Showing papers in "Transactions of the American Mathematical Society in 1994"

Measures of chaos and a spectral decomposition of dynamical systems on the interval

Abstract: Let f: [0, 1] -+ [0, 1] be continuous. For x, y E [0, 1], the upper and lower (distance) distribution functions, Fx*y and Fxy, are defined for any t > 0 as the lim sup and lim inf as n -+ oc of the average number of times that the distance Ifi (x) fi(y)I between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system ?(f) of lower distribution functions which is characterized by the following properties: (1) The elements of ?(f) are mutually incomparable; (2) for any F E ?(f), there is a perfect set PF #8 0 such that FUV = F and FUV 1 for any distinct u, v E PF; (3) if S is a scrambled set for f, then there are F, G in ?(f) and a decomposition S = SF U SG (SG may be empty) such that FUV > F if u, v E SF and FUV > G if u, v E SG . Our principal results are: (1) If f has positive topological entropy, then ?(f) is nonempty and finite, and any F E ?(f) is zero on an interval [0, e], where e > 0 (and hence any PF is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then ?(f) = {F} where F 1. It follows that the spectrum of f provides a measure of the degree of chaos of f . In addition, a useful numerical measure is the largest of the numbers fo(1 F(t))dt, where F E ?(f).

Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity

Abstract: We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) , like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Foures and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.

On superquadratic elliptic systems

Abstract: In this article we study the existence of solutions for the elliptic system \( - \Updelta u = \frac{\partial H}{\partial v}\left( {u,v,x} \right)\,{\text{in}}\Upomega , \) \( - \Updelta v = \frac{\partial H}{\partial v}\left( {u,v,x} \right)\,{\text{in}}\Upomega , \) \( u = 0,v = 0\,{\text{on}}\,\partial \Upomega \)where \( \Upomega \) is a bounded open subset of \( {\mathbb{R}}^{N} \) with smooth boundary \( \partial \Upomega \) and the function H: \( {\mathbb{R}}^{2} \times \overline{\Upomega } \to {\mathbb{R}} \), is of class C 1.

Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )

Abstract: A complete characterization is given of closed shift-invariant subspaces of L2(Rd) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.

Multiplier Hopf algebras

Abstract: In this paper we generalize the notion of Hopf algebra. We consider an algebra A , with or without identity, and a homomorphism A from A to the multiplier algebra M(A ® A) of A ® A . We impose certain conditions on A (such as coassociativity). Then we call the pair {A, A) a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where (Af)(s, t) = f(st) with s, t £ G and f € A . We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a *-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a *-algebra.

Gromov’s compactness theorem for pseudo holomorphic curves

Abstract: We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.

Braid groups and left distributive operations

Abstract: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin's braid group B. . Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator ai is preponderant over all ak with k > i . This ordering is a linear ordering. The first goal of the present paper is to give a proof of the following result, which had been conjectured for several years: Theorem. There is an effective algorithm for deciding whether a given identity is or is not a consequence of the left distributivity identity x(yz) = (xy)(xz). Until recently this question has had a rather unusual status. Conditional solutions were given independently in [6] and [24], where the decision problem was reduced to a specific algebraic hypothesis, one which had been shown by Richard Laver to be a consequence of a very strong set-theoretical axiom, of a type which certainly cannot be derived from the usual axioms of set theory, namely, a large cardinal axiom. The question as to whether a strong axiom of this type was actually needed remained open. Opinions were in fact divided: a connection between large cardinals and a purely finitistic problem of this caliber would seem paradoxical, but it is well known that problems of a combinatorial type can embody surprisingly strong proof principles (see for instance [28]), and some work on free distributive structures has shown that they do give rise to intrinsically complex objects, typically nonprimitive recursive ones (cf. [12, 13]). We will show that in the present case a solution which is purely algebraic in terms of the methods employed and the spirit of the argument can in fact be given. In particular, no unusual set-theoretical axioms are required for this argument. The decision method described in [6] was fully effective, but the proof of the correctness of the algorithm amounted to a direct invocation of this specific algebraic hypothesis which followed from a large cardinal axiom, with no hint of a direct proof. We shall refine this decision method below, introducing uniqueness at each step of the process, and the correctness will then be seen to follow very naturally. Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 20F36, 1 7A30, 1 7A50.

