scispace - formally typeset
Search or ask a question

Showing papers in "Transactions of the American Mathematical Society in 2005"


Journal ArticleDOI
TL;DR: In this article, the denominator theorem of Fomin and Zelevinsky was generalized to any cluster algebra and an algebraic realization and a geometric realization of Cat_C were given.
Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.

499 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that a bounded linear operator T on a complex Hilbert space H is complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H).
Abstract: A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ\T|, where J is an auxiliary conjugation commuting with |T| = √T*T. We consider numerous examples, including the Poincare-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ\T| also extends to the class of unbounded C-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

395 citations


Journal ArticleDOI
TL;DR: Weighted anisotropic Triebel-Lizorkin spaces were introduced and studied with the use of discrete wavelet transforms in this article, where the authors extended the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights.
Abstract: Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the φ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

186 citations


Journal ArticleDOI
TL;DR: Some fundamental analytic properties of the Lengyel-Epstein system are reported, indicating that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states.
Abstract: The first experimental evidence of the Turing pattern was observed by De Kepper and her associates (1990) on the CIMA reaction in an open unstirred gel reactor, almost 40 years after Turing's prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. In this paper we report some fundamental analytic properties of the Lengyel-Epstein system. Our result also indicates that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states. A priori estimates are fundamental to our approach for this nonexistence result. The degree theory was combined with the a priori estimates to derive existence of nonconstant steady states.

156 citations


Journal ArticleDOI
TL;DR: In this paper, Forman's discrete Morse theory is studied from an algebraic viewpoint, and it is shown how this theory can be extended to chain complexes of modules over arbitrary rings.
Abstract: Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.

122 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the problem of determining a bound on the genus of a knot in a 3-manifold is NP-complete, and that determining whether a curve in a metrized PL 3manifolds bounds a surface of area less than a given constant C is also NP-hard.
Abstract: We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area less than a given constant C is NP-hard.

117 citations


Journal ArticleDOI
TL;DR: The invariant of tangle cobordisms is a homomorphism between complexes of bimodules assigned to boundaries of the cobordism, defined up to chain homotopy equivalence.
Abstract: We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the' cobordism.

115 citations


Journal ArticleDOI
Oana Veliche1
TL;DR: In this article, a Tate cohomology theory for complex of left modules over associative rings has been proposed, which is based on the notion of Gorenstein projective dimension (GPD).
Abstract: We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

113 citations


Journal ArticleDOI
TL;DR: In this article, an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, is obtained.
Abstract: Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.

96 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the global well-posedness of the differential equation u tt - Δu + |u| k ∂j(u t ) = |u | p-1 u in Ω x (0,T), where ∂ j is a sub-differential of a continuous convex function j. Under some conditions on j and the parameters in the equations, they obtained several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity.
Abstract: In this article we focus on the global well-posedness of the differential equation u tt - Δu + |u| k ∂j(u t ) = |u| p-1 u in Ω x (0,T), where ∂j is a sub-differential of a continuous convex function j. Under some conditions on j and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k + m, and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that. u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), u 1 ∈ H 1 0 (Ω) are indeed strong solutions.

94 citations


Journal ArticleDOI
TL;DR: Recently, the sharp L 2 -bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao as mentioned in this paper, and they generalize those bilinear restriction estimates to surfaces with curvatures of different signs.
Abstract: Recently, the sharp L 2 -bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of properly embedding H-surfaces in Riemannian three-dimensional three-manifolds of the form M 2 x R, where M 2 is a complete Riemanian surface.
Abstract: The subject of this paper is properly embedded H-surfaces in Riemannian three manifolds of the form M 2 x R, where M 2 is a complete Riemannian surface. When M 2 = R 2 , we are in the classical domain of H-surfaces in R 3 . In general, we will make some assumptions about M 2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 on the behavior of H-surfaces in M 2 x R.

Journal ArticleDOI
TL;DR: In this article, the microlocal properties of partial differential operators with generalized functions as coefficients were investigated and an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols was presented.
Abstract: We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

Journal ArticleDOI
TL;DR: In this article, it was shown that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering.
Abstract: We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.

