Showing papers in "Transactions of the American Mathematical Society in 1993"
TL;DR: In this paper, a well-posedness result in the class W 2,p ∩ W 0 1,p for the Dirichlet problem was proved for the case where Lu = f a and u = 0 on ∂Ω.
Abstract: We prove a well-posedness result in the class W 2,p ∩ W 0 1,p for the Dirichlet problem Lu = f a.e. in Ω, u = 0 on ∂Ω. We assume the coefficients of the elliptic nondivergence form equation that we study are in VMO ∩ L∞
425 citations
TL;DR: The main object of as discussed by the authors is the renormalization of difference operators on post-critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified selfsimilar sets.
Abstract: The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations
399 citations
TL;DR: In this paper, the dependence on the A p norm of w of the operator norms of singular integrals, maximal functions, and other operators in L p (w) was examined.
Abstract: We examine the dependence on the A p norm of w of the operator norms of singular integrals, maximal functions, and other operators in L p (w). We also examine connections between some fairly general reverse Jensen inequalities and the A p and RH p weight conditions
363 citations
TL;DR: In this article, the authors consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior.
Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings
345 citations
TL;DR: In this article, the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings are derived for all x ∈ R.
Abstract: Biadditive mappings B: R × R → R where R is a prime ring with certain additional properties, satisfying B(x, x)x = xB(x, x) for all x ∈ R, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings
339 citations
TL;DR: In this article, it was shown that all the Bass numbers of all local cohomology modules of A are finite, and that each such local cohology module has a finite set of associated prime ideals.
Abstract: Let A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A itself, but with respect to an arbitrary ideal of A. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true
280 citations
TL;DR: The simplest model of the Cauchy problem considered in this article is the following: u t = Δu + u p, x ∈ R n, t > 0, u ≥ 0, p > 1, u| t=0 = φ ∈ C B (R n ), φ ≥ 0 and φ ¬= 0.
Abstract: The simplest model of the Cauchy problem considered in this paper is the following (*) u t = Δu + u p , x ∈ R n , t > 0, u ≥ 0, p > 1, u| t=0 = φ ∈ C B (R n ), φ ≥ 0, φ ¬= 0 It is well known that when 1 (n+2)/n, if φ is «small», (*) has a global classical solution decaying to zero as t → +∞, while if φ is «large», the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on φ for global existence and to study the asymptotic behavior of those global solutions
217 citations
TL;DR: A paraboIic rational map of the Riemann sphere admits a non-atomic h-conformal measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2 as discussed by the authors.
Abstract: A paraboIic rational map of the Riemann sphere admits a non-atomic h-conformal measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent a-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework
212 citations
TL;DR: In this article, the authors use Berezin's dequantization procedure to define a formal *-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kahler manifold.
Abstract: We use Berezin’s dequantization procedure to define a formal *- product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous Kahler manifold M. We prove that this formal *-product isconvergent when M isa hermitian symmetric space. © 1993 American Mathematical Society.
194 citations
TL;DR: In this article, the authors developed a duality theory in the sense of Kothe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace.
Abstract: Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Kothe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Kothe dual of a given Banach space of measurable operators in terms of normality
185 citations
TL;DR: In this paper, the authors clarified the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields and showed the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center.
Abstract: In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III)
TL;DR: In this article, random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability, and the focus is on limit theorems as the number being partitioned approaches ∞.
Abstract: Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches ∞. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct
TL;DR: In this paper, a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets created by a finite set of functions using the wandering subspaces of unitary operators in Hilbert space.
Abstract: Mallat's construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite B-splines is constructed
TL;DR: It is shown that in reasonable settings the super efficient points of a set are norm-dense in the efficient frontier, and a Chebyshev characterization ofsuper efficient points for nonconvex sets and a scalarization theory when the underlying set is convex.
Abstract: We introduce a new concept of efficiency in vector optimization. This concept, super efficiency, is shown to have many desirable properties. In particular, we show that in reasonable settings the super efficient points of a set are norm-dense in the efficient frontier. We also provide a Chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex
TL;DR: In this article, the authors define a self-similar distribution by the same identity but allowing the weights a j to be arbitrary complex numbers, and give necessary and sufficient conditions for the existence of a solution to (*) among distributions of compact support, and show that the space of such solutions is always finite dimensional.
Abstract: A self-similar measure on R n was defined by Hutchinson to be a probability measure satisfying (formule) here S j x = ρ j R j x+b j is a contractive similarity (0 < ρ j < 1, R j orthogonal) and the weights a j satisfy 0 < A j < 1, ∑ j=1 m a j = 1 By analogy, we define a self-similar distribution by the same identity *) but allowing the weights a j to be arbitrary complex numbers We give necessary and sufficient conditions for the existence of a solution to (*) among distributions of compact support, and show that the space of such solutions is always finite dimensional If F denotes the Fourier transformation of a self-similar distribution of compact support, let (formule) where β is defined by the equation ∑ j=1 m ρ j −β |a j | 2 = 1
TL;DR: In this article, the authors proved weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere, and the same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function.
