Showing papers in "Transactions of the American Mathematical Society in 2013"

Wang’s multiplicity result for superlinear (,)–equations without the Ambrosetti–Rabinowitz condition

Abstract: We consider a nonlinear elliptic equation driven by the sum of a p– Laplacian and a q–Laplacian where 1 < q ≤ 2 ≤ p < ∞ with a (p − 1)– superlinear Caratheodory reaction term which doesn’t satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at leat three nontrivial solutions; among them one is positive and one is negative.

Recovery of a source term or a speed with one measurement and applications

Abstract: We study the problem of recovery the source a.t;x/F.x/ in the wave equation in anisotropic medium with a known so that a.0;x/ 6D 0 with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation utt c .x/u D 0with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.

Quantum dimensions and quantum Galois theory

Abstract: The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator algebra modules, the Virasoro vertex operator algebra modules and the lattice vertex operator algebra modules are computed. A criterion for simple current modules of a rational vertex operator algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A full Galois theory for rational vertex operator algebras is established using the quantum dimensions.

Stable blow up dynamics for energy supercritical wave equations

Abstract: We study the semilinear wave equation $\displaystyle \partial _t^2 \psi -\Delta \psi =\vert\psi \vert^{p-1}\psi $ for $ p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $ t=T>0$ given by $\displaystyle \psi ^T(t,x)=c_p (T-t)^{-\frac {2}{p-1}}, $ where $ c_p$ is a suitable constant. We prove that the blow up described by $ \psi ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that leads to a solution which converges to $ \psi ^T$ as $ t\to T-$ in the backward lightcone of the blow up point $ (t,r)=(T,0)$.

Volume growth and stochastic completeness of graphs

Abstract: Given the variable-speed random walk on a weighted graph and a metric adapted to the structure of the random walk, we construct a Brownian motion on a closely related metric graph which behaves similarly to the VSRW and for which the associated intrinsic metric has certain desirable properties. Jump probabilities and moments of jump times for Brownian motion on metric graphs with varying edge lengths, jump conductances, and edge densities are computed. We use these results together with a theorem of Sturm for stochastic completeness, or non-explosiveness, on local Dirichlet spaces to prove sharp volume growth criteria in adapted metrics for stochastic completeness of graphs.

Regularity lemmas for stable graphs

Abstract: We develop a framework in which Szemeredi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemeredi theory. It was known that the "irregular pairs" in the statement of Szemeredi's regularity lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemeredi's regularity lemma for models of stable theories of graphs (i.e. graphs with the non-k�-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. In the other direction, we take a more model-theoretic approach, and give several new Szemeredi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the "indivisibility" condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function ofonly as in the usual Szemeredi regularity lemma.

Some results and open questions on spaceability in function spaces

Abstract: A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y subset of M boolean OR {0}. In this article we prove that, for every infinite dimensional closed subspace X of C[0, 1], the set of functions in X having infinitely many zeros in [0, 1] is spaceable in X. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0, 1] or Muntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0, 1], as well as oscillating and annulling properties of subspaces of C[0, 1].

Geometric grid classes of permutations

Abstract: A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.

A uniform bijection between nonnesting and noncrossing partitions

Abstract: In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner.

String connections and Chern-Simons theory

Abstract: We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connection and that the possible choices form an affine space. Further we discover a new relation between string connections, 3-forms on the base manifold, and degree three differential cohomology. We also discuss in detail the relation between our formulation of string connections and the original version of Stolz and Teichner.

Kumjian-Pask algebras of higher-rank graphs

Abstract: We introduce higher-rank analogues of the Leavitt path algebras, which we call the Kumjian-Pask algebras. We prove graded and Cuntz-Krieger uniqueness theorems for these algebras, and analyze their ideal structure.

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series

Abstract: In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences (\{1\}^a,c,\{1\}^b), (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p. The congruences obtained allow us to find nice p-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p. Moreover, we are also able to provide a new proof of Zagier's formula for \zeta^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.

Paramodular abelian varieties of odd conductor

Abstract: A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on ell-division fields of semistable abelian varieties A, mainly when A[ell] is reducible, by considering extension problems for groups schemes of small rank. Our general results imply, for instance, that the least prime conductor of an abelian surface is 277.

The probability distributions of the first hitting times of Bessel processes

Abstract: We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover all the cases.

