Mathematics Colloquia and Seminars

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Description

Let ${mathcal M}_{n}$ be the set of all $n$ by $n$ matrices over an integral domain $D$. As $D$ modules, we can consider direct sum and tensor of ${mathcal M}$'s. We will look at the embedding problems of $A=oplus_{i=1}^{s} {mathcal M}_i$ into $B=oplus_{i=1}^{t} {mathcal M}'_i$ as $D$ algebras and we will denote by $A hookrightarrow B$. This embedding problem can be interpolated as a {it bin-packing problem} by replacing the size of ${mathcal M}_i$ as the size of a block in the partition of $sum i$. In general, this problem is known as NP-hard. But there are several approches for algorithms. Through the paper, we might restrict ourself to a special case of them is that all the size of ${mathcal M}_{i}$ are power of a fixed number $p$. The following example will demonstrate what we are wondering about. ${mathcal M}_3 oplus {mathcal M}_3$ can not be embedded in ${mathcal M}_5oplus {mathcal M}_2$ but after tensoring with ${mathcal M}_{5}oplus {mathcal M}_3$ the result can be embedded, $i. e. {mathcal M}_{15}oplus{mathcal M}_{15} oplus{mathcal M}_{9}oplus{mathcal M}_{9}$ can be embedded in ${mathcal M}_{25}oplus{mathcal M}_{15} oplus{mathcal M}_{10} oplus {mathcal M}_{6}$.So we define that $A=oplus_{i=1}^{s} {mathcal M}_i$ is {it stably embedded} in $B=oplus_{i=1}^{t} {mathcal M}'_i$ if there is a $D$ algebra $C=oplus_{i=1}^{u} ilde{mathcal M}_i$ such that $Aotimes Chookrightarrow Botimes C$ and it will be denoted by $A overset{s}hookrightarrow B$. We will work out some properties and conjectures(or expectations).