Abstract: We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral
TL;DR: In this paper, an extension operator W which is a bounded mapping from B°(LP(U)) onto B%(Lp(Rd)) is presented. And the authors derive various properties of the Besov spaces, such as interpolation theorems for a pair of B%{Lp{I2}, atomic decompositions for the elements of 5°(lp(f2), and a description of the space by means of spline approximation.
Abstract: We study Besov spaces 5°(LP(I2)), 0
TL;DR: Arino et al. as discussed by the authors gave an alternative proof of recent results of M. Arino and B. Muckenhoupt concerning Hardy's inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces.
Abstract: The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy's inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1, 8] to the values of the parameters which are inaccessible by the methods of these papers
TL;DR: In this paper, the authors introduce a new class of objects, called hairy objects, which share many properties with the Julia sets and the interval: they are topologically unique and admit only one embedding in the plane.
Abstract: The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets-hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family
TL;DR: In this article, a new concept of chaos, ω-chaos, was proposed and proved to be equivalent to positive entropy on the interval, which is the definition of chaos given by Devaney.
Abstract: We present a new concept of chaos, ω-chaos, and prove some properties of ω-chaos. Then we prove that ω-chaos is equivalent to positive entropy on the interval. We also prove that ω-chaos is equivalent to the definition of chaos given by Devaney on the interval
TL;DR: In this paper, the authors investigated the consequences of weak uniformity in Cayley-Bacharach schemes and characterized CB-schemes in terms of the structure of the canonical module of their projective coordinate ring.
Abstract: A set of s points in P d is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s − 1 points has the same Hilbert function. We investigate the consequences of this «weak uniformity.» The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB-scheme X has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points
TL;DR: In this paper, the Minkowski, Brunn-Minkowski and Aleksandrov-Fenchel inequalities for projection and mixed projection bodies are developed for the same purpose.
Abstract: Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies
TL;DR: In this paper, the Ricci flow of R. Hamilton was used to deform stable Riemannian metrics to hyperbolic space forms through Ricci flows. But the Riccis flow was not used to transform negatively pinched RiemANNIAN manifolds to space forms.
Abstract: The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) L 2 -pinched Riemannian manifolds can be deformed to space forms through Ricci flow
TL;DR: The model theory of left R-modules as developed by Ziegler [Z] is in some sense dual to the model theoretic theory of right R -modules as mentioned in this paper.
Abstract: Let R be a ring. A formula φ(x) in the language of left R-modules is called a positive primitive formula (ppf) if it is of the form ∃y(AB) (x/y) = 0 where A and B are matrices of appropriate size with entries in R. We apply Prest's notion of Dφ(x), the ppf in the language of right R-modules dual to φ, to show that the model theory of left R-modules as developed by Ziegler [Z] is in some sense dual to the model theory of right R-modules. We prove that the topologies on the left and right Ziegler spectra are «isomorphic» (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism D between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left R-module U the dual pure-injective indecomposable right R-module DU
TL;DR: In this article, the authors considered the Cauchy problem with general assumptions on the blow-up rate and derived the possible asymptotic behaviours of u(x, t) → (0, T) under general assumptions.
Abstract: Consider the Cauchy problem (p) u t − Δu = u p when x ∈ R N , t > 0, N ≥ 1, u(x, 0) = u 0 (x) when x ∈ R N , where p > 1, and u 0 (x) is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions on u 0 (x), the unique solution of (P), u(x, t), blows up in a finite time, by which we mean that lim sup sup u(x, t) = + ∞. t↑T x ∈ R N In this paper we shall assume that u(x, t) blows up at x = 0, t = T < + ∞, and derive the possible asymptotic behaviours of u(x, t) as (x, t) → (0, T), under general assumptions on the blow-up rate
TL;DR: In this paper, all 7-dimensional nilpotent Lie algebras over C are determined by elementary methods and a multiplication table is given for each isomorphism class.
Abstract: All 7-dimensional nilpotent Lie algebras over C are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter
TL;DR: In this article, it was shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor.
Abstract: It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor. It is also shown that the Normal Chain Conjecture is false and that there exist local noncatenary UFDs
TL;DR: In this paper, the Schrodinger operator for a magnetic potential A and an electric potential q, which are supported in a bounded domain in R n with n ≥ 3, was considered and proved to determine the magnetic field rot(A) and the electric potential 4 simultaneously.
Abstract: We consider the Schrodinger operator for a magnetic potential A and an electric potential q, which are supported in a bounded domain in R n with n ≥ 3. We prove that knowledge of the Dirichlet to Neumann map associated to the Schrodinger operator determines the magnetic field rot(A) and the electric potential 4 simultaneously, provided rot(A) is small in the L∞ topology
TL;DR: In this paper, it was shown that the generalized hypergeometric function 2 F 1 of matrix argument is the series expansion at the origin of a special case of the Jacobi polynomials associated with the root system of type BC.
Abstract: We show that the generalized hypergeometric function 2 F 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type BC. In addition we prove that the Jacobi polynomials of matrix argument correspond to the Jacobi polynomials associated with the root system of type BC. We also give a precise relation between Jack polynomials and the Jacobi polynomials associated with the root system of type A. As a side result one obtains generalized hook-length formulas which are related to Harish-Chandra's c-function and one can prove a conjecture due to Macdonald relating two inner products on a space of symmetric functions