Showing papers in "Transactions of the American Mathematical Society in 1996"

Shellable nonpure complexes and posets. II

Abstract: This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327. 8. Interval-generated lattices and dominance order In this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets. Let F = {I1, I2, . . . , In} be a family of intervals of integers, by which is meant sets of the form [a, b] = {a, a + 1, . . . , b}, a ≤ b. We assume that there are no containments among these intervals, and that they are ordered so that their left and right endpoints are increasing. Let L(F) be the lattice of all sets that are unions of subfamilies of F , ordered by inclusion. Such interval-generated lattices L(F) were introduced and studied by Greene [G]. Define an edge-labeling λ of L(F) as follows. If A → B is a covering and a = max(B \A), then λ(A→ B) = { −a, if (a+ 1) ∈ A and a is the left endpoint of some I ∈ F ,

A Multivariate Faa di Bruno Formula with Applications

Abstract: A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.

On the Well-Posedness of the Kirchhoff String

Abstract: Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation

Prox-regular functions in variational analysis

Abstract: The class of prox-regular functions covers all lsc, proper, convex functions, lower-C2 functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not

Nonsmooth sequential analysis in Asplund spaces

Abstract: We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Frechet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued di erential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

Invariants of piecewise-linear 3-manifolds

Abstract: In this paper we develop a theory for constructing an invariant of closed oriented 3-manifolds, given a certain type of Hopf algebra. Examples are given by a quantised enveloping algebra of a semisimple Lie algebra, or by a semisimple involutory Hopf algebra. The invariant is defined by a state sum model on a triangulation. In some cases, the invariant is the partition function of a topological quantum field theory.

Extremal functions for Moser’s inequality

Abstract: Let Q be a bounded smooth domain in RI, and u(x) a C1 function with compact support in Q. Moser's inequality states that there is a constant co, depending only on the dimension n, such that 1 wn1 . n_ j ee n-U dx < co, where IQI is the Lebesgue measure of Q, and Wn-1 the surface area of the unit ball in Rn. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the sup{ j nw U dx: u E WJ' , 11VuH1n < 1} is attained. Earlier results include Carleson-Chang (1986, Q is a ball in any dimension) and Flucher (1992, Q is any domain in 2-dimensions).

Competitive exclusion and coexistence for competitive systems on ordered Banach spaces

Abstract: The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.

Defect zero p-blocks for finite simple groups

Abstract: We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p-blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (nee the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 < p < 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p-blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p-modularly irreducible representations of Sn, for almost all n. © 1996 American Mathematical Society.

Exact controllability and stabilizability of the Korteweg-de Vries equation

Abstract: In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) atu + uaxu + e3 u = f on the interval 0 0, with periodic boundary conditions (ii) &au(27r, t) = &ku(O, t), k = 0, 1, 2, where the distributed control f _ f (x, t) is restricted so that the "volume" f u(x, t)dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f is allowed to act on the whole spatial domain (0, 27r), it is shown that the system is globally exactly controllable, i.e., for given T > 0 and functions q(x), +(x) with the same "volume", one can alway find a control f so that the system (i)-(ii) has a solution u(x, t) satisfying u(x, 0) 0(x), u(x, T) (x). If the control f is allowed to act on only a small subset of the domain (0, 27r), then the same result still holds if the initial and terminal states, b and 0, have small "amplitude" in a certain sense. In the case of closed loop control, the distributed control f is assumed to be generated by a linear feedback law conserving the "volume" while monotonically reducing f u(x, t)2dx. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

Quadratic forms for the 1-D semilinear Schrödinger equation

Abstract: This paper is concerned with 1-D quadratic semilinear Schr6dinger equations. We study local well posedness in classical Sobolev space HS of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.

An Index Theory For Quantum Dynamical Semigroups

Abstract: W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

Invertibility preserving linear maps on l(x)

Abstract: For Banach spaces X and Y , we show that every unital bijective invertibility preserving linear map between L(X) and L(Y ) is a Jordan isomorphism. The same conclusion holds for maps between CI + K(X) and CI +K(Y ).

Topological centers of certain dual algebras

Abstract: Let A be a Banach algebra with a bounded approximate identity. Let Z1 and Z2 be, respectively, the topological centers of the algebras A** and (A*A)*. In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras L1 (G) and A(G), we study the sets Z1, Z2, the relations between them and with several other subspaces of A** or A*.

