Showing papers in "Transactions of the American Mathematical Society in 2006"
TL;DR: Cluster-tilted algebras as discussed by the authors are the endomorphism algesms of tilting objects in a cluster category, and their representation theory is very close to the representation theory of hereditary algesbras.
Abstract: We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.
390 citations
TL;DR: In this article, a co-fibrantly generated model category struc- ture on the category of small simplicial categories is presented, where weak equivalences are a simplicial analogue of the notion of equivalence of categories.
Abstract: In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
284 citations
TL;DR: In this paper, the concept of frequent hypercyclicity for bounded operators T on separable complex F-spaces was introduced, where T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, the set of integers n such that T n x belongs to U has positive lower density.
Abstract: We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T on separable complex F-spaces: T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, the set of integers n such that T n x belongs to U has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.
265 citations
TL;DR: In this paper, the enumeration of crossing and nestings for matchings and set partitions is studied using a bijection between partitions and vacillating tableaux, and it is shown that if the sets of minimal block elements and maximal block elements have a symmetric joint distribution, then the crossing number and the nesting number of partitions have a similar distribution.
Abstract: We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of k-noncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting). Mathematics Subject Classification. Primary 05A18, secondary 05E10, 05A15.
262 citations
TL;DR: In this paper, the concepts of boundary relations and the corresponding Weyl families are introduced, and fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : (H 2, J(H)) -> (H-2, J (H)), where Gamma need not be surjective and is even allowed to be multivalued.
Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
172 citations
TL;DR: In this article, it was shown that the 2D Navier Stokes Equations in a domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an asymptotically compact random dynamical system in the energy space H.
Abstract: We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS).We prove that for an AC RDS the Ω-limit set ΩB(ω) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a
domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an AC RDS in the energy space H. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.
151 citations
TL;DR: In this paper, the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices is investigated. And applications are given to the symmetry of some highly non-commutative Banach algesbras, to the analysis of twisted convolution and to the theory of localized frames.
Abstract: We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.
131 citations
TL;DR: In this article, the basic infinitesimal deformation theory of abelian categories is developed, which yields a natural generalization of the well-known Deformation Theory of algebras developed by Gerstenhaber.
Abstract: In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems.
131 citations
TL;DR: In this paper, it was shown that the uncentered maximal function M f is continuous and its derivative satisfies the sharp inequality ∥DMf∥ L 1(I) ≤ ∥Df ∥(I).
Abstract: We prove that if f: I ⊂ ℝ → ℝ is of bounded variation, then the uncentered maximal function M f is absolutely continuous, and its derivative satisfies the sharp inequality ∥DMf∥
L1(I) ≤ ∥Df∥(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives. © 2006 American Mathematical Society.
129 citations
TL;DR: In this article, a concentration inequality for the l n p norm on the L n p sphere for p,q > 0 was proved. But the concentration inequality was not used to study the distance between the cone measure and surface measure on the sphere.
Abstract: We prove a concentration inequality for the l n p norm on the l n p sphere for p,q > 0. This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between the cone measure and surface measure on the sphere of l n p . In particular, we obtain a significant strengthening of the inequality derived by Naor and Romik (2003), and calculate the precise dependence of the constants that appeared there on p.
123 citations
TL;DR: In this paper, it was shown that certain linear operators preserve the Polya frequency property and real-rootedness, and apply their results to settle some conjectures and open problems in combinatorics.
Abstract: We prove that certain linear operators preserve the Polya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bona, Brenti and Reiner-Welker.
TL;DR: In this article, the authors show that a normal complete surface with a finitely generated total coordinate ring is projective if and only if any two of its non-factorial singularities admit a common affine neighborhood.
Abstract: Given a variety X with a finitely generated total coordinate ring, we describe basic geometric properties of X in terms of certain combinatorial structures living in the divisor class group of X. For example, we describe the singularities, we calculate the ample cone, and we give simple Fano criteria. As we show by means of several examples, the results allow explicit computations. As immediate applications we obtain an effective version of the Kleiman-Chevalley quasiprojectivity criterion, and the following observation on surfaces: a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non-factorial singularities admit a common affine neighbourhood.
TL;DR: In this article, Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrodinger equation in the case when the inital data is equal to zero were studied.
Abstract: We study Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrodinger equation in the case when the inital data is equal to zero, proving some new estimates for certain exponents and giving counterexamples for some others.
