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  • 1
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    UNCONDITIONALITY IN TENSOR PRODUCTS AND IDEALS OF POLYNOMIALS, MULTILINEAR FORMS AND OPERATORS (2011)
    Oxford University Press
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    Publication Date: 2011-11-24
    Description: We study tensor norms that destroy unconditionality in the following sense: for every Banach space E with unconditional basis, the n -fold tensor product of E (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check whether a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from and destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals never have the Gordon–Lewis property. In some cases we even obtain that the monomial basic sequence can never be unconditional. Analogous problems for multilinear ideals are addressed, and noteworthy differences between the 2-fold and the n -fold ( n ≥ 3) theory are obtained.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 2
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    KING ARTHUR AND HIS KNIGHTS WITH TWO ROUND TABLES (2011)
    Oxford University Press
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    Publication Date: 2011-11-24
    Description: A graph G is r-starred if, for some v V ( G ), a largest pairwise intersecting family of independent r -subsets of V ( G ) may be obtained by taking all such subsets containing v (the r - star at v ). Let G be the disjoint union of powers of cycles; Hilton and Spencer have studied the problem of determining the values of r for which G is r -starred. They conjectured that the property holds for all r , and made a weaker conjecture that this is so for the union of just two cycles. In this paper we prove the second conjecture, showing also that if G is the union of several graphs, each a power of a cycle, then G is α-starred (where α is the independence number of G ), provided that there is a homomorphism from some component of G to each of the other components.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 3
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    ON FEFFERMAN-STEIN INEQUALITY FOR RUBIO DE FRANCIA OPERATORS (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: Given a Calderón–Zygmund singular integral operator, it is an open question whether the Fefferman inequality holds true with u as an arbitrary weight. This inequality is known as the Muckenhoupt–Wheeden conjecture. In this paper we prove that, for a wider class of operators called Rubio de Francia operators, if || u || ≤1, then where the function is slightly bigger than the identity. Applications of this inequality in the setting of rearrangement invariant spaces are given.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 4
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    GL(2, R) STRUCTURES, G2 GEOMETRY AND TWISTOR THEORY (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: A GL(2, R) structure on an ( n + 1)-dimensional manifold is a smooth point-wise identification of tangent vectors with polynomials in two variables homogeneous of degree n . This, for even n = 2 k , defines a conformal structure of signature ( k , k + 1) by specifying the null vectors to be the polynomials with vanishing quadratic invariant. We focus on the case n = 6 and show that the resulting conformal structure in seven dimensions is compatible with a conformal G 2 structure or its non-compact analogue. If a GL(2, R) structure arises on a moduli space of rational curves on a surface with self-intersection number 6, then certain components of the intrinsic torsion of the G 2 structure vanish. We give examples of simple seventh-order ordinary differential equations whose solution curves are rational and find the corresponding G 2 structures. In particular we show that Bryant's weak G 2 holonomy metric on the homology seven-sphere SO(5)/SO(3) is the unique weak G 2 metric arising from a rational curve.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 5
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    TRUNCATED SIMPLICIAL RESOLUTIONS AND SPACES OF RATIONAL MAPS (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: We show that whenever m ≤ n , the space of all continuous rational maps from CP m to CP n has the same homology as the space of all continuous maps between these spaces in dimensions smaller than d (2 n – 2 m + 1) – 1. This improves the result of the paper ‘Spaces of rational maps and the Stone–Weierstrass theorem’ ( Topology 45 (2006), 281–293).
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 6
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    INJECTIVE ENVELOPES AND LOCAL MULTIPLIER ALGEBRAS OF SOME SPATIAL CONTINUOUS TRACE C*-ALGEBRAS (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show herein to be a Kaplansky–Hilbert module over the abelian AW*-algebra C (). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A . In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A .
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 7
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    REAL INTERPOLATION WITH SYMMETRIC SPACES AND SLOWLY VARYING FUNCTIONS (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: We present a real interpolation method defined with slowly varying functions and rearrangement invariant spaces as parameters. We prove generalized Holmstedt-type formulas for this method that are applied to obtain reiteration theorems, including the limiting cases = 0 and = 1. To describe the spaces arising from reiteration theorems in the limiting cases, new interpolation constructions are introduced. Among the techniques used, we establish new Hardy-type inequalities for rearrangement invariant spaces which have independent interest.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 8
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    The Euler Obstruction and Bruce--Roberts' Milnor Number (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 9
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    CAUSTICS OF SURFACES IN THE MINKOWSKI 3-SPACE (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: The caustic of a smooth surface in the Euclidean 3-space is the envelope of the normal rays to the surface. It is also the locus of the centres of curvature (the focal points) of the surface. This is why it is also referred to as the focal set of the surface. It has Lagrangian singularities and its generic models are given in [V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps , Vol. I, Birkhäuser, Boston, 1986]. The aim of this paper is to define the caustic C ( M ) of a smooth surface M embedded in the Minkowski 3-space and to study its geometry. We denote by the locus of degeneracy (LD) the locus of points on M where the metric is degenerate. If M is a closed surface then its LD is not empty. At a point on the LD the ‘normal’ line to M is lightlike and is tangent to M . Also, the focal set of M is not defined at points on the LD. We define the caustic of M as the bifurcation set of the family of distance-squared functions on M . Then C ( M ) coincides with the focal set of M \ LD and provides an extension of the focal set to the LD. We study the local behaviour of the metric on C ( M ).
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 10
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    ON HIGHER RANK INSTANTONS AND THE MONOPOLE COBORDISM PROGRAM (2012)
    Oxford University Press
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    Publication Date: 2012-02-14
    Description: Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg–Witten invariants if the underlying four-manifold is of a simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mariño and Moore had already suggested that there should be a generalization of Witten's conjecture to this type of invariants. In this paper, a generalization of the classical cobordism program to the higher rank situation was studied and vanishing results that give evidence that the generalization of Witten's conjecture should hold were obtained.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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