular the critical polynomial is computed on a finite graph and may be found using the deletion contraction algorithm this allows calculation on a computer and i present such results for the kagome lattice using subgraphs of up to 36 bonds for one of these i find the prediction pc 0 52440572 which differs from the numerical value pc 0 52440503 5 by only 6 9 x 10 7

comparing poincare densities however there is no formula for the poincare density pn of a general plane domain q and so one must resort to estimates many research articles have been devoted to finding and working with such estimates see for example 4 5 15 19 24 and 25 the first tool to establish such estimates is schwarz s lemma which says that every holomorphic map is a contraction with respect to the poincare metric that is ps f ldf i pd z idzl for any holomorphic map f from a riemann surface d to another riemann surface q applying this lemma

on the dynamics of a family of generated renormalization transformations abstract we study the family of renormalization transformations of the generalized d dimensional diamond hierarchical potts model in statistical mechanic and prove that their julia sets and non escaping loci are always connected where d 2 in particular we prove that their julia sets can never be a sierpinski carpet if the parameter is real we show that the julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models moreover the asymptotic formula of the hausdorff dimension of the julia set is calculated as the parameter tends to infinity

conjugate operators that preserve the nonconvergence of bounded martingales we show that the conjugate t of an operator t x y with x and y banach spaces satisfies the following dichotomy either t preserves the nonconvergence of bounded martingales in y or there exists a compact operator k x y such that the kernel n t k fails the radon nikod m property

tukey classification of some ideals on ω and the lattices of weakly compact sets in banach spaces abstract we study the lattice structure of the family of weakly compact subsets of the unit ball b x of a separable banach space x equipped with the inclusion relation this structure is denoted by k b x and also with the parametrized family of almost inclusion relations k l e b x where e 0 this structure is denoted by ak b x tukey equivalence between partially ordered sets and a suitable extension to deal with ak b x are used assuming the axiom of analytic determinacy we prove that separable banach spaces fall into four categories namely k b x is equivalent either to a singleton or to or to the family k q of compact subsets of the rational numbers or to the family c of all finite subsets of the continuum also under the axiom of analytic determinacy a similar classification of ak b x is obtained for separable banach spaces not containing l 1 we prove in zfc that k b x ak b x are equivalent to either 0 k q or c the lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied

on classes of banach spaces admitting small universal spaces we characterize those classes c of separable banach spaces admitting a separable universal space y that is a space y containing up to isomorphism all members of c which is not universal for all separable banach spaces the characterization is a byproduct of the fact proved in the paper that the class nu of non universal separable banach spaces is strongly bounded this settles in the affirmative the main conjecture from argyros and dodos 2007 our approach is based among others on a construction of l spaces due to j bourgain and g pisier as a consequence we show that there exists a family y 1 of separable non universal l spaces which uniformly exhausts all separable banach spaces a number of other natural classes of separable banach spaces are shown to be strongly bounded as well

continuity of capacity of a holomorphic motion abstract we study the behavior of various set functions under holomorphic motions we show that under such deformations logarithmic capacity varies continuously while analytic capacity may not

unconditional families in banach spaces it is shown that for every separable banach space x with non separable dual the space x contains an unconditional family of size x the proof is based on ramsey theory for trees and finite products of perfect sets of reals among its consequences it is proved that every dual banach space has a separable quotient

shellable weakly compact subsets of c 0 1 we show that for every weakly compact subset k of c 0 1 with finite cantor bendixson rank there is a reflexive banach lattice e and an operator t e rightarrow c 0 1 such that k subseteq t b e on the other hand we exhibit an example of a weakly compact set of c 0 1 homeomorphic to omega omega 1 for which such t and e cannot exist this answers a question of m talagrand in the 80 s

operators whose dual has non separable range let x and y be separable banach spaces and t x y be a bounded linear operator we characterize the non separability of t y by means of fixing properties of the operator t