TL;DR: In this article, a nonlinear semi-classical Schrodinger equation for which quadratic oscillations lead to focusing at one point, described by nonlinear scattering operator, is considered.
Abstract: We consider a nonlinear semi-classical Schrodinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian-Kammerer and I. Gallagher for $L^2$-supercritical power-like nonlinearities and more general initial data. The present results concern the $L^2$-critical case, in space dimensions 1 and 2; we describe the set of non-linearizable data, which is larger, due to the conformal invariance. As an application, we precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrodinger equation. The proof relies on linear and nonlinear profile decompositions.
TL;DR: In this paper, an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension has been proved for groups acting on finite dimensional metric spaces.
Abstract: We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.
TL;DR: In this paper, the authors define a random walk loop soup and show that it converges to the Brownian loop soup, making use of the strong approximation result of Komlos, Major, and Tusnady.
Abstract: The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlos, Major, and Tusnady. To make the paper self-contained, we include a proof of the approximation result that we need.
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TL;DR: In this paper, a new classification theorem for smooth embeddings in codimension 2 has been proposed, and the classification space is the rack space and the classifying bundle is the first James bundle.
Abstract: The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle.
We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from de. ning many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for pi(2) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.
Abstract: Let X t be the relativistic α-stable process in R d , a ∈ (0, 2), d > α, with infinitesimal generator H (α) 0 = -((-A + m 2/α ) α/2 - m). We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup T t for this process with generator H (α) 0 - V, V > 0, V locally bounded. We prove that if lim |x|→∞ V(x) = ∞, then for every t > 0 the operator T t is compact. We consider the class V of potentials V such that V > 0, lim |x|→∞ V(x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup T t is IU if and only if lim |x|→∞ V(x)/|x| = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction Φ 1 for T t . In particular, when V(x) = |x| β , β > 0, then the semigroup T t is IU if and only if β > 1. For β > 1 the first eigenfunction Φ 1 (x) is comparable to.
TL;DR: In this paper, the authors conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively.
Abstract: We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These Paley- Wiener theorems are used to extend Dunkl's intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan mo- tion group is established. It is shown how the algebra of radial parts of in- variant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coin- cide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be inter- preted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator. 1. Introduction and overview In recent times the study of special functions associated with root systems has developed to a considerable degree. Starting with a number of conjectures by Mac- donald, and the work of Heckman and Opdam on multivariable hypergeometric functions in the late 1980's, the development of the theory was greatly enhanced by the introduction of rational Dunkl operators by Dunkl (6). Through various intermediate steps of generalization, these operators can even be said to have ulti- mately provided crucial building blocks for Cherednik's work on double affine Hecke algebras and the q-Macdonald conjectures. Originally, before the introduction of Dunkl operators, the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on Lie groups as parameters, and then to develop a theory for Weyl group invariant objects in this more general situation, without the aid of the presence of the group. It was this point of view which underlay the Macdonald conjectures and which led Heckman and Opdam to the development of their theory of hypergeometric functions in higher dimension. One of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators. Apart from an explicit formula for the generalized radial part of the Laplacian—an expression which was in fact the starting point for Heckman and Opdam—the other operators remain somewhat intangible.
TL;DR: In this paper, it was shown that planar periodic Lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing, which is the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations.
Abstract: We show that superpolynomial decay of correlations (rapid mixing) is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations. The proof combines techniques of Dolgopyat and operator renewal theory. It follows from our results that planar periodic Lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing.
TL;DR: In this article, it was shown that every graph G is 2A(G)-matroidally colorable, where G is a simplicial complex of independent sets of a graph.
Abstract: A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a "covering" parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case r = 3 of a matroidal version of Ryser's conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter-the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on "independent systems of representatives" (which are the special case in which the matroid is a partition matroid). In particular, we define a notion of k-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph G is 2A(G)-matroidally colorable. The methods used are mostly topological.
TL;DR: In this paper, the authors studied the one-dimensional case of Northcott's Theorem for the counting function Open image in new window in higher dimensions and proved the existence of a positive constant C = C(d) such that the number of points α in OIN having degree d over the field of rational numbers is finite.
Abstract: Let ℚ denote the field of rational numbers, Open image in new window an algebraic closure of ℚ, and H : Open image in new window the absolute, multiplicative, Weil height. For each positive integer d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number Open image in new window of points α in Open image in new window having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function Open image in new window , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that
Open image in new window
(1)
and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that
Open image in new window
(2)
TL;DR: In this paper, the authors construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation, and show that the density converges towards the solution to the heat equation.
