uniformly factoring weakly compact operators abstract let x and y be separable banach spaces suppose y either has a shrinking basis or y is isomorphic to c 2 n and a is a subset of weakly compact operators from x to y which is analytic in the strong operator topology we prove that there is a reflexive space with a basis z such that every t a factors through z likewise we prove that if a l x c 2 n is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a banach space z with separable dual such that every t a factors through z finally we prove a uniform version of this result in which we allow the domain and range spaces to vary

nonparametric regression with predictors missing at random nonparametric regression with predictors missing at random mar where the probability of missing depends only on observed variables is considered univariate predictor is the primary case of interest a new adaptive orthogonal series estimator is developed large sample theory shows that the estimator is rate minimax and it is also sharp minimax whenever predictors are missing completely at random mcar furthermore confidence bands estimation of nuisance functions including conditional probability of observing the predictor design density and scale and multiple regression are also considered numerical study and a real example show feasibility of the proposed methodology for small samples supplementary materials containing results of the numerical study are available online

index assignment for multichannel communication under failure we consider the problem of constructing multiple description scalar quantizers and describing the achievable rate distortion tuples in that setting we model this as a combinatorial optimization problem of number arrangements in a matrix this approach gives a general technique for deriving lower bounds on the distortion at given channel rates this technique is constructive thus allowing an algorithm that gives an upper bound for the case of two communication channels with equal rates the bounds coincide thus giving the precise lowest achievable distortion at fixed rates the bounds are within a small constant for a higher number of channels to the best of our knowledge this is the first result involving systems with more than two communication channels

classifications of baire 1 functions and spreading models we prove that if for a bounded function f defined on a compact space k there exists a bounded sequence fn of continuous functions with spreading model of order 1 co the index rnd as defined by a kechris and a louveau as a corollary of this result we have that the banach spaces ve k 1 4 defining the notion of nullcoefficient sequences of order 1 4 coi we prove that every bounded sequence fn of continuous functions converging pointwise to a function f with rnd f co is a null coefficient sequence of order as a corollary to this we have the following co spreading model theorem every nontrivial weak cauchy sequence in a banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of co or is a nullcoefficient sequence of order 1

on the bandwidth of a hamming graph the bandwidth of the hamming graph the product kn d of complete graphs has been an open question for many years recently berger wolf and rheingold 1 pointed out that the bandwidth of a numbering of the hamming graph may be interpreted as a measure of the effects of noise in the multi channel transmission of data with that numbering they also gave lower and upper bounds for it in this paper we present better lower and upper bounds showing that the bandwidth of kn d is asymptotic to 2 d nd as d

genericity and amalgamation of classes of banach spaces abstract we study universality problems in banach space theory we show that if a is an analytic class in the effros borel structure of subspaces of c 0 1 of non universal separable banach spaces then there exists a non universal separable banach space y with a schauder basis that contains isomorphs of each member of a with the bounded approximation property the proof is based on the amalgamation technique of a class c of separable banach spaces introduced in the paper we show among others that there exists a separable banach space r not containing l 1 0 1 such that the indices and r n d are unbounded on the set of baire 1 elements of the ball of the double dual r of r this answers two questions of h p rosenthal we also introduce the concept of a strongly bounded class of separable banach spaces a class c of separable banach spaces is strongly bounded if for every analytic subset a of c there exists y c that contains all members of a up to isomorphism we show that several natural classes of separable banach spaces are strongly bounded among them the class of non universal spaces with a schauder basis the class of reflexive spaces with a schauder basis the class of spaces with a shrinking schauder basis and the class of spaces with schauder basis not containing a minimal banach space x

a classification of baire class 1 functions we study in this paper various ordinal ranks of bounded baire class 1 functions and we show their essential equivalence this leads to a natural classification of the class of bounded baire class 1 functions b 1 in a transfinite hierarchy b 1 1 of small baire classes for which for example an analysis similar to the hausdorff kuratowski analysis of 0 2 sets via transfinite differences of closed sets can be carried out the notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a baire class 1 function are introduced and used in this study

on the bandwidth of 3 dimensional hamming graphs this paper presents strategies for improving the known upper and lower bounds for the bandwidth of hamming graphs k n d and 0 1 d our labeling strategy lowers the upper bound on the bandwidth of the continuous hamming graph 0 1 3 from 5 to 4497 a lower bound of 4439 on bw 0 1 3 follows from known isoperimetric inequalities and a related dynamic program is conjectured to raise that lower bound to 4 9 4444 in particular showing the power of our method we prove that the bandwidth of k 6xk 6xk 6 is exactly 101