Divergence of the mock and scrambled Fourier series on fractal measures

Abstract: We study divergence properties of the Fourier series on Cantortype fractal measures, also called the mock Fourier series. We show that in some cases the L1-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, with affine structure, which we call a scrambled Fourier series, have a corresponding Dirichlet kernel whose L1norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence properties are related to the Mahler measure of certain polynomials and to spectral properties of Ruelle operators.

Additivity of the dp-rank

Abstract: The main result is the prove of the linearity of the dp-rank. We also prove that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types and discuss the possible relations between dp-rank and VC-density.

Resolvent at low energy III: The spectral measure

Abstract: The second and third authors were supported by Australian Research Council Discovery grants DP0771826 and DP1095448 and the second author by a Future Fellowship. The first author was partially supported by ANR grant ANR-09-JCJC-0099-01 and by the PICS-CNRS Progress in Geometric Analysis and Applications, and thanks the math department of ANU for its hospitality.

Inverse boundary value problems for the perturbed polyharmonic operator

Abstract: Author(s): Krupchyk, K; Lassas, M; Uhlmann, G | Abstract: We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (-Δ)m, m ≥ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in ℝn, n ≥ 3. Notice that the corresponding result does not hold in general when m = 1. © 2013 American Mathematical Society.

Perelman’s entropy and Kähler-Ricci flow on a Fano manifold

Abstract: In this paper, we extend the method in [TZhu5] to study the energy level $L(\\cdot)$ of Perelman's entropy $\\lambda(\\cdot)$ for K\\\"ahler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $\\lambda(\\cdot)$ in K\\\"ahler class $2\\pi c_1(M)$ under an assumption that the modified Mabuchi's K-energy $\\mu(\\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of K\\\"ahler-Ricci flow in [TZhu3].

Randomness for non-computable measures

Abstract: Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probability measure, we turn our attention to Levin’s neutral measures, for which all sequences are random. We show that every PA degree computes a neutral measure. We also show that a neutral measure has no least Turing degree representation and explain why the framework of the continuous degrees (a substructure of the enumeration degrees studied by Miller) can be used to determine the computational complexity of neutral measures. This allows us to show that the Turing ideals below neutral measures are exactly the Scott ideals. Since X ∈ 2 is an atom of a neutral measure μ if and only if it is computable from (every representation of) μ, we have a complete understanding of the possible sets of atoms of a neutral measure. One simple consequence is that every neutral measure has a Martin-Lof random atom. 1. Defining randomness Let X be an element of Cantor space and μ a Borel probability measure on Cantor space. What should it mean for X to be random with respect to μ? In the case that μ is the Lebesgue measure, then the theory of μrandomness is well developed (for recent treatises on the subject the reader is referred to Downey and Hirschfeldt, and Nies [2, 13]). In fact if μ is a computable measure, then early work of Levin showed that μ-randomness can be seen as essentially a variant on randomness for Lebesgue measure [10]. This leaves the question of how to define randomness if μ is non-computable. We will show that the two approaches that have previously been used to define μ-randomness, for non-computable μ, are equivalent. Later, in Theorem 4.12, we will provide another characterization of μ-randomness using the enumeration degrees. Last compilation: September 8, 2011 Last time the following date was changed: November 22, 2010. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 03D30. The second author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.

Tate (co)homology via pinched complexes

Abstract: For complexes of modules we study two new constructions, which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology and Tate cohomology, which lead to conceptual proofs of balancedness of Tate (co)homology for modules over associative rings. Another application we consider is in local algebra. Under conditions of vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

Abstract: In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.

Gaussian subordination for the Beurling-Selberg extremal problem

Abstract: We determine extremal entire functions for the problem of ma- jorizing, minorizing, and approximating the Gaussian function e x 2 by en- tire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance (3), (4), (10) and (17)), plus a variety of new interesting functions such asjxj for 1 < ; log (x 2 + 2 )=(x 2 + 2 ) , for 0 < ; log x2 + 2 ; andx2n logx2 , forn2 N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomi- als and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

Collapsing of products along the Kähler-Ricci flow

Abstract: Let $X = M \\times E$ where $M$ is an $m$-dimensional K\\\"ahler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^\\infty$ convergence of the normalized K\\\"ahler-Ricci flow on $X$ to a K\\\"ahler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture.

The Bishop-Phelps-Bollobás Theorem for bilinear forms

Abstract: In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × Y is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space C(K) of continuous functions on a compact Hausdorff topological space K and the space K(H) of compact operators on a Hilbert space H. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × L1(μ) fails for any infinite-dimensional L1(μ), a result that was known only when L1(μ) = 1.