Showing papers in "Transactions of the American Mathematical Society in 1998"

Symmetric functional differential equations and neural networks with memory

Abstract: We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

On the best constant for Hardy's inequality in $\mathbb{R}^n$

Abstract: Let Ω be a domain in IR and p ∈ (1,∞). We consider the (generalized) Hardy inequality ∫ Ω |∇u|p ≥ K ∫ Ω |u/δ|p, where δ(x) = dist (x, ∂Ω). The inequality is valid for a large family of domains including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant μp(Ω) = inf ◦ W 1,p(Ω) (∫ Ω |∇u|p / ∫ Ω |u/δ|p) and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth n-dimensional domains, μp(Ω) ≤ cp where cp = (1− 1 p ) is the one-dimensional Hardy constant. Moreover it is shown that μp(Ω) = cp for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for p = 2, it is proved that μ2(Ω) < c2 = 1/4 if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but μp = cp for convex domains. AMS subject classification number: 49R05, 35J70.

Asymptotics for minimal discrete energy on the sphere

Abstract: We investigate the energy of arrangements of N points on the surface of the unit sphere Sd in Rd+1 that interact through a power law potential V = 1/rs, where s > 0 and r is Euclidean distance. With Ed(s,N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N) for Ed(s, N) in the cases when 0 < s < d and 2 ≤ d < s. For s = d, we determine the precise asymptotic behavior of Ed(d,N) as N → ∞. As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy configuration, when s ≥ d ≥ 2. For the unit sphere in R3 (d = 2), we present two conjectures concerning the asymptotic expansion of E2(s,N) that relate to the zeta function ζL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of ζL(s) when 0 < s < 2 (the divergent case).

Self-similar measures and intersections of Cantor sets

Abstract: It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-ca Cantor set and Ca E (1, 1). We show that for any compact set K and for a.e. ag E (0, 1), the arithmetic sum of K and the middle-ae Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential su-premum, as the translation parameter t varies, of the dimension of the intersection of K + t with the middle-ca Cantor set. We also establish a new property of the infinite Bernoutlli convolutions lJ (the distributions of random series E' 0 ?A', where the signs are chosen independently with probabilities (p,1 p)). Let 1 < ql < q2 < 2. For p 2 near 2 and for a.e. A in some nonempty interval, IV is absolutely continuous 2 1 and its density is in LIl but not in Lq2. We also answer a question of Kahane concerning the Fourier transform of vl/2

On free actions, minimal flows, and a problem by ellis

Abstract: We exhibit natural classes of Polish topological groups G such that every continuous action of G on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup E(U) of the universal minimal G-flow U, being trivial, is not isomorphic with the greatest G-ambit. Further refining ouir construction, we obtain a Polish topological group G acting freely on the universal minimal flow U yet such that S(G) and E(U) are not isomorphic. We also display Polish topological groups acting effectively but not -freely orn their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations S(X) of an infinite set, equipped with the pointwise topology, and the unitary group U(X) of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle ?1 forms the universal minimal flow for the topological group Homeo + (?1) of orientation-preservinig haomeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.

Widths of Subgroups

Abstract: We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of any two elements of the collection is infinite and n is maximal possible. We define the width of a finite subgroup to be 0. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic 3-manifolds satisfy the k-plane property for some k.

Stability of multiple-pulse solutions

Abstract: In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of N-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the N-pulses. As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many N-pulses bifurcate for any fixed N > 1. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and N − 1 in the right half plane can be prescribed.

Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Abstract: Generalized Witt algebras, over a field F of characteristic 0, were defined by Kawamoto about 12 years ago Using different notations from Kawamoto’s, we give an essentially equivalent definition of generalized Witt algebras W = W (A,T, φ) over F , where the ingredients are an abelian group A, a vector space T over F , and a map φ : T × A → K which is linear in the first variable and additive in the second one In this paper, the derivations of any generalized Witt algebra W = W (A, T, φ), with the right kernel of φ being 0, are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group H2(W,F ) for any simple generalized Witt algebra is computed The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references

Values of Gaussian hypergeometric series

Abstract: Let p be prime and let GF (p) be the finite field with p elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions 2F1(x) =2 F1 ( φ, φ | x ) and 3F2(x) =3 F2 ( φ, φ, φ , | x ) , where φ and respectively are the quadratic and trivial characters of GF (p). For all but finitely many rational numbers x = λ, there exist two elliptic curves 2E1(λ) and 3E2(λ) for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes p for which 2F1(λ) is zero; however if λ 6= −1, 0, 12 or 2, then the set of such primes has density zero. In contrast, if λ 6= 0 or 1, then there are only finitely many primes p for which 3F2(λ) = 0. Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that 3E2(8) is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating 3F2(4) over every GF (p).

