Classical results of Stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the Camassa Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly. Copyright 2000 Academic Press.
We consider a class of Hopf algebras having as an invariant a generalized Cartan matrix. For a Hopf algebra of this kind, we prove that it is finite dimensional if and only if its generalized Cartan matrix is actually a finite Cartan matrix (under some mild hypothesis). These results allow us to classify all the finite dimensional coradically graded pointed Hopf algebras whose coradical has odd prime dimension p. We also characterize coradically graded pointed Hopf algebras of order p4. Copyright 2000 Academic Press.
We show how various known results concerning the Barnes multiple zeta and gamma functions can be obtained as specializations of simple Features shared by a quite extensive class of Functions. The pertinent functions involve Laplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations. Both of these new approaches to the Barnes Functions give rise to novel integral representations. (C) 2000 Academic Press.
We generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace Beltrami operators on Riemannian manifolds. If 1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then 1nR/(n-1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong Yang's estimate asserts that 12d2, where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for 1 which unifies and sharpens Lichnerowicz and Zhong Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives. Copyright 2000 Academic Press.
We study amoebas associated with Laurent polynomials and obtain new results regarding the number and structure of the connected components of the complement of the amoeba. We also investigate the associated Laurent determinant. In the case of a hyperplane arrangement we perform explicit computations leading to a closed formula for the Laurent determinant. Copyright 2000 Academic Press.
The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G = SL(3, Z) x Z(3) associated to a non-trivial class in H-3(G, Z). (C) 2000 Academic Press.
Galois theory for normal unramified coverings of finite irregular graphs (which may have multiedges and loops) is developed. Using Galois theory we provide a construction of intermediate coverings which generalizes the classical Cayley and Schreier graph constructions. Three different analogues of Artin L-functions are attached to these coverings. These three types are based on vertex variables, edge variables, and path variables. Analogues of all the standard Artin L-functions results for number fields are proved here for all three types of L-functions. In particular, we obtain factorization formulas for the zeta functions introduced in Part I as a product of L-functions. It is shown that the path L-functions, which depend only on the rank of the graph, can be specialized to give the edge L-functions, and these in turn can be specialized to give the vertex L-functions. The method of Bass is used to show that Ihara type quadratic formulas hold for vertex L-functions. Finally, we use the theory to give examples of two regular graphs (without multiple edges or loops) having the same vertex zeta functions. These graphs are also isospectral but not isomorphic. Copyright 2000 Academic Press.
We introduce the notion of representable multicategory, which strands in the same relation to that of monoidal category as fibration does to contravariant pseudo-functor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe represent ability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a several object version of representable multicategories. (C) 2000 Academic Press.
Given a braided tensor *-category with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory we define a crossed product . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between and and closed subgroups of the Galois group Gal(/)=Aut() of , the latter being isomorphic to the compact group associated with by the duality theorem of Doplicher and Roberts. Denoting by the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of , the braiding of extends to a braiding of iff . Under this condition, has no non-trivial degenerate objects iff =. If the original category is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category ¯¯ is called the modular closure of since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2, ). If all simple objects of have dimension one the structure of the category can be clarified quite explicitly in terms of group cohomology. Copyright 2000 Academic Press.