Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Representations of Lie groups

Abstract: With ξ = (ξ1,…,ξ d ), {ξ i } a basis for the Lie algebra ℊ, a basis for the universal enveloping algebra u(ℊ) is given by $$\left[\kern-0.15em\left[ \text{n} \right]\kern-0.15em\right] = \xi ^n = \xi _1^{n1} \cdots \xi _d^{nd}$$ where the product is ordered, since the ξ i do not commute in general. As we saw for Appell systems, it is natural to look at generating functions to see how multiplication by the basis elements ξ i on u looks. We have $$\sum\limits_{n \geqslant 0} {\frac{{A^n }} {{n!}}} \,\xi ^n = \sum {\frac{{\left( {A_1 \xi _1 } \right)^{n_1 } }} {{n_1 !}} \ldots \frac{{\left( {A_d \xi _d } \right)^{n_d } }} {{n_d !}} = } \,e^{A_1 \xi _1 } \ldots e^{A_d \xi _d }$$ (0.1) This is an element of the group, as it is a product of the one-parameter subgroups generated by the basis elements.

Floer homology of cotangent bundles and the loop product

Abstract: We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.

Methods, systems and computer program products for smart card product management

Abstract: Methods, systems and computer program products are provided for managing a smart card product by providing a plurality of generic definitions, at least a portion of which have a predefined relationship to others of the generic definitions, so as to provide a hierarchy of generic definitions. Generic definitions are selected from the plurality of generic definitions and associated with an instance of a card product definition so as to define characteristics of the smart card product associated with the instance of the card product definition. The selected generic definitions are populated with data associated with the smart card product so as to provide a hierarchy of instances of the generic definitions which define the characteristics of the smart card product. The smart card product is managed utilizing the hierarchy of instances of the generic definitions so as to provide the smart card product having the defined characteristics. Systems for managing smart card products utilizing instances of definitions from the hierarchy of definitions are also provided.