Showing papers in "Israel Journal of Mathematics in 2009"

Slice monogenic functions

Abstract: Clifford algebras will be the setting in which we will work throughout this book. They were introduced under the name of geometric algebras by Clifford in 1878. Since then, several people have extensively studied them and nowadays there are, in the literature, several possible ways to introduce Clifford algebras: for example one can use exterior algebras, or present them as a quotient of a tensor algebra or by means of a universal property (see [23], [31], [34], or [75] for a survey on the various possible definitions). In this book, we will adopt an equivalent but more direct approach, using generators and relations.

The road coloring problem

Abstract: A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.

On the spectrum of the Dirichlet Laplacian in a narrow strip

Abstract: We consider the Dirichlet Laplacian Δ∈ in a family of bounded domains {−a < x < b, 0 < y < eh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in e → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ∈.

A class of Finsler metrics with isotropic S-curvature

Abstract: In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form We characterize these metrics with isotropic S-curvature

Dependent first order theories, continued

Abstract: A dependent theory is a (first order complete theory) T which does not have the independence property. A major result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them. We introduce “strongly dependent” and look at definable groups; and also at dividing, forking and relatives.

Perverse sheaves on affine flags and langlands dual group

Abstract: This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a simple algebraic group in terms of the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra ℍ; of the anti-spherical module for ℍ; and of the space of Iwahori-invariant Whittaker functions. As a byproduct we obtain some new properties of central sheaves introduced in [G].

Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions

Abstract: We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled nonlocal problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation.

Schmidt’s game on fractals

Abstract: We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some.

On sequentially Cohen-Macaulay complexes and posets

Abstract: The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra.

Bohr’s theorem for holomorphic mappings with values in homogeneous balls

Abstract: Let X, Y be complex Banach spaces. Let G be a bounded balanced domain in X and B Y be the unit ball in Y. Assume that B Y is homogeneous. Let f: G → B Y be a holomorphic mapping. In this paper, we show that, if P = f(0), then we have Σ =0 ∞ ‖ D φP (P)[D k f(0)(z k )]‖/(k!‖D φP (P)‖) < 1 for z ∈ (1/3)G, where φP ∈ AutB Y ) such that φP (P) = 0. Moreover, we show that the constant 1/3 is best possible, if B Y is the unit ball of a J*-algebra. The above result was proved by Liu and Wang in the case that G = B Y is one of the four classical domains in the sense of Hua. This result generalises a classical result of Bohr.

Categoricity, amalgamation, and tameness

Abstract: Theorem: For each 2 ≤ k < ω there is an $$ L_{\omega _1 ,\omega } $$ -sentence ϕk such that (1) ϕk is categorical in μ if μ≤ℵk−2; (2) ϕk is not ℵk−2-Galois stable (3) ϕk is not categorical in any μ with μ>ℵk−2; (4) ϕk has the disjoint amalgamation property (5) For k > 2 (a) ϕk is (ℵ0, ℵk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵk−3; (b) ϕk is ℵm-Galois stable for m ≤ k − 3 (c) ϕk is not (ℵk−3, ℵk−2).

On the hyperplane conjecture for random convex sets

Abstract: Let N ≥ n + 1, and denote by K the convex hull of N independent standard gaussian random vectors in ℝn. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes.

Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

Abstract: We obtain asymptotic formulae for the number of primes p ≤ x for which the reduction modulo p of the elliptic curve $$ E_{a,b} :Y^2 = X^3 + aX + b $$ satisfies certain “natural” properties, on average over integers a and b such that |a| ⩽ A and |b| ⩽ B, where A and B are small relative to x. More precisely, we investigate behavior with respect to the Sato-Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer m.

On solutions of the Ricci curvature equation and the Einstein equation

Abstract: We consider the pseudo-Euclidean space (Rn, g), with n ≥ 3 and gij = δij ei, ei = ±1, where at least one ei = 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij (xi, xj) depends on xi and xj. We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on Rn, on the n-dimensional torus Tn and on cylinders Tk×Rn-k, that solve the Ricci equation or the Einstein equation.

Existential questions in (relatively) hyperbolic groups

Abstract: We study the decidability of the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. Our tools are Rips and Sela’s canonical representatives for these groups, and solvability of equations with rational constraints (involving finite state automata) in free groups and free products.

Completely monotone sequences and universally prestarlike functions

Abstract: We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions.

Sch’nol’s theorem for strongly local forms

Abstract: We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ- or Kirchhoff boundary conditions.