WAVELETS OF MULTIPLICITY r

Abstract: A multiresolution approximation (Km)m€Z of L2(R) is of multi- plicity r > 0 if there are r functions x, ... , r whose translates form a Riesz basis for V$ . In the general theory we derive necessary and sufficient conditions for the translates of x, ... , r, y/x, ... , y/r to form a Riesz basis for V\ . The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V0 and its orthogonal complement W0 in Vx . The general theory is applied in the construction of spline wavelets with mul- tiple knots. Algorithms for the construction of these wavelets for some special cases are given.

The profile near blowup time for solution of the heat equation with a nonlinear boundary condition

Abstract: This paper studies the blowup profile near the blowup time for the heat equation ut = Au with the nonlinear boundary condition un = uP on Oi x [0, T). Under certain assumptions, the exact rate of the blowup is established. It is also proved that the blowup will not occur in the interior of the domain. The asymptotic behavior near the blowup point is also studied.

The structure of hyperfinite Borel equivalence relations

Abstract: We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.

Approximation properties for group *-algebras and group von Neumann algebras

Abstract: Let G be a locally compact group, let C,' (G) (resp. VN(G)) be the C*-algebra (resp. the von Neumann algebra) associated with the left regular representation I of G, let A(G) be the Fourier algebra of G, and let MOA(G) be the set of completely bounded multipliers of A(G) . With the completely bounded norm, MOA(G) is a dual space, and we say that G has the approximation property (AP) if there is a net {ua} of functions in A(G) (with compact support) such that ua -a 1 in the associated weak *-topology. In particular, G has the AP if G is weakly amenable (< A(G) has an approximate identity that is bounded in the completely bounded norm). For a discrete group F, we show that F has the AP X C,' (F) has the slice map property for subspaces of any C*-algebra X VN(JT) has the slice map property for a-weakly closed subspaces of any von Neumann algebra (Property Se) . The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group extensions. We also obtain some results concerning crossed products. For example, we show that the crossed product M ?a G of a von Neumann algebra M with Property S, by a group G with the AP also has Property S, .

Some cubic modular identities of Ramanujan

Abstract: There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is $(\sum_{n,m=-\infty}^{\infty} q^{n^2+nm+m^2})³ = (\sum_{n,m=-\infty}^{\infty} ω^{n-m}q^{n²+nm+m²})³ + (\sum_{n,m=-\infty}^{\infty} q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³.$ Here $ω = exp(2π i/3).$ In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.

The structure of a random graph at the point of the phase transition

Abstract: Consider the random graph models G(n, #edges = M) and G(n, Prob(edge) = p) with M = M(n) = (1 + )n-1/3)n/2 and p = p(n) = (1 +An-1/3)/n . For 1 > -1 define an i-component of a random graph as a component which has exactly I more edges than vertices. Call an i-component with I > 1 a complex component. For both models, we show that when A is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order n2/3, and the maximum value of I is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order n2/3, and deletion of all suspended trees results in a "smoothed" graph of size of order n /3, with the maximum vertex degree 3. The total number of branching vertices, i.e., of degree 3, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a 3-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n /3. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdos and Renyi back in 1960. It is proven that there exists p(A), the limiting planarity probability, with 0 < p(A) < 1, p(-oo) = 1, p(oo) = 0. In particular, G(n, M) (G(n, p), resp.) is almost surely nonplanar iff (M n/2)n-213 -* 00 ((np I)n-1/3) -+ oo, resp.).

Intersection bodies and the Busemann-Petty problem

Abstract: It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in ¿-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d < 4 , and that suitably smooth axis-convex bodies of revolution are intersection bodies when d < 4. These results show that the Busemann-Petty problem has a negative answer for d > 5 and a positive answer for d = 3 and d = 4 when the body with smaller sections is a body of revolution.

Holomorphic motions and Teichmüller spaces

Abstract: We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane C extends to a holomorphic motion of C. As a consequence we prove that every holomorphic map of the unit disc into Teichmuller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmuller metric.

On a two-dimensional elliptic problem with large exponent in nonlinearity

Abstract: A semilinear elliptic equation on a bounded domain in R2 with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the con- strained minimizing problem possesses nice asymptotic behavior as the nonlin- ear exponent, serving as a parameter, gets large. We shall prove that cp , the minimum of energy functional with the nonlinear exponent equal to p , is like (&Ke)lf2p~^2 as p tends to infinity. Using this result, we shall prove that the variational solutions remain bound- ed uniformly in p . As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of -A. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.

