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We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (fi) in ( ∑ Lpi ) 2 , where pi ↓ 1, so that for all ε > 0 and 1 < q 2, every subsequence of (fi) admits a block basis (1+ ε)-equivalent to the Haar basis for Lq . We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so. Publishe...

We consider variational problems defined on domains ‘weakly’ connected through a separation hyperplane (‘sieve plane’) by an ε-periodically distributed ‘contact zone’. We study the asymptotic behaviour as ε tends to 0 of integral functionals in such domains in the nonlinear and vector-valued case, showing that the asymptotic memory of the sieve is described by a nonlinear ‘capacitarytype’ formu...

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that pro...

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that pro...

We construct an explicit family of linear rank-metric codes over any field Fh that enables efficient list decoding up to a fraction ρ of errors in the rank metric with a rate of 1− ρ− ε, for any desired ρ ∈ (0, 1) and ε > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding bey...

Properties of categories enriched over the category metric spaces are investigated and applied to a study well-known constructions Banach spaces. We prove e.g. that weighted limits colimits exist in metric-enriched iff ordinary ε-(co)equalizers given by ε-(co)isometries for all ε. An object is called approximately injective w.r.t. morphism h:A→A′ morphisms from A into it arbitrarily close those...

In this supplementary material, we give additional details on the two polynomial filters discussed in the main paper. First, we give an example to illustrate how the choice of the degree in the extend McWeeny filter method affects the inflexion point and the rank estimated. Next, we discuss some details on the practical implementation of the Chebyshev polynomial filter method. In section 4, we ...

The communication complexity of F with unbounded error is the limit of the ε-error randomized complexity of F as ε → 1/2. Communication complexity with weakly bounded error is defined similarly but with an additive penalty term that depends on 1/2− ε . Explicit functions are known whose two-party communication complexity with unbounded error is exponentially smaller than with weakly bounded err...

We introduce the space BLD(Ω) consisting of all fields u : Ω → Rn defined on a domain Ω ⊂ Rn, whose symmetric gradient satisfies ∫ Ω |ε(u)|ln(1 + |ε(u)|) dx < ∞. These fields of bounded logarithmic defomation form a proper subspace of the class BD(Ω) consisting of all functions having bounded deformation. With the help of Reshetnyak’s representation formulas we prove that BLD(Ω) is compactly em...