Journal ArticleDOI
TL;DR: In this article, it was shown that for all p ≠ 1/2, the mixing time of this shuffling is O(N^2), as conjectured by Diaconis and Ram (2000).
Abstract: Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p ≠ 1/2, the mixing time of this card shuffling is O(N^2), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

Journal ArticleDOI
TL;DR: In this paper, the authors derived the Weyl exponent for the spectral asymptotics of the Laplacians of the harmonic calculus for finite atomless measures with compact support on the real line.
Abstract: Differentiation of functions wrt finite atomless measures with compact support on the real line is introduced The related harmonic calculus is similar to that of the classical Lebesgue case As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians wrt linear Cantor-type measures with arbitrary weights

Journal ArticleDOI
TL;DR: In this article, it was shown that compact, simply connected homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics, and that these spaces admit simply connected Euclidean spaces as well.
Abstract: We show that compact, simply connected homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics.

Journal ArticleDOI
TL;DR: In this article, a lower bound for the Bergman distance in smooth pseudoconvex domains due to Diederich and Ohsawa was improved using the pluricomplex Green function and an L 2 -estimate for the ∂ operator of Donnelly and Fefferman.
Abstract: We improve a lower bound for the Bergman distance in smooth pseudoconvex domains due to Diederich and Ohsawa. As the main tool we use the pluricomplex Green function and an L 2 -estimate for the ∂-operator of Donnelly and Fefferman.

Journal ArticleDOI
TL;DR: In this paper, the authors extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002) to the setting of interpolation scales.
Abstract: We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions

Journal ArticleDOI
TL;DR: In this paper, the inhomogeneous nonlinear Shrodinger equation (INLS-equation) was studied in the critical and supercritical cases p ≥ 4/N, with N ≥ 2, and it was shown that standing-wave solutions of the INLS equation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small Є > 0.
Abstract: This paper is concerned with the inhomogeneous nonlinear Shrodinger equation (INLS-equation)iu_t + Δu + V(Єx)│u│^pu = 0, x Є R^N. In the critical and supercritical cases p ≥ 4/N, with N ≥ 2, it is shown here that standing-wave solutions of (INLS-equation) on H^1(R^N) perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small Є > 0.

Journal ArticleDOI
TL;DR: In this paper, the authors study uniform and non-tangentially accessible domains in homogeneous groups of steps 2 and 3, and show that C 1,1 domains in groups of step 2 are not tengentially accessible.
Abstract: We study John, uniform and non-tangentially accessible domains in homogeneous groups of steps 2 and 3. We show that C 1,1 domains in groups of step 2 are non-tangentially accessible and we give an explicit condition which ensures the John property in groups of step 3.

Journal ArticleDOI
TL;DR: In this article, a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor is introduced, called the (p, q)-curvatures.
Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.
Abstract: Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.

Journal ArticleDOI
Xiaonan Ma1
TL;DR: In this article, the comportement de metrique de Quillen par immersions d'orbifold is analyzed, and a formule de Bismut-Lebeau is presented.
Abstract: Dans cet article, on calcule le comportement de metrique de Quillen par immersions d'orbifold. On etend ainsi une formule de Bismut-Lebeau au cas d'orbifold.

Journal ArticleDOI
TL;DR: In this article, an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces is given, and a variation formula in integral geometry with interest in its own is obtained.
Abstract: We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

Journal ArticleDOI
TL;DR: In this paper, the authors published a paper in the Transactions of the American Mathematical Society in volume 357, issue 12, published by the AAMS, which is a journal of the Mathematical Association of America.
Abstract: First published in Transactions of the American Mathematical Society in volume 357, issue 12, published by the American Mathematical Society.

Journal ArticleDOI
TL;DR: In this article, a dual form of the normal property for closed convex cones is derived, and the dual normal property is the property (G) introduced by Jameson, which is a dual characterization of the strong conical hull intersection property.
Abstract: We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a. characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets.

Journal ArticleDOI
TL;DR: In this paper, the authors study canard solutions near non-generic turning points, which are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch.
Abstract: This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. "Canard solutions" are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the "control curve".

Journal ArticleDOI
TL;DR: A necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds was given in this paper.
Abstract: We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary and showed that the complex dimensions of the string can only lie on the real line.
Abstract: In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.