Showing papers in "Transactions of the American Mathematical Society in 1992"

On explosions of solutions to a system of partial differential equations modelling chemotaxis

Abstract: A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of paramaters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values

Intersection theory of moduli space of stable N-pointed curves of genus zero

Abstract: We give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the Chow ring

On compactly supported spline wavelets and a duality principle

Abstract: Let ...⊂V −1 ⊂V 0 ⊂V 1 ⊂... be a multiresolution analysis of L 2 generated by the mth order B-spline N m (x). We exhibit a compactly supported basic wavelet ψ m (x) that generates the corresponding orthogonal complementary wavelet subspaces ...,W − 1 ,W 0 ,W 1 , .... Consequently, the two finite sequences that describe the two-scale relations of N m (x) and ψ m (x) in terms of N m (2x−j), j∈Z, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases {N m (x−j)} and {ψ#59 m (x−j)}, relative to {N m (x−j)} and {ψ m (x−j)}, respectively

Liouvillian first integrals of differential equations

Abstract: Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form

Motion of level sets by mean curvature. II

Abstract: We give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface. Our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

On q-analogues of the Fourier and Hankel transforms

Abstract: For H. Exton's q-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson's q-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskii. As a specialization we get q-cosines and q-sines which admit q-analogues of the Fourier-cosine and Fourier-sine transforms

Global existence, large time behavior and life span of solutions of a semilinear parabolic cauchy problem

Abstract: We investigate the behavior of the solution u(x, t) of (...) where Δ = Σ 1=1 n ∂ 2 /∂ xi 2 is the Laplace operator, p > 1 is a constant, T > 0, and φ is a nonnegative bounded continuous function in R n . The main results are for the case when the initial value φ has polynomial decay near x = ∞. Assuming φ ∼ λ(1+|x|) #75a with λ, a > 0, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution u(x, t) are answered in terms of simple conditions on λ, a, p and the space dimension n

Groups of piecewise linear homeomorphisms

Abstract: In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups

Continuation theorems for periodic perturbations of autonomous systems

Abstract: It is first shown in this paper that, whenever it exists, the coincidence degree of the left-hand member of an autonomous differential equation x' - g(x) = 0, in the space of periodic functions with fixed period omega, can be computed in terms of the Brouwer degree of g. This result provides efficient continuation theorems specially for omega-periodic perturbations of autonomous systems. Extensions to differential equations in flow-invariant ENR's are also given.

Uniqueness of Radial Solutions of Semilinear Elliptic Equations

Abstract: E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one finite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coefficients. One particular example covered is Δu+u p ± u = 0, with p > 1. The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong

Homological theory of idempotent ideals

Abstract: Let Λ be an artin algebra U and a two-sided ideal of Λ. Then U is the trace of a projective Λ-module P in Λ. We study how the homological properties of the categories of finitely generated modules over the three rings Λ/U, Λ and the endomorphism ring of P are related. We give some applications of the ideas developed in the paper to the study of quasi-hereditary algebras

A damped hyperbolic equation on thin domains

Abstract: For a damped hyperbolic equation in a thin domain in R3 over a bounded smooth domain in R2 , it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.

On the positive solutions of semilinear equations Δ+-ℎ^{}=0 on the compact manifolds

Abstract: In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations Δu+λu-hu p =0 on compact Riemannian manifolds as well as on bounded smooth domains in R n with homogeneous Dirichlet or Neumann boundary conditions

Expansions of chromatic polynomials and log-concavity

Abstract: In this paper we present several results and open problems about log-concavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial

A global lojasiewicz inequality for algebraic varieties

Abstract: Let X be the locus of common zeros of polynomials f 1 ,..., f κ in n complex variables. A global upper bound for the distance to X is given in the form of a Lojasiewicz inequality. The exponent in this inequality is bounded by d min(n,k) where d=max(3, deg f i ). The estimates are also valid over an algebraically closed field of any characteristic

A model in which GCH holds at successors but fails at limits

Abstract: Starting with GCH and a _9'3K-hypermeasurable cardinal, a model is produced in which 21 = A+ if A is a successor cardinal and 21 = A++ if A is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing.

On Auslander-Reiten components of blocks and self-injective biserial algebras

Abstract: We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here

On twistor spaces of anti-self-dual hermitian surfaces

Abstract: We consider a complex surface M with anti-self-dual hermitian metric h and study the holomorphic properties of its twistor space Z We show that the naturally defined divisor line bundle [X] is isomorphic to the -1/2 power of the canonical bundle of Z, if and only if there is a Kahler metric of zero scalar curvature in the conformal class of h This has strong consequences on the geometry of M, which were also found by C Boyer [3] using completely different methods We also prove the existence of a very close relation between holomorphic vector fields on M and Z in the case that M is compact and Kahler

Generation and propagation of interfaces in reaction-diffusion systems

Abstract: This paper is concerned with the asymptotic behavior, as e0, of the solution (u e , υ e ) of the second initial-boundary value problem of the reaction-diffusion system: u t e − eΔu e = 1/e f(u e , υ e) ≡ 1/e [u e (1 − u e2 ) − υ e υ t e − Δυ e = u e − γυ e where γ > 0 is a constant. When υ ∈ (−2√3/9, 2√3/9), f is bistable in the sense that the ordinary differential equation u t = f(u, υ) has two stable solutions u = h − (υ) and u = h + (υ) and one unstable solution u = h 0 (υ), where h − (υ), h 0 (υ), and h + (υ) are three solutions of the algebraic equation f(u, υ) = 0

The index of a Brauer class on a Brauer-Severi variety

Abstract: Let D and E be central division algebras over k; let K be the generic splitting field of E; we show that the index of D⊗ k K is the minimum of the indices of D⊗E ○ × i as i varies. We use this to calculate the index of D under related central extensions and to construct division algebras with special properties

Moderate deviations and associated Laplace approximations for sums of independent random vectors

Abstract: Let {Xj} be an i.i.d. sequence of Banach space valued r.v.'s and let Sn = Z, 1 Xj . For certain positive sequences bn oo, we determine the exact asymptotic behavior of E exp{(b2/n)D(Sn/bn)}, where D is a smooth function. We also prove a large deviation principle for {2Y(Sn/bn)} .

Growth series of some wreath products

Abstract: The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions. n=0 Since only one generating set will be associated with each group below, the generating set associated with fr(x) will be obvious. Let H be a group with a finite generating set Sh ■ These will be fixed for the rest of the paper. The Cayley graph of the pair (H, Sh) is, as usual, the directed graph whose vertices are the elements of H and there is an edge from a vertex hx Xoa vertex h2 if and only if h2 = hxh for some h in SH US^1. In particular, the edge from hx to h2 has an opposite edge from h2 to hx . Let C be the graph gotten from the Cayley graph of (H, Sh) simply by identifying opposite edges. In other words, C might be called the undirected Cayley graph of (H,SH). hex K also be a group with a finite generating set Sk ■ These will also be fixed for the rest of the paper. It is possible to form what might be called a restricted direct product group P, which consists of all functions p from the vertices of C to K such that there are only finitely many vertices v in C with p(v) t? 1. This is a subgroup of the direct product group whose elements consist of all functions from the vertex set of C to K. The group P admits H as a group of automorphisms by means of the action of H on C. The resulting semidirect product P x H is the restricted wreath product K\H.

Studying links via closed braids. V. The unlink

Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the r-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most 1. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theo- rem 1 shows that exchange moves are the only obstruction to reducing a closed n-braid representative of the r-component unlink to the standard closed r-braid representative, through a sequence of braids of nonincreasing braid index. Let K be an oriented link type in oriented S3, K a representative of K, and A an unknotted curve in S3. K is a closed n-braid with braid axis A if K winds monotonically about A with total winding number n. "Mono- tonically" means that if A is the z-axis in R3, parametrized by cylindrical coordinates (r(t), 0(t), z(t)), then 0(t) is a strictly increasing function of t. This is equivalent to the assertion that if {Ht: t E (0, 27X)} is a fibration of S3 - A by meridian discs then K meets each Ht transversally in exactly n coherently oriented intersections. Alexander proved in (A) that every link could be so-represented (in many ways). The theorem which is known as Markov's theorem describes how the various closed braid representatives of a given link type are related. Markov's Theorem was announced in (Ma), with a working outline for a proof. It has been important in recent years because of its central

Toeplitz and Hankel operators on Bergman spaces

Abstract: In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C n whose symbols are bounded measurable functions. We giv e necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Topelitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H ∞(Ω); for these symbols we describe when the Toeplitz or Hankel operators are compact

On the structure of twisted group C*-algebras

Abstract: We first give general structural results for the twisted group algebras C*(G, σ) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, σ) as the sections of a C*-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ is a discrete subgroup of a solvable Lie group G, the K-groups K * (C*(Γ, σ)) are isomorphic to certain twisted K-groups K*(G/Γ, δ(σ)) of the homogeneous space G/Γ, and we discuss how the twisting class δ(σ) ∈ H 3 (G/Γ, Z) depends on the cocycle σ

Differentials of complex interpolation processes for Köthe function spaces

Abstract: We continue the study of centralizers on Kothe function spaces and the commutator estimates they generate (see [29]). Our main result is that if X is a super-reflexive Kothe function space then for every real centralizer Ω on X there is a complex interpolation scale of Kothe function spaces through X inducing Ω as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a (UMD)-space

Growth rates, _{}-homology, and volumes of hyperbolic 3-manifolds

Abstract: It is shown that if M is a closed orientable irreducible 3-manifold and n is a nonnegative integer, and if H 1 (M, Z p ) has rank ≥ n+2 for some prime p, then every n-generator subgroup of π 1 (M) has infinite index in π 1 (M), and is in fact contained in infinitely many finite-index subgroups of π 1 (M). This result is used to estimate the growth rates of the fundamental group of a 3-manifold in terms of the rank of the Z p -homology

Log concave sequences of symmetric functions and analogs of the jacobi-trudi determinants

Abstract: We prove that various sequences of elementary and complete ho- mogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of «/-binomial coefficients and 17-Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot (G-V 85).

Approximation of Jensen measures by image measures under holomorphic functions and applications

Abstract: We show that Jensen measures defined on C n or more generally on a complex Banach space X can be approximated by the image of Lebesgue measure on the torus under X-valued polynomials defined on C. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar

Cesàro summability of double Walsh-Fourier series

Abstract: We introduce quasi-local operators (these include operators of Calder6n-Zygmund type), a hybrid Hardy space HO of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type ( , 1) . As an application, we show that Cesatro means of the double Walsh-Fourier series of a function f converge a.e. when f belongs to HO. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all f E L1 provided summability takes place in some positive cone.