Showing papers in "Transactions of the American Mathematical Society in 2001"
TL;DR: In this article, the authors study R-modules and complexes of such, with excellent duality properties, and count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself.
Abstract: Let R be a commutative Noetherian ring. We study R–modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes “lying between” these extremes is incentive.
257 citations
TL;DR: In this article, a weakly defective irreducible projective variety X is defined as one whose intersection with a general (k + 1)-tangent hyperplane has no isolated singularities at the k + 1 points of tangency.
Abstract: A projective variety X is k-weakly defective' when its intersection with a general (k + 1)-tangent hyperplane has no isolated singularities at the k + 1 points of tangency. If X is k-defective, i.e. if the k-secant variety of X has dimension smaller than expected, then X is also k-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of k-defective surfaces.
228 citations
TL;DR: In this paper, a tractors with canonical linear connections for all parabolic geometries are presented, which are analogous to a covariant derivative, iterable and defined on all vector bundles on parabolic geometry.
Abstract: Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.
215 citations
TL;DR: In this article, the multiplier ideal associated to an arbitrary monomial ideal in C^n was calculated and applications to the calculation of log canonical thresholds were discussed, as well as the application of this multiplier ideal to the problem of counting the number of canonical thresholds.
Abstract: In this note we calculate the multiplier ideal associated to an arbitrary monomial ideal in C^n. We discuss applications to the calculation of log canonical thresholds.
199 citations
TL;DR: In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given, and the homogeneous hypersurfaces in these spaces are obtained by computing the cohomogeneity for all hyperpolic actions.
Abstract: An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.
190 citations
TL;DR: In this paper, it was shown that the Korteweg-de-vries equation (0.1) can be approximated arbitrarily well by solving a finite number of linear problems.
Abstract: The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem (0.1) {η t + η x + ηη x + η xxx = 0, for x, t ≥ 0. (0.1) {η t + η x + ηη x + η xxx = 0, for x, t ≥ 0 η(x,0) = o(x), η(0,t) = h(t), studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data o in the class H s (R + ) for s > 3 and boundary data h in H (1+s)/3 loc (R + ), whereas global well-posedness is shown to hold for o ∈ H s (R + ), h ∈ H 12 loc (R + ) when 1 3. In addition, it is shown that the correspondence that associates to initial data O and boundary data h the unique solution u of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.
178 citations
TL;DR: In this paper, a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k ∈z) of absolutely regular stochastic processes (Zn)n∈z), which occur naturally as orbits of chaotic dynamical systems, was developed.
Abstract: In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.
153 citations
TL;DR: In this article, the authors constructed natural relative compactifications for the relative Jacobian over a family of reduced curves, which admit a universal sheaf, after an etale base change.
Abstract: We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our method consists of considering the functor $F$ of relatively simple, torsion-free, rank 1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have good properties. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an etale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.
152 citations
TL;DR: In this paper, it was shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure in E. This latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities.
Abstract: Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure of zero in E. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each R-module is introduced and studied. This theory gives rise to an ideal of R which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of R in general, and shown to equal the test ideal under the hypothesis that 0E = 0 fg∗ E in every local ring of R.
149 citations
TL;DR: In this article, the conditional variational principle is used to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.
Abstract: We establish a \" conditional \" variational principle, which unifies and extends many results in the multifractal analysis of dynam-ical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new mul-tifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy. Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the en-tropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the \" mixed \" multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the \" non-mixed \" multifractal spectra.
145 citations
TL;DR: In this paper, it was shown that the initial value problem associated to the derivative nonlinear Schrödinger equation is ill-posed for data in Hs(R), s < 1/2.
Abstract: Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in Hs(R), s < 1/2. This result implies that best result concerning local well-posedness for the IVP is in Hs(R), s ≥ 1/2. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
TL;DR: In this paper, it was shown that model categories of a broad class can be replaced up to Quillen equivalence by simplicial model categories, which can be seen as a simplification of model categories.
Abstract: In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories.
TL;DR: In this paper, it was shown that the necessity of the nuclear dominance condition for the existence of a process having its sample paths in a reproducing kernel Hilbert space (RKHS) turns out to be incorrect.
Abstract: A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either 0 or 1. Driscoll also found a necessary and sufficient condition for that probability to be 1. Doing away with Driscoll’s restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem – specifically the necessity of the condition – turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available. Using Fortet’s tools along with some new ones, we correct Fortet’s theorem and then find the generalization of Driscoll’s result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet’s work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.
TL;DR: Morel as discussed by the authors showed that the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes and proved that this is always the case when the category is a Grothendieck category, as has also been done by Morel.
Abstract: In this paper, we try to determine when the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes. We prove that this is always the case when the abelian category is a Grothendieck category, as has also been done by Morel. But this model structure is not very useful for dening derived tensor products. We therefore consider another method for constructing a model structure, and apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.
TL;DR: In this paper, the three bilinearities uv, uv and uv for functions u, v : R2×[0, T ] 7−→ C are sharply estimated in function spaces Xs,b associated to the Schrodinger operator i∂t+∆.
Abstract: The three bilinearities uv, uv, uv for functions u, v : R2×[0, T ] 7−→ C are sharply estimated in function spaces Xs,b associated to the Schrodinger operator i∂t+∆. These bilinear estimates imply local wellposedness results for Schrodinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrodinger equation (and certain generalizations) are also obtained.
TL;DR: In this article, it was shown that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus, if the substitution is unimodular and satisfies a certain combinatorial condition.