Abstract: We construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation. When the relaxation time tends to zero, we show that the density converges towards the solution to the heat equation.
TL;DR: In this paper, a representation theorem for weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms is given.
Abstract: We consider maps of the family of convex bodies in Euclidean d-dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms. The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.
TL;DR: In this article, a classification of the rank two p-local finite groups for odd p is given, and the analysis of possible saturated fusion systems in terms of the outer automorphism group of the possible F-radical subgroups is given.
Abstract: In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.
TL;DR: In this paper, the singularities of linear systems of P with assigned singularities were translated into a geometric one concerning the singularity of linear system of P. The singularity problem was then solved when the degree of the form f is greater than the number of variables.
Abstract: The Waring problem for homogeneous forms asks for additive decomposition of a form f into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper we answer this question when the degree of f is greater than the number of variables. To do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of P" with assigned singularities.
TL;DR: In this paper, the authors developed a geometric description of the geodesies in Randers spaces of constant curvature, and showed that these curves are given by composing Geodesies of the Riemannian metric with the flow generated by W. This claim is formalized by Theorem 2.
Abstract: Geodesies in Randers spaces of constant curvature are classified. Randers metrics have received much attention lately as solutions to Zermelo's problem of navigation; largely because this navigation structure provides the frame work for a complete classification of constant flag curvature Randers spaces. (Flag curvature is the Finslerian analog of Riemannian sectional curvature. See (BR04).) Briefly, a Randers metric is of constant flag curvature if and only if it solves Zer melo's problem of navigation on a Riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety W. See Subsection 1.1 for a sketch of the navigation problem, and Theorem 3 for an explicit statement of the classification result. The aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature. Intuitively, these paths minimize travel time across a Riemannian landscape under windy conditions. Presently we will show that these curves are given by composing geodesies of the Riemannian metric with the flow generated by W. This claim is formalized by Theorem 2. Geodesies on surfaces of constant, nonpositive curvature are illustrated in Section 3. We then turn, in Section 4, to the constant flag curvature K = 1 Randers metrics on Sn. The case of the sphere is especially interesting; it is possible to endow this closed manifold with a metric whose geodesies display distinctly non-Riemannian behaviors. For example: (1) A metric is projectively flat if every point admits coordinates in which the geodesies are straight lines. Belt r ami's theorem assures us that a Riemannian metric is of constant sectional curvature if and only if it is projectively flat. In contrast few Randers spaces of constant flag curvature are projectively flat. There are infinitely many nonisometric Randers metrics of constant, positive
TL;DR: In this paper, it was shown that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial P(z) of degree d = 4, if Δ m P m (z) = 0 for all m ≥ 1, then Δm P m+1 (z), p m + 1 (p m), pm = 0 when m > > 0.
Abstract: Let z = (z 1 ,···,z n ) and let A = Σ n i=1 ∂ 2 ∂z 2 i be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial P(z) of degree d = 4, if Δ m P m (z) = 0 for all m ≥ 1, then Δ m P m+1 (z) = 0 when m > > 0, or equivalently, Δ m P m+1 (z) = 0 when m > 3 2(3 n-2 -1). It is also shown in this paper that the condition Δ m P m (z) = 0 (m ≥ 1) above is equivalent to the condition that P(z) is Hessian nilpotent, i.e. the Hessian matrix Hes P(z) = (∂ 2 P ∂z i ∂z j ) is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.
TL;DR: In this article, a simple algebraic group over the complex numbers containing a Borel subgroup B is defined, and two conjectures where ideal exponents arise are proved. But these conjectures depend on the assumption that B-stable ideal I in the nilradical of the Lie algebra of B is known.
Abstract: Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1 , m 2 ,..., m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n , B n , C n and some other types. When / = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
TL;DR: In this article, the authors develop a dualization of the Fraisse limit construction from model theory and indicate its surprising connections with the pseudo-arc, and obtain Mioduszewski's theorem on surjective universality of the pseudoarc among chainable continua and a theorem on projective homogeneity of the pseudarc.
Abstract: The aim of the present work is to develop a dualization of the Fraisse limit construction from model theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Fraisse limits, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem on projective homogeneity of the pseudo-arc (which generalizes a result of Lewis and Smith on density of homeomorphisms of the pseudo-arc among surjective continuous maps from the pseudo-arc to itself). We also get a new characterization of the pseudo-arc via the projective homogeneity property.