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  • Articles  (228)
  • 2010-2014  (228)
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  • Articles  (228)
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  • 11
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    SMOOTH MODULI SPACES OF ASSOCIATIVE SUBMANIFOLDS (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: Let M 7 be a smooth manifold equipped with a G 2 -structure , and Y 3 be a closed compact -associative submanifold. McLean [Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747] proved that the moduli space M Y , of the -associative deformations of Y has vanishing virtual dimension. In this paper, we perturb into a G 2 -structure in order to ensure the smoothness of M Y , near Y . If Y is allowed to have a boundary moving in a fixed coassociative submanifold X , it was proved in Gayet and Witt [Deformations of associative submanifolds with boundary, Adv. Math. 226 (2011), 2351–2370] that the moduli space M Y , X of the associative deformations of Y with boundary in X has finite virtual dimension. We show here that a generic perturbation of the boundary condition X into X ' gives the smoothness of M Y , X ' . In another direction, we use Bochner's technique to prove a vanishing theorem that forces M Y or M Y , X to be smooth near Y . For every case, some explicit families of examples will be given.
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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  • 12
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    BIFURCATION OF CLIFFORD TORI IN BERGER 3-SPHERES (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: We study the intrinsic and the extrinsic geometry of Clifford tori in Berger spheres , 〉0, and use these results to prove several local rigidity and bifurcation results in this context.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 13
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    DIFFERENTIAL INCLUSIONS, NON-ABSOLUTELY CONVERGENT INTEGRALS AND THE FIRST THEOREM OF COMPLEX ANALYSIS (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: In the theory of complex-valued functions of a complex variable, arguably the first striking theorem is that pointwise differentiability implies C regularity. As mentioned in Ahlfors [ Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable , 3rd edn, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978], there have been a number of studies [Porcelli and Connell, A proof of the power series expansion without Cauchy's formula, Bull. Amer. Math. Soc. 67 (1961), 177–181; Plunkett, A topological proof of the continuity of the derivative of a function of a complex variable, Bull. Amer. Math. Soc. 65 (1959), 1–4] proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note, we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration, and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 14
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    AN OUTER ADMISSIBLE ACTION OF A COMMUTATIVE LOCALLY COMPACT GROUP ON THE GENERIC DYNAMICS FACTOR (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: The generic dynamics factor A is a small, monotone complete, hyperfinite wild factor of Type III obtained as a quotient of the Borel *-envelope of the Fermion algebra. In our earlier work with Wright, we constructed infinitely many outer actions of any Polish group equipped with a dense normal subgroup on A. These results are algebraic and in another work with Wright, we constructed an outer admissible action of the additive group R of real numbers and the torus T on A. Let G be any locally compact second countable commutative Hausdorff group. In this paper, we shall show that there exists an appropriate parameterization, for A, that makes it possible to generalize the results of the works above, and to construct an admissible outer action of G on A. When an action α of G on A is admissible, we can define the small monotone cross product A x α G and the dual action of Ĝ in such a way that , the Takesaki's duality principle holds true and this is a new parameterization of the algebra A.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 15
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    IS THE MISSING AXIOM OF MATROID THEORY LOST FOREVER? (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic (MSOL) for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in MSOL and show that it is possible to finitely axiomatize matroids using only sentences in this collection. Moreover, we can also axiomatize representability over any fixed finite field (assuming Rota's conjecture holds). We prove that it is not possible to finitely axiomatize representability, or representability over any fixed infinite field, using sentences from the collection.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 16
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    SLICING CORANK 1 MAP GERMS FROM C2 TO C3 (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: The classification and the geometry of corank 1 map germs f :(C 2 , 0)-〉(C 3 , 0) have been studied by David Mond. Normal forms of such maps f ( x , y )=( x , p ( x , y ), q ( x , y )) suggest, at least in some cases, that they could be seen as 1-parameter unfoldings of the plane curves x ( y )=( p ( x , y ), q ( x , y )). If a certain genericity condition is satisfied, then the transverse slice curve 0 contains information on the geometry of f . We introduce invariants C , J , T related to the Reidemeister moves (codimension 1 transitions) that appear in a stable perturbation of 0 . We compare them with Mond's invariants of f and obtain interesting geometric results. For instance, f is finitely determined if and only if C , J , T 〈 and given a 1-parameter family f t , it is Whitney equisingular if and only if the three invariants are independent of t .
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 17
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    CONSTRUCTING CO-HIGGS BUNDLES ON CP2 (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP 1 x CP 1 -〉CP 2 , is naturally a co-Higgs bundle, with the twisted endomorphism, or ‘Higgs field’, also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an eight-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with non-zero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger.
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    Electronic ISSN: 1464-3847
    Topics: Mathematics
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  • 18
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    ERROR TERM IMPROVEMENTS FOR VAN DER CORPUT TRANSFORMS (2014)
    Oxford University Press
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    Publication Date: 2014-11-20
    Description: We improve the error term in the van der Corput transform for exponential sums, * a ≤ n ≤ b g ( n ) exp(2 i f ( n )). For many functions g and f , we can extract a simple explicit main term for the error. In addition, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [ a , b ] with better estimates than previous results. As a result of these refinements, we are able to provide an improvement to the classical Kusmin–Landau inequality.
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    Topics: Mathematics
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  • 19
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    EINSTEIN MANIFOLDS WITH SKEW TORSION (2014)
    Oxford University Press
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    Publication Date: 2014-08-27
    Description: This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew-symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar -curvature, it is necessarily compact and it has finite first fundamental group 1 . The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G 2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein–Sasaki manifold and any seven-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.
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    Topics: Mathematics
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  • 20
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    CLASSIFICATION OF CONGRUENCES FOR MOCK THETA FUNCTIONS AND WEAKLY HOLOMORPHIC MODULAR FORMS (2014)
    Oxford University Press
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    Publication Date: 2014-08-27
    Description: Let f ( q ) denote Ramanujan's mock theta function It is known that there are many linear congruences for the coefficients of f ( q ) and other mock theta functions. We prove that if the linear congruence a ( mn + t ) 0 (mod ) holds for some prime ≥ 5, then | m and ((24 t – 1)/) != (–1/). We prove analogous results for the mock theta function ( q ) and for a large class of weakly holomorphic modular forms which includes -quotients. This extends work of Radu, in which he proves a conjecture of Ahlgren and Ono for the partition function p ( n ).
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    Topics: Mathematics
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