Abstract: We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.
TL;DR: In this article, the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms has been shown, and some precise renorming theorems have been proved for Banach spaces.
Abstract: We prove some rather precise renorming theorems for Banach spaces with Szlenk index wo. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.
TL;DR: In this paper, the authors considered the problem of optimally transporting mass on a complete, connected, Riemannian manifold, assuming absolute continuity of the Borel probability of the mass.
Abstract: Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures μ + ¬= μ - , find the measurepreserving map s: M → M between them which minimizes the average distance transported Set on a complete, connected, Riemannian manifold M - and assuming absolute continuity of μ + - an optimal map will be shown to exist Aspects of its uniqueness are also established
TL;DR: In this article, a non-commutative generalization of the A-polynomial of a knot is introduced using the Kauffman bracket skein module of the knot complement, based on the relationship between skein modules and character varieties.
Abstract: The paper introduces a noncommutative generalization of the A-polynomial of a knot This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus The generalized version of the A-polynomial, called the noncommutative A-ideal, consists of a finitely generated ideal of polynomials in the quantum plane Some properties of the noncommutative A-ideal and its relationships with the A-polynomial and the Jones polynomial are discussed The paper concludes with the description of the examples of the unknot, and the right- and left-handed trefoil knots
TL;DR: In this article, a discrete analogue of the classical Brunn-Minkowksi inequality for finite subsets of the integer lattice is obtained, which is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and in fact a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets.
Abstract: A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.
TL;DR: In this paper, a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra was given, and it was shown that such a ring is a homomorphic image of a finite-dimensional Gorenstein ring.
Abstract: The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
TL;DR: In this paper, the authors study the category pro-SSet of simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion, and establish a model structure on this category, which is closely related to the strict structure of Edwards and Hastings.
Abstract: We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.
TL;DR: In this article, it was shown that the test ideal is the annihilator of the tight closure relations and plays a crucial role in tight closure theory, and this was generalized to wider classes of singularities.
Abstract: We study tight closure and test ideals in rings of characteristic p 0 using resolution of singularities. The notions of F -rational and F regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal Q-Gorenstein ring of characteristic p 0, the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings. The notion of the tight closure of an ideal in a commutative ring of prime characteristic was defined by Hochster and Huneke [HH1] in terms of the asymptotic behavior of the ideal under iteration of the Frobenius map. Tight closure enables us to define the notions of F -regular rings [HH1] and F -rational rings [FW]. Namely, a ring of characteristic p > 0 is called F -regular (resp. F -rational) if all ideals (resp. all ideals generated by a system of parameters) are tightly closed in all of its local rings. Although these concepts are defined quite ring-theoretically, they have been suspected to have a mysterious correspondence with some classes of singularities in characteristic zero defined via resolution of singularities. Surprisingly, recent results by Smith [S2], Watanabe [W3], Mehta and Srinivas [MS], and the author [Ha] conclude that a ring in characteristic zero has at most rational (resp. log terminal) singularities if and only if its reduction modulo p is F -rational (resp. F -regular and Q-Gorenstein) for p 0. The aim of this paper is to generalize these results to wider classes of singularities. To do this we use a fairly standard technique of “reduction modulo p,” starting from a singularity in characteristic zero. Let (R,m) be a d-dimensional normal local ring which is reduced from characteristic zero to characteristic p 0, together with a resolution of singularities f : X → SpecR. When the non-(F -)rational locus of (R,m) is isolated, we actually proved in [Ha] that the tight closure of the zero Received by the editors July 27, 1999. 2000 Mathematics Subject Classification. Primary 13A35, 14B05; Secondary 13A02, 14B15.
TL;DR: In this paper, the authors consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic) and show that all their minors are formal power series in edge weights with nonnegative coefficients.
Abstract: We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
TL;DR: In this paper, it was shown that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential of a given potential under certain conditions.
Abstract: We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others.
TL;DR: In this paper, the authors present a paper by the American Mathematical Society in which the authors propose a method to solve a set of problems in the context of algebraic geometry.
Abstract: First published in Transactions- American Mathematical Society in Vol.353, No.10, pp.4235-4260, published by the American Mathematical Society
TL;DR: In this article, the authors define the notion of a peripheral splitting of a group and prove an accessibility result for relatively hyperbolic groups with connected boundary, where the peripheral subgroups are the maximal parabolic subgroups.
Abstract: We define the notion of a "peripheral splitting" of a group. This
is essentially a representation of the group as the fundamental group of a bipartite
graph of groups, where all the vertex groups of one colour are held
fixed - the "peripheral subgroups". We develop the theory of such splittings
and prove an accessibility result. The theory mainly applies to relatively hyperbolic
groups with connected boundary, where the peripheral subgroups are
precisely the maximal parabolic subgroups. We show that if such a group admits
a non-trivial peripheral splitting, then its boundary has a global cut point.
Moreover, the non-peripheral vertex groups of such a splitting are themselves
relatively hyperbolic. These results, together with results from elsewhere, show
that under modest constraints on the peripheral subgroups, the boundary of
a relatively hyperbolic group is locally connected if it is connected. In retrospect,
one further deduces that the set of global cut points in such a boundary
has a simplicial treelike structure.
TL;DR: In this article, the authors present an algorithmic method for computing a projective resolution of a module over an algebra over a field, and apply this resolution to the study of the Ext-algebra of the algebra.
Abstract: In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.