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Acta Mathematica, 2008
ISSN:
0001-5962 (Print)
1871-2509 (Online)
In this year (11 articles)
Volume 203, Issue 1
[1] Super-potentials of positive closed currents, intersection theory and dynamics
Tien-Cuong Dinh Université Pierre et Marie Curie – Paris 6, Institut de Mathématiques de Jussieu Nessim Sibony Université Paris-Sud, Mathématique – Bâtiment 425
TBD mathscidoc:1701.332005
Acta Mathematica, 203, (1), 1-82, 2007.5
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We introduce a notion of super-potential for positive closed currents of bidegree ($p$,$p$) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.
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[2] Liouville theorems for the Navier–Stokes equations and applications
Gabriel Koch Department of Mathematics, University of Chicago Nikolai Nadirashvili LATP, CMI, CNRS, Université de Provence Gregory A. Seregin Mathematical Institute, Oxford University Vladimir Šverák University of Minnesota, 236 Vincent Hall, 206 Church St. SE, Minneapolis, MN, U.S.A.
TBD mathscidoc:1701.332006
Acta Mathematica, 203, (1), 83-105, 2007.10
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We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in$R$^{$n$}× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form$u$($x$,$t$) =$b$($t$), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].
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[3] La somme des chiffres des carrés
Christian Mauduit Institut de Mathématiques de Luminy CNRS-UMR 6206, 163 avenue de Luminy, Case 907, Marseille Cedex 9, France Joël Rivat Institut de Mathématiques de Luminy CNRS-UMR 6206, 163 avenue de Luminy, Case 907, Marseille Cedex 9, France
TBD mathscidoc:1701.332007
Acta Mathematica, 203, (1), 107-148, 2007.8
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Volume 203, Issue 2
[1] Quantum decay rates in chaotic scattering
Stéphane Nonnenmacher Institut de Physique Théorique, CNRS URA 2306, CEA-Saclay Maciej Zworski University of California, Berkeley, Department of Mathematics, Berkeley, CA, U.S.A.
TBD mathscidoc:1701.332008
Acta Mathematica, 203, (2), 149-233, 2007.8
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We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.
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[2] Rigidity of escaping dynamics for transcendental entire functions
Lasse Rempe Department of Mathematical Sciences, University of Liverpool
TBD mathscidoc:1701.332009
Acta Mathematica, 203, (2), 235-267, 2008.3
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We prove an analog of Böttcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $ \mathcal{B} $ . More precisely, let$f$and$g$be entire functions with bounded sets of singular values and suppose that$f$and$g$belong to the same parameter space (i.e., are$quasiconformally equivalent$in the sense of Eremenko and Lyubich). Then$f$and$g$are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane.
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[3] A Selberg integral for the Lie algebra A_{$n$}
S. Ole Warnaar School of Mathematics and Physics, The University of Queensland
TBD mathscidoc:1701.332010
Acta Mathematica, 203, (2), 269-304, 2007.9
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A new$q$-binomial theorem for Macdonald polynomials is employed to prove an A_{$n$}analogue of the celebrated Selberg integral. This confirms the $ \mathfrak{g} ={\rm{A}}_{n}$ case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra $ \mathfrak{g} $ .
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Volume 204, Issue 1
[1] Primes in tuples II
Daniel A. Goldston Department of Mathematics, San José State University János Pintz Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Cem Yalçin Yıldırım Department of Mathematics, Bog̃aziçi University
TBD mathscidoc:1701.332011
Acta Mathematica, 204, (1), 1-47, 2007.11
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We prove that $$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$ where$p$_{$n$}denotes the$n$th prime. Since on average$p$_{$n$+1}−$p$_{$n$}is asymptotically log_{$n$}, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences$p$−$p$′ between primes which includes the small gap result above.
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[2] Quantum cohomology of$G/P$and homology of affine Grassmannian
Thomas Lam Department of Mathematics, University of Michigan Mark Shimozono Department of Mathematics, Virginia Tech
TBD mathscidoc:1701.332012
Acta Mathematica, 204, (1), 49-90, 2008.2
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Let$G$be a simple and simply-connected complex algebraic group,$P$⊂$G$a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the homology$H$_{*}(Gr_{$G$}) of the affine Grassmannian Gr_{$G$}of$G$. As a consequence, all three-point genus-zero Gromov–Witten invariants of$G$/$P$are identified with homology Schubert structure constants of$H$_{*}(Gr_{$G$}), establishing the equivalence of the quantum and homology affine Schubert calculi.
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[3] The mean field traveling salesman and related problems
Johan Wästlund Department of Mathematical Sciences, Chalmers University of Technology
TBD mathscidoc:1701.332013
Acta Mathematica, 204, (1), 91-150, 2008.4
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The edges of a complete graph on$n$vertices are assigned i.i.d. random costs from a distribution for which the interval [0,$t$] has probability asymptotic to$t$as$t$→0 through positive values. In this so called pseudo-dimension 1 mean field model, we study several optimization problems, of which the traveling salesman is the best known. We prove that, as$n$→∞, the cost of the minimum traveling salesman tour converges in probability to a certain number, approximately 2.0415, which is characterized analytically.
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Volume 204, Issue 2
[1] Estimates for maximal functions associated with hypersurfaces in ℝ^{3}and related problems of harmonic analysis
Isroil A. Ikromov Department of Mathematics, Samarkand State University Michael Kempe Mathematisches Seminar, C.A.-Universität Kiel Detlef Müller Mathematisches Seminar, C.A.-Universität Kiel
TBD mathscidoc:1701.332014
Acta Mathematica, 204, (2), 151-271, 2008.2
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We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces$S$of finite type in 3-dimensional Euclidean space. For$p$> 2, we prove that if no affine tangent plane to$S$passes through the origin and$S$is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if$p$>$h$($S$), where$h$($S$) denotes the so-called height of the surface$S$(defined in terms of certain Newton diagrams). For non-analytic$S$we obtain the same statement with the exception of the exponent$p$=$h$($S$). Our notion of height$h$($S$) is closely related to A. N. Varchenko’s notion of height$h$($ϕ$) for functions$ϕ$such that$S$can be locally represented as the graph of$ϕ$after a rotation of coordinates.
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[2] Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane
Michael T. Lacey School of Mathematics, Georgia Institute of Technology Eric T. Sawyer Department of Mathematics & Statistics, McMaster University Ignacio Uriarte-Tuero Mathematics Department, University of Missouri
TBD mathscidoc:1701.332015
Acta Mathematica, 204, (2), 273-292, 2008.6
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Let $ E \subset \mathbb{C} $ be a compact set, $ g:\mathbb{C} \to \mathbb{C} $ be a$K$-quasiconformal map, and let 0 <$t$< 2. Let $ {\mathcal{H}^t} $ denote$t$-dimensional Hausdorff measure. Then $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$
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Impact Factor
3.719
Available
1882 - 2016
Volumes
228
Issues
301
Articles
2167