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 $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner Property

Abstract: A ranked partially ordered set is said to be Sperner if it has no antichain bigger than its largest rank. A necessary and sufficient condition for a ranked partially ordered set to be rank symmetric, rank unimodal and strongly Sperner is presented. This condition involves representations of $\mathfrak{sl} ( 2,\mathbb{C} )$. It is used to provide a new, short proof that this combination of properties is preserved under the product operation. The sufficient part of this condition is also used to provide new, simpler proofs that certain combinatorially interesting partially ordered sets are rank symmetric, rank unimodal and strongly Sperner.

Computation of matrix chain products. Part II

Abstract: This paper considers the computation of matrix chain products of the form $M_1 \times M_2 \times \cdots \times M_{n - 1} $. If the matrices are of different dimensions, the order in which the product is computed affects the number of operations. An optimum order is an order which minimizes the total number of operations. We present some theorems about an optimum order of computing the matrices. Based on these theorems, an $O(n\log n)$ algorithm for finding an optimum order will be presented in Part II.