On tensor categories attached to cells in affine Weyl groups, III

Abstract: We prove a weak version of Lusztig’s conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine) related to monodromic sheaves on the base affine space (both finite and affine), and explain its relation to Lusztig’s classification of character sheaves.

Classes of strictly singular operators and their products

Abstract: V. D. Milman proved in [20] that the product of two strictly singular operators on L p [0, 1] (1 ⩽ p < 1) or on C[0, 1] is compact. In this note we utilize Schreier families $$ \mathcal{S}_\xi $$ in order to define the class of $$ \mathcal{S}_\xi $$ -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of $$ \mathcal{S}_\xi $$ -hereditarily indecomposable Banach spaces and we examine the operators on them.

Simplicity of c ∗ -algebras associated to row-finite locally convex higher-rank graphs

Abstract: In a previous work, the authors showed that the C*-algebra C*(Λ) of a row-finite higher-rank graph Λ with no sources is simple if and only if Λ is both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing’s “removing sources” construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent.

Real characters and degrees

Abstract: Several classical theorems on character degrees are revisited from the point of view of the real characters.

Lipschitz structure of quasi-banach spaces

Abstract: We show that the Lipschitz structure of a separable quasi-Banach space does not determine, in general, its linear structure. Using the notion of the Arens-Eells p-space over a metric space for 0 < p ≤ 1 we construct examples of separable quasi-Banach spaces which are Lipschitz isomorphic but not linearly isomorphic.

On groups of central type, non-degenerate and bijective cohomology classes

Abstract: A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] ∈ H 2(G, ℂ*) (G acts trivially on ℂ*). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties. Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π ∈ Z 1(Q,Ǎ), where Ǎ = Hom(A, ℂ*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z 2(G, ℂ*), where G:= A × Q. Hence, the semidirect product G is of central type. In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class [π] ∈ H 1(Q,Ǎ) as above, we construct non-degenerate classes [cπ] ∈ H 2(G,ℂ*) for certain extensions 1 → A → G → Q → 1 which are not necessarily split. We thus strictly extend the above family of central type groups.

Brauer algebras of simply laced type

Abstract: The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A_(n − 1) on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A_(n − 1), D_n , E_6, E_7, E_8). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.

On contraction groups of automorphisms of totally disconnected locally compact groups

Abstract: We prove that recent results of Baumgartner and Willis on contraction groups of automorphisms of metrizable totally disconnected locally compact groups (Israel J. Math. 142 (2004), 221–248) remain true for non-metrizable groups.

Cotorsion pairs generated by modules of bounded projective dimension

Abstract: We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an ℵ0-noetherian ring Q of little finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and to Cohen Macaulay noetherian commutative rings. Our results allow us to give a positive answer to an open problem on the structure of divisible modules of projective dimension one over commutative domains posed in [23, Problem 6, p. 139]. We also give some insight on the structure of modules of finite weak dimension, giving a counterexample to [25, Open Problem 3, p. 187].

Continuous and random Vapnik-Chervonenkis classes

Abstract: Nous demontrons que si T est une theorie dependante, sa randomisee de keisler T R l’est aussi. Pour faire cela nous generalisons la notion d’une classe de Vapnik-Chervonenkis a des familles de fonctions a valeurs dans [0, 1] (dyune classe de Vapnik-Chervonenkis continue), et nous caracterisons les familles de fonctions ayant cette propriete par la vitesse de croissance de la largeur moyenne d’une famille de compacts convexes associes.

On the distribution of the residues of small multiplicative subgroups of \( \mathbb{F}_p \)

Abstract: Distributional properties of small multiplicative subgroups of \( \mathbb{F}_p \) are obtained. In particular, it is shown that if H < \( \mathbb{F}_p^* \) is of size larger than polylogarithmic in p, then, letting β < 1 be a fixed exponent, most elements of any coset aH (a ∈ \( \mathbb{F}_p^* \), arbitrary) will not fall into the interval [−p β, p β] ∈ \( \mathbb{F}_p \). The arguments are based on the theory of heights and results from additive combinatoric.

Cotorsion pairs associated with Auslander categories

Abstract: We prove that the Auslander category determined by a semidualizing module is the left half of a perfect cotorsion pair. We also prove that the Bass class determined by a semidualizing module is preenveloping.

Products of primes in conjugacy class sizes and irreducible character degrees

Abstract: We prove that if the product pq of distinct primes p and q divides the degree of some irreducible complex character of a finite group G, then pq divides the size of some conjugacy class of G.