interpolating hereditarily indecomposable banach spaces a banach space x is said to be hereditarily indecomposable h i if for any pair of closed subspaces y z of x with y z 0 y z is not a closed subspace throughout this section by the term subspace we mean a closed infinite dimensional subspace of x the h i spaces form a new and as we believe fundamental class of banach spaces the celebrated example of a banach space with no unconditional basic sequence due to w gowers and b maurey gm is the first construction of a h i space it is easily seen that every h i space does not contain any unconditional basic sequence actually the concept of h i spaces came after w johnson s observation that this was a property of the gowers maurey example to describe even further the peculiar structure of a h i space we recall an alternative definition of such a space a banach space x is a h i space if and only if for every pair of subspaces y z and e 0 there exist y y z z with y z 1 and y z e thus h i spaces are structurally irrelevant to classical banach spaces in particular to hilbert spaces other constructions of h i spaces already exist we mention argyros and deliyanni s construction of h i spaces which are asymptotic spaces ad2 v ferenczi s example of a uniformly convex h i space f2 and h i modified asymptotic spaces contained in adkm other examples of banach spaces which are h i or which have a h i subspace are given in g1 h os1 the construction of such a space requires several steps and it uses two fundamental ideas the first is tsirelson s recursive definition of saturated norms ts and the second is maurey rosenthal s construction of weakly null sequences without unconditional basic subsequences mr an important ingredient in the gowersmaurey construction is the schlumprecht space this is a tsirelson type banach space which is arbitrarily distortable and has been used in the solution of important problems thus beyond its use in the constructions of h i spaces it plays a central role in the solution of the distortion problem for hilbert spaces os the essential difference between schlumprecht and tsirelson spaces became more transparent in ad2 where the mixed tsirelson spaces were introduced it is natural to expect that h i spaces share special and interesting properties not found in the previously known banach spaces indeed the following theorem is proven in

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

banach spaces with separable duals it is proved that every banach space with a separable dual em beds into a space with a shrinking basis it follows that every separable re flexive space can be embedded in a reflexive space with a basis

a note on banach spaces containing c0 or c it is shown that every separable banach space x containing a subspace isomorphic to c0 has a subspace y with basis such that xy c0c and the latter space has a shrinking basis and an unconditional fdd moreover it is shown that x c has a basis if x has the bounded approximation property

optimal sequences of continuous functions converging to a baire 1 function a proof of rosenthal s co index conjecture is given our approach uses optimal sequences of continuous functions converging to a baire 1 function their existence is obtained by the optimal sequences theorem stated and proved here for a sequence of functions and a countable ordinal the variation is also introduced if pointwise converges to f the relation between and is completely clarified finally optimal sequences associated to a baire 1 function are defined and their existence for every baire 1 function is provided

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

ξ completely continuous operators and ξ schur banach spaces abstract for each ordinal 0 1 we introduce the notion of a completely continuous operator and prove that for each ordinal 0 1 the class v of completely continuous operators is a closed injective operator ideal which is not surjective symmetric or idempotent we prove that for distinct 0 1 the classes of completely continuous operators and completely continuous operators are distinct we also introduce an ordinal rank v for operators such that v a 1 if and only if a is completely continuous and otherwise v a is the minimum countable ordinal such that a fails to be completely continuous we show that there exists an operator a such that v a if and only if 1 1 and there exists a banach space x such that v i x if and only if there exists an ordinal 1 such that finally prove that for every 0 1 the class a l v a is 1 1 complete in l the coding of all operators between separable banach spaces this is in contrast to the class v l which is 2 1 complete in l

a universal reflexive space for the class of uniformly convex banach spaces we show that there exists a separable reflexive banach space into which every separable uniformly convex banach space isomorphically embeds this solves problems raised by j bourgain b in 1980 and by w b johnson in 1977 jo we also give intrinsic characterizations of separable reflexive banach spaces which embed into a reflexive space with a block q hilbertian and or a block p besselian finite dimensional decomposition

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

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

index assignment for two channel quantization this paper concerns the design of a multiple description scalar quantization mdsq system for two identical channels for an unbounded discrete information source this translates to the combinatorial problem of finding an arrangement of the integers into the infinite plane square grid so that each row and each column contains exactly n numbers such that the difference between any two numbers in the same row or column is at most d with d to be minimized for a given n the best previous lower and upper bounds on the lowest d were n sup 2 3 o n and n sup 2 2 o n we give new lower and upper bounds both of the form 3n sup 2 8 o n we also consider minimizing the maximal variance in any row or column and show that it must be at least n sup 4 60 o n sup 3 and that it does not have to be more than 3n sup 4 160 o n sup 3