Composition of Binary Quadratic Forms over Number Fields

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.Comment: 16 pages, a small change made in signs in the bijection (because of a further use of the result

Clifford Algebraic Remark on the Mandelbrot Set of Two--Component Number Systems

We investigate with the help of Clifford algebraic methods the Mandelbrot set over arbitrary two-component number systems. The complex numbers are regarded as operator spinors in D\times spin(2) resp. spin(2). The thereby induced (pseudo) normforms and traces are not the usual ones. A multi quadratic set is obtained in the hyperbolic case contrary to [1]. In the hyperbolic case a breakdown of this simple dynamics takes place.Comment: LaTeX, 27 pages, 6 fig. with psfig include

On generating series of finitely presented operads

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.Comment: Minor changes; references to recent articles by Berele and by Belov, Bokut, Rowen, and Yu are adde

On the geometry of a class of N-qubit entanglement monotones

A family of N-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the three-tangle, and some of the four, five and N-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form ${\bf C}^{2^N}\simeq{\bf C}^L\otimes{\bf C}^l$ with $L=2^{N-n}\geq l=2^n$. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes in ${\bf C}^L$ that can be realized as the zero locus of quadratic polinomials in the complex projective space of suitable dimension via the Plucker embedding. The invariants are neatly expressed in terms of the Plucker coordinates of the Grassmannian.Comment: 7 pages RevTex, Submitted to Physical Review