On the discriminant of a hyperelliptic curve

Abstract: The minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves. Over number fields this leads to a higher genus version of Szpiro's Conjecture. Analytically, the discriminant is shown to be related to Siegel modular forms of higher degree.

Singular polynomials for finite reflection groups

Abstract: The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of G that occur in this kernel.

Centered bodies and dual mixed volumes

Abstract: We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in R\" by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central (n l)-slices. It implies Lutwak's affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry. 0. Introduction Let R\" be the «-dimensional Euclidean space. A convex body K c R\" is a compact convex subset with nonempty interior. Let 3? denote the set of convex bodies in R\", and let 5£e denote the symmetric convex bodies with respect to the origin. As Minkowski noted in the case of R3, the (n — 1)dimensional volume of the image of the orthogonal projection of K g 5? onto a (n l)-subspace is the support function of another convex body UK, called the projection body of K. Projection bodies have received considerable attention in recent years (see, for example, [4, 11, 16, 24, 36, 38, 39, 45, 47]). Instead of considering the projection of a convex body, Busemann [6] proved that the (n l)-dimensional volume of the intersection of K g 3£e with a (n 1)subspace of Rn is the radial function of another convex body IK, called by Lutwak the intersection body of K. The intersection body, which may be viewed as the dual of the projection body, is an important tool in understanding the sections of convex bodies. Lutwak studied intersection bodies by bis dual mixed volumes. He denned intersection bodies for star bodies and related them to the spherical Radon transform. Many analogous theorems to those on projection bodies and mixed volumes were proved, see [25-28]. A slight extension of the definition of intersection bodies was given by using measures (see [18]). It states Received by the editors May 6,1992 and, in revised form, December IS, 1993. 1991 Mathematics Subject Classification. Primary 52A20, 52A40, 53C45.

Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis

Abstract: In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.

On the generalized Benjamin-Ono equation

Abstract: We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation tu + ukdxU = 0, k e , in Sobolev spaces HS(R). For small data and higher nonlinearities (k > 2) new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrodinger equations (or systems) of the form atU -i2U + P(u, OxU, ui, axi) = 0.

Amenable actions of groups

Abstract: The equivalence between different characterizations of amenable actions of a locally compact group is proved. In particular, this answers a question raised by R. J. Zimmer in 1977.

Mean value inequalities in Hilbert space

Abstract: We establish a new mean value theorem applicable to lower semicontinuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance, and generalized solutions to partial differential equations. RtSUMt. On etablit un nouveau theoreme de la valeur moyenne qui s'applique aux fonctions semicontinues inferieurement sur un espace de Hilbert. On deduit plusieurs consequences du resultat portant, par exemple, sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles.

Müntz systems and orthogonal Müntz-Legendre polynomials

Abstract: The Muntz-Legendre polynomials arise by orthogonalizing the Muntz system {xIo, x'l~, . .. } with respect to Lebesgue measure on [0, 1] . In this paper, differential and integral recurrence formulae for the Muntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Muntz space on [0, 1], which implies that in this case the orthogonal Muntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1) . Some inequalities for Muntz polynomials are also investigated, most notably, a sharp L2 Markov inequality is proved.

Geometric consequences of extremal behavior in a theorem of Macaulay

Abstract: F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded k-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay's theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zeroschemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve.

Projective resolutions and Poincaré duality complexes

Abstract: Let k be a field lof characteristic p > 0 and let G be a finite group. We investigate the structure of the cohomology ring H*(G, k) in relation to certain spectral sequences determined by systems of homogeneous parameters for the cohomology ring. Each system of homogeneous parameters is associated to a complex of projective fcG-modules which is homotopically equivalent to a Poincaré duality Complex. The initial differentials in the hypercohomology spectral sequence of the complex are multiplications by the parameters, while the higher differentials are matric Massey products. If the cohomology ring is Cohen-Macaulay, then the duality of the complex assures that the Poincaré series for the cohomology satisfies a certain functional equation. The structure of the complex also implies the existence of cohomology classes which are in relatively large degrees but are not in the ideal generated by the parameters. We consider several other questions concerned with the minimal projective resolutions and the convergence of the spectral sequence.

Graphs with the circuit cover property

Abstract: A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover