双线性映射的增长性
Growth of bilinear maps
摘要：
对于非负系数的双线性映射$*:\mathbb R^d\times \mathbb R^d\to \mathbb R^d$和正条目的向量$s\in \mathbb R^d$，在针对给定$n$使用$*$的$n-1$应用组合$s$的$n$实例的指数数量的方式中，我们对所有所得向量上的最大条目感兴趣。渐近行为是当$n$趋于无穷大时，该最大项的$n$次根收敛于增长率$λ$。本文通过一种特殊的结构--线性模式证明了这种极限的存在性。我们还提出了一个问题的可能性之间的关系的结构和是否$λ$是代数。
英文摘要：
For a bilinear map $*:\mathbb R^d\times \mathbb R^d\to \mathbb R^d$ of nonnegative coefficients and a vector $s\in \mathbb R^d$ of positive entries, among an exponentially number of ways combining $n$ instances of $s$ using $n-1$ applications of $*$ for a given $n$, we are interested in the largest entry over all the resulting vectors. An asymptotic behavior is that the $n$-th root of this largest entry converges to a growth rate $λ$ when $n$ tends to infinity. In this paper, we prove the existence of this limit by a special structure called linear pattern. We also pose a question on the possibility of a relation between the structure and whether $λ$ is algebraic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
