Showing papers in "Transactions of the American Mathematical Society in 2000"
TL;DR: In this article, the existence and multiplicity of solutions for the quasi-linear partial differential equation (QPDE) was studied using variational methods, where the variational approach requires that 1 < p < n, p ≤ q ≤ p ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ = np n−p, which we assume throughout.
Abstract: We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡ p∗(0) = np n−p , which we assume throughout. However, the situations differ widely with q and r, and the interesting cases occur either at the critical Sobolev exponent (r = p∗) or in the Hardy-critical setting (s = p = q) or in the more general Hardy-Sobolev setting when q = n−s n−pp. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case p = 2, especially those corresponding to singularities (i.e., when 0 < s ≤ p).
487 citations
TL;DR: In this article, the objects of a category may be viewed as models for homotopy theories, and it is shown that the category of such models has a well-behaved internal homobject.
Abstract: We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.
410 citations
TL;DR: In this paper, a general framework within which previously isolated results can now be properly understood was provided, and several previously conjectured evaluations, including an intriguing conjecture of Don Zagier were proved.
Abstract: Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.
403 citations
TL;DR: In this article, a local theory was developed for the property of a distance function being continuously differentiable outside of C on some neighborhood of a point x ∈ C, which is equivalent to the prox-regularity of C at x, a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation.
Abstract: Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation Additional characterizations are provided in terms of dC being locally of class C 1+ or such that dC + σ| · |2 is convex around x for some σ > 0 Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x, and leads to a formula for PC by way of NC The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting
401 citations
TL;DR: In this article, the computation of normal forms for Partial Functional Differential Equations (PFDEs) near equilibria has been studied and the analysis is based on the theory previously developed for autonomous functional differential equations and on the existence of center (or other invariant) manifold.
Abstract: The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.
228 citations
TL;DR: In this paper, the wave equation in the hyperbolic space HI and the Strichartz type estimates in the Minkowski space were obtained. But the results of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear HH problems with small data were not discussed.
Abstract: The aim of this article is twofold. First we consider the wave equation in the hyperbolic space HI and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in Rn x 1R which extend the ones of Georgiev, Lindblad, and Sogge.
220 citations
TL;DR: In this paper, a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains was developed, which resembles in many ways that of the Dirichlet problem.
Abstract: We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish Lp-Lq-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.
167 citations
TL;DR: A linearly ordered structure is weakly o-minimal if all definable sets in one variable are the union of finitely many convex sets in the structure as mentioned in this paper.
Abstract: A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
159 citations
TL;DR: In this paper, the authors characterized the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mpi (n+ ki) ≥ 0, where deg pi = 2ki or 2ki − 1 (1 ≤ i ≤ m).
Abstract: Let γ ≡ γ(2n) denote a sequence of complex numbers γ00, γ01, γ10, . . . , γ0,2n, . . . , γ2n,0 (γ00 > 0, γij = γji), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel measure μ on C such that γij = ∫ zizj dμ (0 ≤ i + j ≤ 2n) and suppμ ⊆ K. For K ≡ KP a semi-algebraic set determined by a collection of complex polynomials P = {pi (z, z)}mi=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mpi . We prove that there exists a rankM (n)-atomic representing measure for γ(2n) supported in KP if and only if M (n) ≥ 0 and there is some rankpreserving extension M (n+ 1) for which Mpi (n+ ki) ≥ 0, where deg pi = 2ki or 2ki − 1 (1 ≤ i ≤ m).
154 citations
TL;DR: In this paper, it was shown that local derivations from a von Neumann algebra into any dual bimodule are derivations, and that these results do not extend to the algebra C1[0, 1] of continuously differentiable functions on [0,1].
Abstract: Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any C∗-algebra A into any Banach A-bimodule X. Most of the work is involved with establishing this result when A is a commutative C∗-algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra C1[0, 1] of continuously differentiable functions on [0, 1]. We also give an automatic continuity result, that is, we show that local derivations on C∗-algebras are continuous even if not assumed
154 citations
TL;DR: In this article, the authors developed a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e.g., a free boundary problem describing a model of a stationary tumor.
Abstract: In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in e. 1. THE MODEL AND MAIN RESULT In this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e. The method works not only for boundary value problems but also for free boundary problems. In this latter context it can be used to establish long time existence of transient solutions, and also to study the existence of spatially asymmetric steady solutions. Since free boundary problems are typically more challenging than their boundary-value counterparts, we shall concentrate here on a free boundary problem from developmental biology, namely, a model of tumor growth. To further exemplify the generality of our approach an instance of a boundary value problem (in a fixed domain) is presented in the last section of the paper. A variety of other problems are amenable to the same analysis, including, in particular, the Hele-Shaw model of fluid flow [11]. Within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas (see. e.g., [1, 5, 6, 8, 12, 13] and the references cited there). The main objective of these models has been to qualitatively describe, under various simplifying assumptions, the growth and stability of tumor tissue. Analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments. On the other hand, the description of the stationary (dormant) configurations that arise from the models has only been addressed in the case of spherical tumors, but otherwise it remains largely unexplored. In this paper we develop a method for establishing analyticity of Received by the editors August 17, 1999. 1991 Mathematics Subject Classification. Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15.
TL;DR: In this paper, a special class of compact complex nilmanifolds with nilpotent complex structure is considered, called compact compact complex (CCN) with compact complex structure (CCS) and it is shown that the Dolbeault cohomology H ∗, ∗ ∂̄ (G) is canonically isomorphic to the ∆-cohomology H∗,∗ ∆ ∆ (gC) of the bigraded complex (G, ∆) of complex valued left invariant differential forms.
Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ\\G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H∗,∗ ∂̄ (Γ\\G) is canonically isomorphic to the ∂̄–cohomology H∗,∗ ∂̄ (gC) of the bigraded complex (Λ∗,∗(gC)∗, ∂̄) of complex valued left invariant differential forms on the nilpotent Lie group G.
TL;DR: In this paper, it was shown that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus.
Abstract: We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the n-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.
TL;DR: In this paper, a generalization of spectral entire functions of spherical exponential type and Lagrangian splines on manifolds is considered and an analog of the PaleyWiener theorem is given.
Abstract: We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the PaleyWiener theorem is given. We also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.
TL;DR: A cobordism theory for manifolds with corners was proposed in this article, which gives a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating chromatic status of the elements so realized.
Abstract: This work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.
TL;DR: In this article, it was shown that p is locally Lipschitz on which p is bounded; in particular, if M and N are compact, then p is globally Lipschnitz on Uy.
Abstract: Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uv the total space of the unit normal bundle of N. For each v C Uv, let p(v) denote the distance from N to the cut point of N on the geodesic -y, with the velocity vector tyv(0) = v. The continuity of the function p on Uv is well known. In this paper we prove that p is locally Lipschitz on which p is bounded; in particular, if M and N are compact, then p is globally Lipschitz on Uy. Therefore, the canonical interior metric 6 may be introduced on each connected component of the cut locus of N, and this metric space becomes a locally compact and complete length space. Let N be an immersed submanifold of a complete CC Riemannian manifold M and 7 : Uv -N the unit normal bundle of N. For each positive integer k and vector v C Uv, let a number k (v) denote the parameter value of fy, where 7yv denotes the geodesic for which the velocity vector is v at t = 0, such that 'yv(Ak(v)) is the k-th focal point (conjugate point for the case in which N is a point) of N along -y, counted with focal (or conjugate) multiplicities. A unit speed geodesic segment : [0, a] -+ M emanating from N is called an N-segment if t = d(N, y(t)) on [0, a]. By the first variation formula, an N-segment is orthogonal to N. A point 7y(to) on an N-segment 7y, v e Uv, is called a cut point of N if there is no N-segment properly containing y[O, to]. For each v C Uv, let p(v) denote the distance from N to the cut point on -y of N. Whitehead [27] investigated the structure of the conjugate locus and the cut locus of a point on a real analytic Finsler manifold. He determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood (cf. also [25]) and proved the continuity of the function p. In compact symmetric spaces, T. Sakai [19] and M. Takeuchi [23] determined the detailed structure of the cut locus of a point. The detailed structure of the cut locus of a point in a 2-dimensional Riemannian manifold has been investigated by Poincare, Myers, and others [7], [11], [13]. Hartman first tried to show the absolute continuity of the function p when M is 2-dimensional. He proved in [8] that if p is of bounded variation, then p is absolutely continuous. Recently, Hebda [11] and the first named author [13] independently proved Ambrose's problem by showing that p is absolutely continuous on a closed arc on which p is bounded when N is a point in a 2-dimensional Riemannian manifold. Therefore, the cut locus of a point in a compact 2-dimensional Received by the editors October 14, 1998 and, in revised form, April 13, 1999. 2000 Mathematics Subject Classification. Primary 53C22; Secondary 28A78. Supported in part by a Grant-in-Aid for Scientific Research from The Ministry of Education, Science, Sports and Culture, Japan. (2000 American Mathematical Society
TL;DR: In this paper, the authors present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algesbras similar to U(sl2) introduced by S. P. Smith.
Abstract: We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to U(sl2) introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of U(sl2) by finding sets of generators for the group of automorphisms.
TL;DR: In this article, a special case of Schmüdgen's Theorem for polynomials positive on a compact interval is discussed. But it is only on the real line and not on the non-compact interval.
Abstract: This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If h(x), p(x) ∈ R[x] such that {α ∈ R | h(α) ≥ 0} = [−1, 1] and p(x) > 0 on [−1, 1], then there exist sums of squares s(x), t(x) ∈ R[x] such that p(x) = s(x) + t(x)h(x). Explicit degree bounds for s and t are given, in terms of the degrees of p and h and the location of the roots of p. This is a special case of Schmüdgen’s Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval [0,∞) are also considered.
TL;DR: In this paper, the authors address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. They reformulate the Harbourne and Hirschowitz conjecture by explicitly listing those systems which have unexpected dimension.
Abstract: In this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. A conjecture of Harbourne and Hirschowitz implies that when d > 3m, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple (-1)-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m < 12.
TL;DR: In this paper, it was shown that the failure of the nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson E(n)∗-homology but nontrivial vn-periodic homotopy groups.
Abstract: In telescopic homotopy theory, a space or spectrum X is approximated by a tower of localizations LnX, n ≥ 0, taking account of vn-periodic homotopy groups for progressively higher n. For each n ≥ 1, we construct a telescopic Kuhn functor Φn carrying a space to a spectrum with the same vn-periodic homotopy groups, and we construct a new functor Θn left adjoint to Φn. Using these functors, we show that the nth stable monocular homotopy category (comprising the nth fibers of stable telescopic towers) embeds as a retract of the nth unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving “infinite Ln-suspension spaces.” We deduce that Ravenel’s stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel’s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson E(n)∗-homology but nontrivial vn-periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is K(n)-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the E(n)∗-localizations and K(n)∗-localizations of infinite loop spaces in terms of E(n)∗-localizations of spectra under suitable conditions. We also determine the E(n)∗-localizations and K(n)∗-localizations of arbitrary Postnikov H-spaces.
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TL;DR: In this paper, an analogue of Jantzen's sum formula for the cyclotomic q-Schur algebra is presented, which is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.
Abstract: The cyclotomic q-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen's sum formula for the cyclotomic q-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.
TL;DR: Brauer algebras arise in representation theory of orthogonal or symplectic groups as mentioned in this paper, and are shown to be iterated inflations of group algesbras of symmetric groups.
Abstract: Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.
Abstract: We consider the linear heat equation on the half-line with a Dirichlet boundary control We analyze the null-controllability problem More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the L2 boundary control We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials {eit }j> in which the usual summability condition on the inverses of the eigenvalues does not hold Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time
TL;DR: In this article, the authors give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases.
Abstract: We give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: Given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases.
TL;DR: In this paper, the authors extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds, and generalize Colding's volume convergence results and extend the Cheeger-Colding splitting theorem.
Abstract: We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding’s volume convergence results and extend the Cheeger-Colding splitting theorem.
TL;DR: In this paper, an Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals and it is observed that the mimimum principle frames symmetry of crystals.
Abstract: An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.
TL;DR: In this article, the shadow and small Weyl group invariants of simple weight 0-modules were introduced for simple Lie superalgebras, and the classifications of the finite cuspidal modules over certain simple superalgebra classes were reduced to finite simple weight modules.
Abstract: Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight 0-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of go, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.
TL;DR: In this paper, the existence of filtration pairs for isolated invariant sets of continuous maps was proved and it was shown that up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated set.
Abstract: In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.