Mathematics as science

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: −∆p u = uq + µ, Fk [−u] = uq + µ, u ≥ 0, on Rn , or on a bounded domain Ω ⊂ Rn . Here ∆p is the p-Laplacian defined by ∆p u = div ( u| u|p−2 ), and Fk [u] is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D2 u (k = 1, 2, . . . ,...

58p dontetvui 17-01-2013 58 10   Download

Bowen’s formula relates the Hausdorff dimension of a conformal repeller to the zero of a ‘pressure’ function. We present an elementary, self-contained proof to show that Bowen’s formula holds for C 1 conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class.

55p dontetvui 17-01-2013 55 7   Download

For an -adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class. Let F be a perfect field and U be a separated and smooth scheme of finite type purely of dimension d over F . In this paper, we study ramification of a finite ´tale scheme V over U along the boundary, by introducing a map (0.1)...

65p dontetvui 17-01-2013 34 7   Download

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Amp`re equation when the inhomogee neous term is only assumed to be H¨lder continuous. As a consequence of our o approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.

38p dontetvui 17-01-2013 48 8   Download

The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. ...

20p dontetvui 17-01-2013 49 7   Download

We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber.

48p dontetvui 17-01-2013 35 6   Download

We show that every subset of SL2 (Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL2 (Z/pZ), every element of SL2 (Z/pZ) can be expressed as a product of at most O((log p)c ) elements of A ∪ A−1 , where c and the implied constant are absolute. 1. Introduction 1.1. Background. Let G be a finite group. Let A ⊂ G be a set of generators of G. By definition, every g ∈ G can be expressed as a product of elements of...

24p dontetvui 17-01-2013 44 7   Download

Let p 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincar´ inequality. Then there exists e ε 0 such that (X, d, μ) admits a (1, q)-Poincar´ inequality for every q p−ε, e quantitatively. 1. Introduction Metric spaces of homogeneous type, introduced by Coifman and Weiss [7], [8], have become a standard setting for harmonic analysis related to singular integrals and Hardy spaces. Such metric spaces are often referred to as a metric measure space with a doubling measure. An advantage of working...

26p dontetvui 17-01-2013 54 6   Download

This paper is about a class of strange attractors that have the dual property of occurring naturally and being amenable to analysis. Roughly speaking, a rank one attractor is an attractor that has some instability in one direction and strong contraction in m−1 directions, m here being the dimension of the phase space. The results of this paper can be summarized as follows. Among all maps with rank one attractors, we identify, inductively, subsets Gn, n = 1, 2, 3, · · · ,

133p dontetvui 17-01-2013 46 6   Download

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces (i.e., surfaces without self-intersection). In fact, we will prove considerably more. The heart of our argument is very general and should apply to a variety of situations, as will be more apparent once we describe the main steps of the proof later in the introduction.

34p dontetvui 17-01-2013 49 9   Download

A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. ...

117p dontetvui 17-01-2013 41 7   Download

We prove analogues for hypergraphs of Szemer´di’s regularity lemma and e the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemer´di theorem of Furstenberg e and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, R¨dl, Schacht and Skokan. o 1.

51p noel_noel 17-01-2013 68 7   Download

Introduction and statement of results Let E ⊂ Rn , and m ≥ 1. We write C m (E) for the Banach space of all real-valued functions ϕ on E such that ϕ = F on E for some F ∈ C m (Rn ). The natural norm on C m (E) is given by ϕ C m (E) = inf{ F C m (Rn ) : F ∈ C m (Rn ) and F = ϕ on E} . Here, as usual, C m (Rn ) is the space of real-valued functions on Rn with continuous and bounded derivatives through order m; and F C m (Rn ) = |β|≤m x∈Rn max sup |∂ β F (x)| .

58p noel_noel 17-01-2013 53 6   Download

The purpose of this paper is to prove that the stable homotopy category of algebraic topology is ‘rigid’ in the sense that it admits essentially only one model: Rigidity Theorem. Let C be a stable model category. If the homotopy category of C and the homotopy category of spectra are equivalent as triangulated categories, then there exists a Quillen equivalence between C and the model category of spectra. Our reference model is the category of spectra in the sense of Bousfield and Friedlander [BF, §2] with the stable model structure. The point of the rigidity theorem is that its...

28p noel_noel 17-01-2013 52 6   Download

We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst C r Axiom A flows, r ≥ 2, we show that there is a C 2 -open, C r -dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A flows that holds in a C r , as opposed to H¨lder, topology. o

24p noel_noel 17-01-2013 53 7   Download

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = Ar . “Twisted” Dirichet series are considered, which contain the series of [4] as a special case.

25p noel_noel 17-01-2013 65 7   Download

We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero. ...

31p noel_noel 17-01-2013 33 6   Download

In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come ˇ in pairs.

50p noel_noel 17-01-2013 46 6   Download

We prove the Bers density conjecture for singly degenerate Kleinian surface groups without parabolics. 1. Introduction In this paper we address a conjecture of Bers about singly degenerate Kleinian groups. These are discrete subgroups of PSL2 C that exhibit some unusual behavior: • As groups of projective transformations of the Riemann sphere C they act properly discontinuously on a topological disk whose closure is all of C. • As groups of hyperbolic isometries their action on H3 is not convex cocompact. ...

18p noel_noel 17-01-2013 41 5   Download

To any two graphs G and H one can associate a cell complex Hom (G, H) by taking all graph multihomomorphisms from G to H as cells. In this paper we prove the Lov´sz conjecture which states that a if Hom (C2r+1 , G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r +1 vertices. The proof requires analysis of the complexes Hom (C2r+1 , Kn ). For even n, the obstructions to graph colorings are provided by the presence of torsion...

44p noel_noel 17-01-2013 58 8   Download