Skein theory for the ADE planar algebras

We give generators and relations for the planar algebras corresponding to $ADE$ subfactors. We also give a basis and an algorithm to express an arbitrary diagram as a linear combination of these basis diagrams

On the comparison between compound louvered-vortex generator fins and X-shaped louvered fins

A recent evolution in heat exchanger design is the use of compound designs. One of the designs under study is a combination between a louvered fin and vortex generators. Several possible placements of the vortex generators are studied. These compound designs are compared with the X-shaped louvered fin, which maximizes the louvered area. It is shown that the X-shaped louvered fin exhibits the same heat transfer enhancement mechanism as the compound design, with respect to the rectangular louvered fin. The X-shaped louvered fin outperforms all of the compound designs

Group Theoretical Structure of N=1N=1 and N=2N=2 Two-Form Supegravity

We clarifies the group theoretical structure of $N=1$ and $N=2$ two-form supergravity, which is classically equivalent to the Einstein supergravity. $N=1$ and $N=2$ two-form supergravity theories can be formulated as gauge theories. By introducing two Grassmann variables $\theta^A$ ($A=1,2$), we construct the explicit representations of the generators $Q^i$ of the gauge group, which makes to express any product of the generators as a linear combination of the generators $Q^iQ^j=\sum_k f^{ij}_k Q^k$. By using the expression and the tensor product representation, we explain how to construct finite-dimensional representations of the gauge groups. Based on these representations, we construct the Lagrangeans of $N=1$ and $N=2$ two-form supergravity theories.Comment: Latex file, 15

Compatible quadratic Poisson brackets related to a family of elliptic curves

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained.Comment: 17 pages, Latex, major change

Linear combinations of generators in multiplicatively invariant spaces

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega,\mathcal{H})$ that are invariant under multiplications of (some) functions in $L^{\infty}(\Omega)$. In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in $L^2(\mathbb{R}^d)$.Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematic