Showing papers in "Journal of The London Mathematical Society-second Series in 2011"

Abelian complexity of minimal subshifts

Abstract: In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerden's theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words, we prove something stronger: for every Sturmian word ω and positive integer k, each sufficiently long factor of ω begins with an Abelian k-power.

Hyperplane sections in arithmetic hyperbolic manifolds

Abstract: In this paper, we prove that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homology group of the

Random soups, carpets and fractal dimensions

Abstract: We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the “conformal loop ensembles” defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.

The number of Ks,t‐free graphs

Abstract: Denote by fn(H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdős conjectured that whenever H contains a cycle, fn(H) = 2 (1+o(1)) ex(n,H), yet it is still open for every bipartite graph, and even the order of magnitude of log2 fn(H) was known only for C4, C6, and K3,3. We show that for all s and t satisfying 2 ≤ s ≤ t, fn(Ks,t) = 2 O(n), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Turan number ex(n,Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K2,t-free graphs of order n have at least 1/12 ·ex(n,K2,t) and at most (1−c) ex(n,K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramseyand Turan-type problems.

On the Brauer group of diagonal quartic surfaces

Abstract: We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Mizukami.

Classification of abelian complex structures on 6-dimensional Lie algebras

Abstract: We classify the 6-dimensional Lie algebras that can be endowed with an abelian com- plex structure and parameterize, on each of these algebras, the space of such structures up to holo- morphic isomorphism.

Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations

Abstract: Positive solutions of systems of Hammerstein integral equations are studied by using the theory of the fixed-point index for compact maps defined on cones in Banach spaces. Criteria for the fixed-point index of the Hammerstein integral operators being 1 or 0 are given. These criteria are generalizations of previous results on a single Hammerstein integral operator. Some of criteria are new and involve the first eigenvalues of the corresponding systems of linear Hammerstein operators. The existence and estimates of the first eigenvalues are given. Applications are given to systems of fractional differential equations with two-point boundary conditions. The Green’s functions of the boundary value problems are derived and their useful properties are provided. As illustrations, the existence of nonzero positive solutions of two specific such boundary value problems is studied.

Heat kernel estimates for Δ+Δα/2 in C1, 1 open sets

Abstract: We consider a family of pseudo differential operators { Δ+ a α Δ α/2 ; a ∈ (0, 1)} on R d for every d 1 that evolves continuously from Δ to Δ + Δ α/2 ,w hereα ∈ (0, 2). It gives rise to a family of Levy processes {X a ,a ∈ (0, 1)} in R d ,w hereX a is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + a α Δ α/2 with zero exterior condition in a family of open subsets, including bounded C 1,1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a ∈ (0, 1) so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of X a in bounded C 1,1 open sets in R d , which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for Δ + Δ α/2 in C 1,1 open sets and their applications', Illinois J. Math., to appear) using a completely different approach.

Intersection patterns of curves

Abstract: Keywords: Ramsey-Type Theorems ; Crossing Patterns ; Graphs ; Orders ; Sets Note: Professor Pach's number: [215] Reference DCG-ARTICLE-2008-022doi:10.1112/jlms/jdq087View record in Web of Science Record created on 2008-11-18, modified on 2017-05-12

Generalized contact structures

Abstract: We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the three-dimensional Heisenberg group and its co-compact quotients.

Algebro-geometric aspects of Heine–Stieltjes theory

Abstract: The goal of the paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk i=1 Qi ...

The sum-of-digits function of polynomial sequences

Abstract: Let q≥2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the re-partition of the sequence (sq(P(n)))n∈ in arithmetic progressions when P∈[XS is such that P()⊂. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence ( sq(P(n)))n∈ for ∈ , provided that q is a large enough prime number co-prime with the leading coefficient of P.

Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains

Abstract: We characterize, using the Bergman kernel, Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C n , generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing

Proper actions and proper invariant metrics

Abstract: We show that if a (locally compact) group $G$ acts properly on a locallycompact $\sigma$-compact space $X$ then there is a family of $G$-invariantproper continuous finite-valued pseudometrics which induces the topology of$X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and onlyif there exists a $G$-invariant proper compatible metric on $X$.

Centroids and the rapid decay property in mapping class groups

Abstract: We study a notion of an equivariant, Lipschitz, permutation- invariant centroid for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).