Duality and Polynomial Testing of Tree Homomorphisms

Abstract: Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

The ergodic theory of discrete isometry groups on manifolds of variable negative curvature

Abstract: This paper studies the ergodic theory at infinity of an arbitrary discrete isometry group Γ acting on any Hadamard manifold H of pinched variable negative curvature. Most of the results obtained by Sullivan in the constant curvature case are generalized to the case of variable curvature. We describe connections between measures supported on the limit set of Γ, dynamics of the geodesic flow and the geometry of M = H/Γ. We explore the relationship between the growth exponent of the group, the Hausdorff dimension of the limit set and the topological entropy of the geodesic flow. The equivalence of various descriptions of an analogue of the Hopf dichotomy is proved. As applications, we settle a question of J. Feldman and M. Ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points. At the end of this paper, we apply our results to the study of some rigidity problems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with infinite covolume will be published in subsequent papers by the author. Notations • H is a Hadamard manifold with pinched sectional curvature −K 2 ≤ K ≤ −K 1 , K2 ≥ K1 > 0. • Γ is a (non-elementary) discrete isometry group acting on H freely and properly discontinuously (which is called Fuchsian). • ∂H is the ideal boundary of H. • M = H/Γ is a complete Riemannian manifold. • SH (or SM) is the unit tangent bundle. • g is the geodesic flow on SH (or SM). • v(t) is the unique geodesic in SH (resp. SM) with the initial velocity v(0) = v ∈ SH (resp. SM). • v(∞) (resp. v(−∞)) is the asymptotic class of the geodesic v(t) (resp. v(−t)). • L(Γ) is the limit set of Γ on ∂H. • L(Γ) is the radical limit set of Γ in L(Γ). • Ω = Ω(Γ) is the nonwandering set of the geodesic flow on SM Received by the editors May 7, 1995. 1991 Mathematics Subject Classification. Primary 58F17; Secondary 58F11, 58F15, 20H10. Research at MSRI supported by NSF Grant #DMS 8505550. Also partially supported by NSF Grant #DMS 9403870 and SFB 170. c ©1996 American Mathematical Society

Berezin Quantization and Reproducing Kernels on Complex Domains

Abstract: Let Ω be a non-compact complex manifold of dimension n, ω = ∂∂Ψ a Kähler form on Ω, and Kα(x, y) the reproducing kernel for the Bergman space Aα of all analytic functions on Ω square-integrable against the measure e−αΨ|ωn|. Under the condition Kα(x, x) = λαe αΨ(x) F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on (Ω, ω) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = Cn and Ω a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → +∞. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in Cn. Along the way, we also fix two gaps in Berezin’s original paper, and discuss, for Ω a domain in Cn, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure |ωn|.

Presentation and central extensions of mapping class groups

Abstract: We give a presentation of the mapping class groupM of a (possibly bounded) surface, considering either all twists or just non-separating twists as generators. We also study certain central extensions of M. One of them plays a key role in studying TQFT functors, namely the mapping class group of a p1-structure surface. We give a presentation of this extension.

Foxby duality and Gorenstein injective and projective modules

Abstract: In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appropriate to call the modules with G-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611–633 and were shown to have properties predicted by Auslander’s results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of G-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local CohenMacaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite G-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby’s duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators

Abstract: New unique characterization results for the potential V(x) in connection with Schrodinger operators on R and on the half-line [0,∞)are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrodinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrodinger operators with confining potentials on the entire real line.

Cohomology of the complement of a free divisor

Abstract: We prove that if D is a "strongly quasihomogeneous" free divisor in the Stein manifold X, and U is its complement, then the de Rham cohomology of U can be computed as the cohomology of the complex of meromorphic differential forms on X with logarithmic poles along D, with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

Hyponormality and spectra of Toeplitz operators

Abstract: This paper concerns algebraic and spectral properties of Toeplitz operators Tφ, on the Hardy space H2(T), under certain assumptions concerning the symbols φ ∈ L∞(T). Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at Tφ, for each quasicontinuous φ. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

Composition operators between Bergman and Hardy spaces

Abstract: We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.

Combinatorial $B_n$-analogues of Schubert polynomials

Abstract: Combinatorial $B_n$-analogues of Schubert polynomials and corresponding symmetric functions are constructed from an exponential solution of the $B_n$-Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.

The dynamical properties of Penrose tilings

Abstract: The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of R2 by translation. We show that this action is an almost 1:1 extension of a minimal R2 action by rotations on T4, i.e., it is an R2 generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on T4. The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.

Curvature invariants, differential operators and local homogeneity

Abstract: We first prove that a Riemannian manifold (M, g) with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold (M, g) whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

Abstract: A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan’s equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for secondorder Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

Strong laws for L- and U-statistics

Abstract: Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems; of Hoeffding and of Helmers for lid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.