Showing papers in "Bulletin of the American Mathematical Society in 1992"
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Abstract: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions
5,267 citations
TL;DR: In this paper, the authors present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems, in particular in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method.
Abstract: In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation,
$$y_t-6y^2y_x+y_{xxx}=0,\qquad -\infty
728 citations
TL;DR: In this article, a measure-theoretic condition for a property to hold "almost everywhere" on an infinite-dimensional vector space, with particularemphasis on function spaces such as C k and L p, was proposed.
Abstract: We present a measure-theoretic condition for a property to hold «almost everywhere» on an infinite-dimensional vector space, with particularemphasis on function spaces such as C k and L p . Like the concept of «Lebesgue almost every» on an infinite-dimensional spaces, our notion of «prevalence» is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of «open and dense» or «generic» when one desires a probabilistic result on the likelihood of a given property on a function space
458 citations
TL;DR: In this paper, Sunada's theorem was used to construct a pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, can one hear the shape of a drum?
Abstract: We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, can one hear the shape of a drum? In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold
353 citations
TL;DR: It is shown that the study of algorithms not only increases the understanding of algebraic number fields but also stimulates the curiosity about them.
Abstract: In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the urea. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them
177 citations
TL;DR: In this paper, certain solvable extensions of $H$-type groups provide noncompact counterexamples to the Lichnerowicz conjecture, which asserted that Riemannian spaces must be rank 1 symmetric spaces.
Abstract: Certain solvable extensions of $H$-type groups provide noncompact counterexamples to the so-called Lichnerowicz conjecture, which asserted that ``harmonic'' Riemannian spaces must be rank 1 symmetric spaces.
173 citations
TL;DR: In this paper, the authors give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling, and give a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice.
Abstract: We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice
164 citations
TL;DR: In this article, it was proved that the free $m$-generated Burnside groups of exponent $n$ are infinite provided that $m>1, $n\ge2^{48}$.
Abstract: It is proved that the free $m$-generated Burnside groups $\Bbb{B}(m,n)$ of exponent $n$ are infinite provided that $m>1$, $n\ge2^{48}$.
146 citations
TL;DR: Recently, the first author of as discussed by the authors gave a quasipolynomial upper bound for ∆(d, n) for linear programming with pivot rules, which requires n √ d (or less) arithmetic operations for every linear programming problem with d variables and n constraints.
Abstract: The diameter of the graph of a d-dimensional polyhedron with n facets is at most nlog d+2 Let P be a convex polyhedron. The graph of P denoted by G(P ) is an abstract graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge (= 1-dimensional face) of P . The diameter of the graph of P is denoted by δ(P ). Let ∆(d, n) be the maximal diameter of the graphs of d-dimensional polyhedra P with n facets. (A facet is a (d− 1)-dimensional face.) Thus, P is the set of solutions of n linear inequalities in d variables. It is an old standing problem to determine the behavior of the function ∆(d, n). The value of ∆(d, n) is a lower bound for the number of iterations needed for Dantzig’s simplex algorithm for linear programming with any pivot rule. In 1957 Hirsch conjectured [2] that ∆(d, n) ≤ n−d. Klee and Walkup [6] showed that the Hirsch conjecture is false for unbounded polyhedra. They proved that for n ≥ 2d,∆(d, n) ≥ n − d + [d/5]. This is the best known lower bound for ∆(d, n). The statement of the Hirsch conjecture for bounded polyhedra is still open. For a recent survey on the Hirsch conjecture and its relatives, see [5]. In 1967 Barnette proved [1, 3] that ∆(d, n) ≤ n3. An improved upper bound, ∆(d, n) ≤ n2, was proved in 1970 by Larman [7]. Barnette’s and Larman’s bounds are linear in n but exponential in the dimension d. In 1990 the first author [4] proved a subexponential bound ∆(d, n) ≤ 2 √ (n−d) . The purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for ∆(d, n). Such a bound was proved by the first author in March 1991. The proof presented here is a substantial simplification that was subsequently found by the second author. See [4] for the original proof and related results. The existence of a polynomial (or even linear) upper bound for ∆(d, n) is still open. Recently, the first author found a randomized pivot rule for linear programming which requires an expected n √ d (or less) arithmetic operations for every linear programming problem with d variables and n constraints. 1991 Mathematics Subject Classification. Primary 52A25, 90C05. Received by the editors July 1, 1991 The first author was supported in part by a BSF grant by a GIF grant. The second author was supported by an AFOSR grant c ©1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page
140 citations
TL;DR: In this paper, the authors survey Galois theory of function fields with non-zero caracteristic properties and its relation to the structure of finite permutation groups and matrix groups.
Abstract: The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.
104 citations
TL;DR: In this paper, it was shown that for all n ≥ 10 there exists a tiling of R n by unit n-cubes such that no two ncubes have a complete facet in common.
Abstract: O. H. Keller conjectured in 1930 that in any tiling of R n by unit n-cubes there exist two of them having a complete facet in common. 0. Perron proved this conjecture for n ≤ 6. We show that for all n ≥ 10 there exists a tiling of R n by unit n-cubes such that no two n-cubes have a complete facet in common
TL;DR: The main theorem of as mentioned in this paper states that each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A such that m(p+q)=m(p)+m(q) whenever p and q are orthogonal projections.
Abstract: Let A be a von Neumann algebra with no direct summand of Type I 2 , and let P(A) be its lattice of projections. Let X be a Banach space. Let m:P(A)→X be a bounded function such that m(p+q)=m(p)+m(q) whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A
TL;DR: A characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin this paper, has been shown to be a combinatorial characterization.
Abstract: We describe a characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in E 3 all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832
TL;DR: In this paper, the authors survey existence and regularity results for semi-linear wave equations and give a self-contained, slightly simplified proof for the $u^5$-Klein Gordon equation.
Abstract: We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the $u^5$-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.
TL;DR: A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given in this article, where the main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
Abstract: A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
TL;DR: The notion of the automorphic dual of matrix algebraic groups was introduced in this article, which is defined as the part of the unitary dual that corresponds to arithmetic spectrum.
Abstract: We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to deduce arithmetic vanishing theorems of ``Ramanujan'' type as well as to give a new construction of automorphic forms.
TL;DR: A survey of some results of the pcf-theory and their applications to cardinal arithmetic can be found in this article, where the limitations on independence proofs are discussed and the status of two axioms that arise in the new setting is discussed.
Abstract: We present a survey of some results of the pcf-theory and their applications to cardinal arithmetic. We review basics notions (in section 1), briefly look at history in section 2 (and some personal history in section 3). We present main results on pcf in section 5 and describe applications to cardinal arithmetic in section 6. The limitations on independence proofs are discussed in section 7, and in section 8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in section 9.
TL;DR: In the last few years various types of trees have been the subject of much investigation, but this activity has not been exposed much to the wider mathematical community as mentioned in this paper, and this article attempts to fill this gap and explain various aspects of the recent work on generalized trees.
Abstract: To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much investigation, but this activity has not been exposed much to the wider mathematical community. This article attempts to fill this gap and explain various aspects of the recent work on generalized trees.
The subject is very appealing for it mixes very na\\\"{\\i}ve geometric considerations with the very sophisticated geometric and algebraic structures. In fact, part of the drama of the subject is guessing what type of techniques will be appropriate for a given investigation: Will it be direct and simple notions related to schematic drawings of trees or will it be notions from the deepest parts of algebraic group theory, ergodic theory, or commutative algebra which must be brought to bear? Part of the beauty of the subject is that the na\\\"{\\i}ve tree considerations have an impact on these more sophisticated topics and that in addition, trees form a bridge between these disparate subjects.
TL;DR: In this paper, it was shown that an extremal quasiconformal mapping can be relaxed to a map of lesser dilatation when lifted to a sufficiently large Riemann surface.
Abstract: About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds which can be built up in an inductive way from 3-balls, i.e. Haken manifolds. Thurston’s construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichmuller space to itself. Thurston proposes to find the fixed point by iterating this map. Here we outline Thurston’s construction, and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently
TL;DR: In this paper, it was shown that the set of characters for which the corresponding local system has nontrivial cohomology in a given degree is a union of finitely many components that are translates of algebraic subgroups of the fundamental group of a compact Kahler manifold.
Abstract: Let $X$ be a compact K\"ahler manifold. The set $\cha(X)$ of one-dimensional complex valued characters of the fundamental group of $X$ forms an algebraic group. Consider the subset of $\cha(X)$ consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree $d$. This set is shown to be a union of finitely many components that are translates of algebraic subgroups of $\cha(X)$. When the degree $d$ equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is shown that the number of equivalence classes (under a natural equivalence relation) of holomorphic maps, with connected fibers, of $X$ onto smooth curves of a fixed genus $>1$ is a topological invariant of $X$. In fact it depends only on the fundamental group of $X$.
TL;DR: In this paper, the authors consider the Yang-Mills repulsive force in 3 + 1 space-time dimensions and show how one can prove rigorously the existence of a globally defined smooth static solution, with the associated Einstein metric being asymptotically flat and the total mass finite.
Abstract: This talk will consider the Einstein/Yang-Mills equations in 3 + 1 space-time dimensions and show how one can prove rigorously the existence of a globally defined smooth static solution, with the associated Einstein metric being asymptotically flat and the total mass finite. Physically this means that the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in space-time.
TL;DR: In this paper, the existence of a homoclinic orbit of the Lorenz equations has been proved and a shooting technique has been developed to obtain a one-to-one correspondence between a set of solutions and the set of all infinite sequences of 1's and 3's.
Abstract: We announce and outline a proof of the existence of a homoclinic orbit of the Lorenz equations. In addition, we develop a shooting technique and two key conditions, which lead to the existence of a one-to-one correspondence between a set of solutions and the set of all infinite sequences of 1's and 3's
TL;DR: In this article, a set of invariants for a finite group is described, which arise naturally from Frobenius' early work on the group determinant and provide an answer to a question of Brauer.
Abstract: A set of invariants for a finite group is described. These arise naturally from Frobenius' early work on the group determinant and provide an answer to a question of Brauer. Whereas it is well known that the ordinary character table of a group does not determine the group uniquely, it is a consequence of the results presented here that a group is determined uniquely by its ``3-character'' table.
TL;DR: Shoen and Uhlenbeck as discussed by the authors showed that tangent maps can be defined at singular points of energy minimizing maps, but these are not unique, even for generic boundary conditions, and examples are discussed which have isolated singularities with a continuum of distinct tangent map.
Abstract: Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum of distinct tangent maps.
TL;DR: In this article, the method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and q integral/Sums.
Abstract: The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and q integral/sums
TL;DR: In this article, a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative Bailey chain concept in the setting of basic hypergeometric series very well-poised on unitary A or symplectic C groups is presented.
Abstract: We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative Bailey chain concept in the setting of basic hypergeometric series very well-poised on unitary A or symplectic C groups. The classical case, corresponding to A 1 or equivalently U(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities
TL;DR: In this paper, it was shown that the remarkable homogeneous space M =Diff(S 1 )/PSL(2, R) sits as a complex analytic and Kahler submanifold of the Universal Teichmuller Space.
Abstract: In previous work it had been shown that the remarkable homogeneous space M=Diff(S 1 )/PSL(2, R) sits as a complex analytic and Kahler submanifold of the Universal Teichmuller Space. There is a natural immersion Π of M into the infinite-dimensional version (due to Segal) of the Siegel space of period matrices. That map Π is proved to be injective, equivariant, holomorphic, and Kahler-isometric (with respect to the canonical metrics). Regarding a period mapping as a map describing the variation of complex structure, we explain why Π is an infinite-dimensional period mapping
TL;DR: In this article, the authors give a criterion for a holomorphic mapping between compact complex manifolds to have unobstructed deformations, i.e., for the local moduli space of the mapping to be smooth.
Abstract: Let $f$ be a holomorphic mapping between compact complex manifolds. We give a criterion for $f$ to have {\it unobstructed deformations}, i.e. for the local moduli space of $f$ to be smooth: this says, roughly speaking, that the group of infinitesimal deformations of $f$, when viewed as a functor, itself satisfies a natural lifting property with respect to infinitesimal deformations. This lifting property is satisfied e.g. whenever the group in question admits a `topological' or Hodge-theoretic interpretation, and we give a number of examples, mainly involving Calabi-Yau manifolds, where that is the case.
TL;DR: In this paper, the same nilpotent orbit may be shared by more than one simple group and the symmetry groups of that geometry are encoded in the symplectic and algebraic geometry.
Abstract: The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be "shared" by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic