Product (mathematics)

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

Zonoids with minimal volume-product- a new proof

Abstract: A new and simple proof of the following result is given: The product of the volumes of a symmetric zonoid A in RI and of its polar body is minimal if and only if A is the Minkowski sum of n segments.

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Abstract: Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory ${\mathcal M}(G,R)$ of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that ${\mathcal M}(G,R)$ is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in ${\mathcal M}(G,R).$ We prove that the Krull--Schmidt theorem holds in many cases.

Fast backtrack-free product configuration using a precompiled solution space representation

Abstract: In this paper we describe a two-phase approach to interactive product configuration. In the first phase, a compressed symbolic representation of the set of valid configurations (the solution space) is compiled offline. In the second phase, this representation is embedded in an online configurator and utilized for fast, complete, and backtrack-free interactive product configuration. The main advantage of our approach compared to online search-based approaches is that we avoid searching for valid solutions in each iteration of the interactive configuration process. The computationally hard part of the problem is fully solved in the offline phase given that the produced symbolic representation is small. The employed symbolic representation is Binary Decision Diagrams (BDDs). More than a decade of research in formal verification has shown that BDDs often compactly encode formal models of systems encountered in practice. To our experience this is also the case for product models. Often the compiled BDD is small enough to be embedded directly in hardware. Our research has led to the establishment of a spin-off company called Configit Software A/S. Configit has developed software for writing product models in a strongly typed language and has patented a particularly efficient symbolic representation called Virtual Tables.