Inverse systems, Gelfand–Tsetlin patterns and the weak Lefschetz property

Abstract: Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of fat points and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [J. Eur. Math. Soc. 2010, arXiv:0803.0892] allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Mir\'o-Roig-Nagel [arXiv:1008.2149] for related results.

The algebra of integro‐differential operators on a polynomial algebra

Abstract: We prove that the algebra $\mI_n:=K\langle x_1, ..., x_n, \frac{\der}{\der x_1},...,\frac{\der}{\der x_n}, \int_1, ..., \int_n\rangle $ of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension $n$ and of Gelfand-Kirillov dimension $2n$. Its weak homological dimension is $n$, and $n\leq \gldim (\mI_n)\leq 2n$. All the ideals of $\mI_n$ are found explicitly, there are only finitely many of them ($\leq 2^{2^n}$), they commute ($\ga \gb = \gb\ga$) and are idempotent ideals ($\ga^2= \ga$). The number of ideals of $\mI_n$ is equal to the {\em Dedekind number} $\gd_n$. An analogue of Hilbert's Syzygy Theorem is proved for $\mI_n$. The group of units of the algebra $\mI_n$ is described (it is a huge group). A canonical form is found for each integro-differential operators (by proving that the algebra $\mI_n$ is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra $\mA_n$ (but $\GK (\mA_n) =3n$, note that $\mI_n\subset \mA_n$). It is proved that the algebras $\mI_n$ and $\mA_n$ are ideal equivalent.

Spectral points of definite type and type π for linear operators and relations in Krein spaces

Abstract: Spectral points of type …+ and type …i for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. It turns out that these spectral points are stable under compact perturbations and perturbations small in the gap metric.

Multipliers of locally compact quantum groups via Hilbert C*-modules

Abstract: A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $\alpha,\beta:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (\beta(t)|\alpha(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to a proof that the (unbounded) antipode acts on the space of completely bounded multipliers, in a way which interacts naturally with our representation result. The dual of the universal quantum group (in the sense of Kustermans) can be identified with a subalgebra of the completely bounded multipliers, and we show how this fits into our framework. Finally, this motivates a certain way to deal with two-sided multipliers.

Stable polynomial division and essential normality of graded Hilbert modules

Abstract: The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the stable decomposition $h = \sum a_i f_i$ may be obtained by carefully applying techniques from computational algebra. We show that when the algebra of polynomials in $d$ variables is given the natural $\ell^1$ norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that a module $M$ that has the stable division property (with respect to the appropriate norm) is $p$-essentially normal for $p > \dim(M)$, as conjectured by Douglas. This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal. Finally, we reduce the problem of determining whether all graded submodules of the $d$-shift Hilbert module are essentially normal, to the problem of determining whether all ideals generated by quadratic scalar valued polynomials are essentially normal.

Degrees of irreducible morphisms and finite-representation type

Abstract: We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of $n$ irreducible morphisms between indecomposable modules lies in the $n+1$-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path.

Clifford theory for tensor categories

Abstract: A graded tensor category over a group G will be called a strongly G-graded tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the module categories over a strongly G-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of G.

On generalizing Lutz twists

Abstract: We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that $R^{2n+1}$ has at least three distinct contact structures. This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end.

A new Garside structure for the braid groups of type (e, e, r)

Abstract: We describe a new presentation for the complex reflection groups of type~$(e,e,r)$ and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation which is shown to give rise to a Garside structure. A detailed study of the combinatorics of this structure leads us to describe it as~\emph{post-classical}.

On the distribution of angles between the N shortest vectors in a random lattice

Abstract: We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n→∞. Moreover, we interpret the result in terms of eig ...

Hardy type spaces on certain noncompact manifolds and applications

Abstract: In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M),X^2(M), . . . of new Hardy spaces onM, the sequence Y^1(M), Y^2(M), . . . of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace-Beltrami operator L on M. Under an additional geometric condition, we prove also an endpoint result for the first-order Riesz transform ∇L^{−1/2}. In this case, the kernels of the operators L^iu and ∇L^{−1/2} are singular both on the diagonal and at infinity. In particular, these results apply to Riemannian symmetric spaces of the noncompact type