Function spaces not containing ℓ<Subscript>1</Subscript>

For Ω bounded and open subset of andX a reflexive Banach space with 1-symmetric basis, the function spaceJF _X (Ω) is defined. This class of spaces includes the classical James function space. Every member of this class is separable and has non-separable dual. We provide a proof of topological nature thatJF _X (Ω) does not contain an isomorphic copy of ℓ₁. We also investigate the structure of these spaces and their duals.     

The Szlenk index and local ℓ<Subscript>1</Subscript>-indices

We introduce two new local ₁-indices of the same type as the Bourgain ₁-index; the ⁺₁-index and the ⁺₁-weakly null index. We show that the ⁺₁-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ₁. The ⁺₁-weakly null index has the same form as the Bourgain ₁-index: if it is countable it must take values for some <₁. The different ₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ₁-index, via the ⁺₁-weakly null index. We show that I(C(^{))=^1++1}.     

Containment of c<Subscript>0</Subscript> and ℓ<Subscript>1</Subscript> in π<Subscript>1</Subscript>(<Emphasis Type="Italic">E,F</Emphasis>)

Suppose π₁(E, F) is the space of all absolutely 1-summing operators between two Banach spacesE andF. We show that ifF has a copy of c₀, then π₁ (E, F) will have a copy of c₀, and under some conditions ifE has a copy of ℓ₁ then π₁ (E, F) would have a complemented copy of ℓ₁.     

Integral polynomials on Banach spaces not containing ℓ<Subscript>1</Subscript>

We give new characterizations of Banach spaces not containing ℓ₁ in terms of integral and p-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.     

Cuts,Trees and ℓ<Subscript>1</Subscript>-Embeddings of Graphs*

Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ₁ space. The main results are:1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor.2. A constant-distortion embedding of outerplanar graphs into the restricted class of ₁-metrics known as dominating tree metrics. A lower bound of (log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics is also presented. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of ₁-embeddability.* A preliminary version of this work appeared in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp. 399–408. This work was done while the author was at the University of California, Berkeley. Supported in part by NSF grants CCR-9505448 and CCR-9820951.     

ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Diagonally Dominant Symmetric Operators

The existence of _p-universal diagonally dominant symmetric liner operators on separable Hilbert spaces is discussed. Also, properties of these operators and their characterization in terms of their spectra are investigated.     

Moduli spaces ℳ<Subscript>2,1</Subscript> and ℳ<Subscript>3,1</Subscript>

The cell structure of the spaces ℳ_2,1 and ℳ_3,1 is considered. These are the spaces of complex curves of genus 2 and 3 with one marked point. For the space ℳ_2,1, nine cells of the highest dimension 8 are described and their adjacency is studied. For the space ℳ_3,1, a list of all 1726 cells of the highest dimension 14 (with orientation) is obtained. The list of adjacent couples of cells is also obtained. These lists can be found on the web.     

On Perturbations of ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-localized Frames

In this paper, we give some sufficient conditions under which perturbations preserve ? _p -localized frames. Using an arbitrary given sequence, we provide a simple way for constructing ? _p -localized sequences.     

The point of continuity property in Banach spaces not containing ℓ<Subscript>1</Subscript>

We obtain a local characterization of the point of continuity property for bounded subsets in Banach spaces not containing basic sequences equivalent to the standard basis of ℓ₁ and, as a consequence, we deduce that, in Banach spaces with a separable dual, every closed, bounded, convex and nonempty subset failing the point of continuity property contains a further subset which can be seen inside the set of Borel regular probability measures on the Cantor set in a weak-star dense way. Also, we characterize in terms of trees the point of continuity property of Banach spaces not containing ℓ₁, by proving that a Banach space not containing ℓ₁ satis- fies the point of continuity property if, and only if, every seminormalized weakly null tree has a boundedly complete branch.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

New multiple <Subscript>6</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>6</Subscript> summation formulas and related conjectures

Three new summation formulas for ₆ ψ ₆ bilateral basic hypergeometric series attached to root systems are presented. Remarkably, two of these formulae, labelled by the A_2n−1 and A_2n root systems, can be reduced to multiple ₆ φ ₅ sums generalising the well-known van Diejen sum. This latter sum serves as the weight-function normalisation for the BC _n q-Racah polynomials of van Diejen and Stokman. Two ₈ φ ₇-level extensions of the multiple ₆ φ ₅ sums, as well as their elliptic analogues, are conjectured. This opens up the prospect of constructing novel A-type extensions of the Koornwinder–Macdonald theory.     

The Algorithm Z and Ramanujan’s <Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript> summation

We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan’s ₁ ψ ₁ summation formula.     

Topological recursion relation for <Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>2</Subscript> in

We give the definition of the full potential with descendants, which is a genus expansion of the Barannikov-Kontsevich solution of the WDVV equation. This potential satisfies the Getzler relation, which comes from the geometry of the moduli space of curves Open image in new window .     

Operators on <Emphasis Type="Italic">C</Emphasis>[0,1] preserving copies of asymptotic ℓ<Subscript>1</Subscript> 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]).     

Stabilization of Linear Switched Delay Systems: ℋ<Subscript>2</Subscript> and ℋ<Subscript>∞</Subscript> Methods

This paper presents new results pertaining to the delay-dependent stability and control synthesis of a class of linear switched continuous-time systems with time-varying delays. A new state transformation is introduced to exhibit the delay-dependent dynamics in the slow-time scale. For stability, we construct an appropriate selective Lyapunov functional to derive delay-dependent LMI-based sufficient conditions under arbitrary switching and without relying to overbounding. For the control synthesis, we design switched feedback schemes based on quadratic ℋ₂, ℋ_∞ and simultaneous ℋ₂/ℋ_∞ performance criteria. Under the developed transformation, it is established that both the instantaneous and delayed feedback control yield identical results. Numerical examples are presented to illustrate the analytical development.     

Approximation Representations for Δ<Subscript>2</Subscript> Reals

We study ₂ reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of approximation representation and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ₂ reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.)Mathematics Subject Classification (2000): 03F60, 03D30     

Convergence Theorems for ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Norm Minimizers with Respect to <Emphasis Type="Italic">p</Emphasis>

Let C be a fixed compact convex subset of and let x_p be the unique minimal _p-norm element in C for any . In this paper, we study the convergence of x_p as p or , respectively. We characterize also the limit point as the minimal element of C with respect to the lexical minimax order relation or the lexical minitotal order relation, respectively.Communicated by D. G. LuenbergerThe author thanks Professor H. Komiya for valuable advice and an anonymous referee for kind comments.     

<Emphasis Type="Italic">q</Emphasis>-extensions of Dougall’s bilateral <Subscript>2</Subscript>H<Subscript>2</Subscript>-series

Transformations for a bilateral ₂ ψ ₂-series are investigated by means of Abel’s lemma on summation by parts. Two q-extensions of Dougall’s classical identity for bilateral ₂H₂-sum are established. As by-products, the Rogers–Fine identity is recovered and a new proof is presented for Bailey’s identity of bilateral well-poised ₆ ψ ₆-series.     

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.