Showing papers in "Duke Mathematical Journal in 1998"

On fluctuations of eigenvalues of random Hermitian matrices

Abstract: 1,156,645. Ornamenting. E. I. DU PONT DE NEMOURS & CO. 14 June, 1967 [15 June, 1966], No. 27377/67. Heading B6G. [Also in Divisions B5 and C3] Decorative articles are obtained by printing and laminating surface-receptive polyvinyl fluoride film with an adhesive/printing ink vehicle composition comprising an organic solvent solution of (1) a vinyl polymer having attached to the main chain thereof monovalent radicals of the formula wherein Z is a tetravalent aliphatic hydrocarbon radical of 2 or 3 carbon atoms, R is H or C 1 -C 8 alkyl, R 1 is H, OH or C 1 -C 8 alkyl and R 11 is H or C 1 -C 8 non-tertiary alkyl or C 1 -C 8 #- hydroxyalkyl, the amino nitrogen content of said vinyl polymer being at least 0A01%, based on the weight of the vinyl polymer, and (2) an epoxy ether or epoxy thioether reaction product characterized in that the precursor epoxy ether (or thioether) is partially reacted with an ethylenically unsaturated compound of the formula wherein R 3 is H or carboxyl or C 1 -C 4 alkyl, R 4 is an alkyleneoxy, alkyleneoxy carbonyl, divalent aromatic, other divalent aliphatic or divalent alicyclic radical, or when A is R 4 may be a single bond, A is -O-, -TH-, and R 7 is H or a carboxy, carboalkoxy, hydroxy, sulphhydryl, sulphonyl, amino, non-hindering alkyl or non-hindering alkyl-aryl group, said reaction product having unhindered ethylenic unsaturation, an epoxide equivalent of from 50 to 3000, and a bromine number of from 0A1 to 100; said components (1) and (2) being present in a ratio, by weight, of from 1 : 99 to 99 : 1. Preferred vinyl polymers contain 80-97% by weight of units of (meth)-acrylate esters of C 1 - C 8 aliphatic alcohols, and have the amino-ester groups present in amino-ethyl, amino-propyl or amino-hydroxypropyl methacrylate units, the amino group generally being introduced by modification of a suitable preformed polymer (e.g. by imination of a polymer containing carboxylic acid groups). The epoxy-(thio)- ether reaction product is generally derived from an epoxy compound containing on average more than one epoxy group per molecule; the preferred precursor epoxy-compounds being diphenylol-propane diglycidyl ether resins (and corresponding thio - derivatives). The other component of the epoxy reaction product is preferably an olefinically unsaturated carboxylic acid. The polyvinyl fluoride film is rendered surface receptive by chemical, electrical or flame-treatment methods. It is then printed with the aforesaid solution and laminated to a substrate composed of a thermosetting resin, optionally reinforced with, e.g. glass fibres. The laminating/printing solution may contain a pigment, e.g. a brown pigment or titanium dioxide. In the examples, the polyvinyl fluoride film is printed with the solution in a pattern by stamping a series of \" X's \" into a template, spraying the solution over the X's and drying; after laminating with a polyester/ styrene/glass fibre mat there are obtained exterior trim panels suitable for use in station wagons.

Description of the $n$-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type, I: Integration of the Lamé equations

Abstract: The classical problem of description of n-Orthogonal Curvilinear Coordinate System in flat Euclidean space is solved by the Inverse Scattering Method. The developed method allows to describe Hamiltonian and semi-Hamiltonian Integrable systems of Hydrodynamic type.

Bonnet pairs and isothermic surfaces

Abstract: In this note we classify all Bonnet pairs on a simply connected domain. Our main intent was to apply what we call a quaternionic function theory to a concrete problem in differential geometry. The ideas are simple: conformal immersions into quaternions or imaginary quaternions take the place of chart maps for a Riemann surface. Starting from a reference immersion we construct all conformal immersions of a given (simply connected) Riemann surface (up to translational periods) by spin transformations. With this viewpoint in mind we discuss how to construct all Bonnet pairs on a simply connected domain from isothermic surfaces and vice versa. Isothermic surfaces are solutions to a certain soliton equation and thus a simple dimension count tells us that we obtain Bonnet pairs which are not part of any of the classical Bonnet families. The corresponcence between Bonnet pairs and isothermic surfaces is explicit and to each isothermic surface we obtain a 4-parameter family of Bonnet pairs.

Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

Abstract: We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables $x_1,x_2,...$ in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.

Affine Hecke algebras and raising operators for Macdonald polynomials

Abstract: We introduce certain raising and lowering operators for Macdonald polynomials (of type $A_{n-1}$) by means of Dunkl operators. The raising operators we discuss are a natural $q$-analogue of raising operators for Jack polynomials introduced by L.Lapointe and L.Vinet. As an application we prove the integrality of double Kostka coefficients. Double analog of the multinomial coefficients are introduced.

Commutators of free random variables

Abstract: Let A be a unital C∗-algebra, given together with a specified state φ : A → C. Consider two selfadjoint elements a, b of A, which are free with respect to φ (in the sense of the free probability theory of Voiculescu). Let us denote c := i(ab− ba), where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper [12].

Une suite exacte en l2-cohomologie

Abstract: © Séminaire de Théorie spectrale et géométrie (Grenoble), 1995-1996, tous droits réservés. L’accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: Intermediate solutions

Abstract: where A = cy=, 5 is the Laplace operator, R is a bounded smooth domain in R”, E > 0 is a constant, and the exponent p satisfies 1 < p < 2 for n 2 3 and 1 < p < 0;) for n = 2. We are especially interested in the properties of solutions of (1.1) as E tends to 0. In particular, we shall establish the existence of a “spike-layer’’ solution, and determine the location of the peak as well as the profile of the spike.

Flows on homogeneous spaces and Diophantine properties of matrices

Abstract: We generalize the notions of badly approximable (resp. singular) systems of m linear forms in n variables, and relate these generalizations to certain bounded (resp. divergent) trajectories in the space of lattices in Rm+n.

The identity component of the isometry group of a compact Lorentz manifold

Abstract: This result may be compared with a Theorem of E. Ghys [Ghy] (see also [Bel]), asserting a similar conclusion, but assuming that M has dimension 3, and that the action is just volume preserving and locally free. The statement there, is that the action of AG may be extended to an action of a finite cover of PSL(2,R), or to an action of the solvable 3-dimensional Lie group SOL. Here we have another motivation. We want to understand the structure of Lie groups acting isometrically on compact Lorentz manifold. The first results in the subject are due to [Zim] and [Gro]. A “final” result is due to [A-S] and [Zeg1], independently. Necessary and sufficient conditions were given in order that a Lie group acts isometrically (and locally faithfully) on a compact Lorentz manifold. Note however, that if a group acts in such a fashion, then its subgroups also act in the same way. For instance, all known examples of isometric actions of AG are obtained by viewing it as a subgroup of SL(2,R). So a natural question is: what are the maximal (connected) Lie groups acting isometrically on a compact Lorentz manifold? Equivalently:

Chord diagram invariants of tangles and graphs

Abstract: The notion of a chord diagram emerged from Vassiliev's work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This knot invariant establishes an isomorphism of a projective limit of algebras generated by the Vassiliev equivalence classes of knots onto the algebra of chord diagrams. Kontsevich originally deened his invariant of knots via a multiple integral given by an explicit but complicated analytic expression. This expression, however beautiful, does not reveal the combinatorial nature of the invariant. (A similar situation would occur if the linking number of knots were introduced via the Gauss integral formula without a combinatorial calculation). A combinatorial reformulation of the Kontsevich integral appeared in the works of Bar-Natan BN94], Cartier Car93], Le and Murakami LM93], Piunikhin Piu95] (see also Kas95, Chapter XX]). On the algebraic side, this reformulation uses the notions of braided and innnitesimal symmetric categories as well as the notion of an associator introduced by Drinfeld Dri89] in his study of quasitriangular quasi-Hopf algebras. On the geometric side, one uses categories of tangles, as introduced by Yetter and Turaev (see Tur94]). Note also that a counterpart of the Kontsevich knot invariant in the theory of braids was discovered earlier by Kohno Koh85] who considered an algebraic version of chord diagrams. In this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we x a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coeecients in R, cf. Section 2. The set of morphisms in T (R) has a canonical ltration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I m+1 are exactly the Vassiliev invariants of degree m for framed oriented tangles. Completing T (R) at the ideal I, we obtain the pro-unipotent completion b T (R) = lim ?m T (R)=I m+1. Our main result (Corollary 2.5) states that, if R contains the eld Q of rational numbers, then b T (R) is isomorphic to a suitable completion b A(R) of the category …