Kruzkov’s estimates for scalar conservation laws revisited

Abstract: We give a synthetic statement of Kružkov-type estimates for multidimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in h1/4 known for finite volume methods on unstructured grids. Les estimations de Kružkov pour les lois de conservation scalaires revisitées Résumé Nous donnons un énoncé synthétique des estimations de type de Kružkov pour les lois de conservation scalaires multidimensionnelles. Nous l’appliquons pour obtenir d’estimations nombreuses pour problèmes différents d’approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en h1/4 connue pour les méthodes de volumes finis sur des maillages non structurés.

The real field with convergent generalized power series

Abstract: We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable. This expansion is polynomially bounded.

Stability results on interpolation scales of quasi-Banach spaces and applications

Abstract: We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented.

Limit theorems for random transformations and processes in random environments

Abstract: I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

Hyperbolic groups and free constructions

Abstract: We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are virtually cyclic. The results are applied, in particular, to show that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.

On the classification of irregular surfaces of general type with nonbirational bicanonical map

Abstract: The present paper is devoted to the classification of irregular surfaces of general type with pg > 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3, and therefore pg = q = 3 and K2 = 6. Furthermore we obtain some results towards the classification of minimal surfaces with pg = q = 3. Such surfaces have 6 < Kz < 9, and we show that Kz = 6 if and only if S is the symmetric product of a curve of genus 3. We also classify the minimal surfaces with pg = q = 3 with a pencil of curves of genus 2, proving in particular that for those one has Kz = 8.

The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature

Abstract: In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in R' as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in C11(Q) if the given boundary data extends to a locally strictly convex c2 function ori Q. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.

On the conjectures of J. Thompson and O. Ore

Abstract: If G is a finite simple group of Lie type over a field containing more than 8 elements (for twisted groups 'Xn (ql) we require q > 8, except for 2B2(q2), 2G2(q2), and 2 F4 (q2), where we assume q2 > 8), then G is the square of some conjugacy class and consequently every element in G is a commutator.

On the subgroup structure of exceptional groups of Lie type

Abstract: We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q > 9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis

Abstract: The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete The 1-dimensional vibrating string satisfying wtt − wxx = 0 on the unit interval x ∈ (0, 1) is an infinite dimensional harmonic oscillator Consider the boundary conditions: at the left end x = 0, the string is fixed, while at the right end x = 1, a nonlinear boundary condition wx = αwt − βw3 t , α, β > 0, takes effect This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type un+1 = F (un), where F is the nonlinear reflection relation We say that the PDE system is chaotic if the mapping F is chaotic as an interval map Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities We then define a rotation number, following JP Keener [11], and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of MI Malkin [17] to determine the chaotic regime of α for the nonlinear reflection relation F , thereby rigorously proving chaos Nonchaotic cases for other values of α are also classified Such cases correspond to limit cycles in nonlinear second order ODEs Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics Received by the editors July 20, 1995 and, in revised form, October 16, 1996 1991 Mathematics Subject Classification Primary 35L05, 35L70, 58F39, 70L05 The first and third authors’ work was supported in part by NSF Grant DMS 9404380, Texas ARP Grant 010366-046, and Texas A&M University Interdisciplinary Research Initiative IRI 96-39 Work completed while the first author was on sabbatical leave at the Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, ROC The second author’s work was supported in part by Grant NSC 83-0208-M-007-003 from the National Council of Science of the Republic of China c ©1998 American Mathematical Society

Subelliptic harmonic maps

Abstract: We study a nonlinear harmonic map type system of subelliptic PDE In particular, we solve the Dirichlet problem with image contained in a convex ball

The possible orders of solutions of linear differential equations with polynomial coefficients

Abstract: We find specific inforrnation about the possible orders of transeendental solutions of equations of the form f(n) +Pn _l(z)f(n-l) ++Po(z)f0, where Po(Z),Pl(Z)? aPnL(Z) are polynomials with po(z) g 0. Several examples are given.

The Santaló-regions of a convex body

Abstract: Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52

Recognizing constant curvature discrete groups in dimension 3

Abstract: We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space IHI3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the 2-sphere.

Based algebras and standard bases for quasi-hereditary algebras

Abstract: Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly fullbased. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.