$J^{\,r}(1,1)$上接触系紧化的小增长向量
Small growth vectors of the compactifications of the contact systems on $J^{\,r}(1,1)$
摘要：
众所周知，生活在实喷流空间$J^{\,r}(1,1)$，$r \ge 2$上的正则接触系统的紧化是局部泛Goursat分布，$Δ^r$，生活在具有开放稠密喷流（$J^{\,r}(1,1)$ -like）部分的紧流形（称为Goursat怪物）上。根据Jean（1996）的结果，对于每个$r \ge 2$，可以递归地计算第$r$个怪物的任意点处\，$Δ^r$的{ \it小增长向量\，}。结果是通过执行一系列$r$操作得到的，这些操作是根据所讨论的$Δ^r$的局部几何，从固定递归规则集$\{\,1,\,2,\,3\,\}$（在本文中分别称为S，T，G）中进行的。通过局部普适性的\，$Δ^r$，因此能够计算{ \it所有\，}所有现有的Goursat分布的小增长向量。在Mormul（2004）的工作中，提出了Jean递归序列的显式解。这些解决方案揭示了一种令人惊讶的复杂的基本算术--一种类似G{<$} del的编码，用正整数的整齐序列对三个字母表$\{$ G，\，S，\，T $\}$进行编码。然而，这些公式，虽然表征真正存在的对象，似乎从稀薄的空气，和证明被推迟到另一个出版物。在目前的贡献，我们提交证明我们的公式从2004年。然而，这不是一个简单的检查，我们的候选人满足让的递归。在这种方式下，我们检索并重新生成那些令人惊讶的公式。
英文摘要：
It is well known that the compactifications of the canonical contact systems living on real jet spaces $J^{\,r}(1,1)$, $r \ge 2$, are locally universal Goursat distributions, $Δ^r$, living on compact manifolds (called Goursat monsters) having open dense jet-like ($J^{\,r}(1,1)$-like) parts. By virtue of the results of Jean (1996), one was able, for each $r \ge 2$, to recursively compute the {\it small growth vector\,} of \,$Δ^r$ at any point of the $r$-th monster. The result was got by performing series of $r$ operations taken, in function of the local geometry of \,$Δ^r$ in question, from the set of fixed recursive rules $\{\,1,\,2,\,3\,\}$ (called in the present text S,\,T,\,G, respectively). By the local universality of \,$Δ^r$ one was thus able to compute {\it all\,} small growth vectors of all existing Goursat distributions. In the work of Mormul (2004) proposed were explicit solutions to the series of Jean's recurrences. The solutions uncovered a surprisingly involved underlying arithmetics -- a kind of G{ö}del-like encoding of words over a three-letter alphabet $\{$G,\,S,\,T$\}$ by neat sequences of positive integers. Yet, those formulas, though characterizing really existing objects, appeared as if from thin air, and proofs were postponed to another publication. In the present contribution we submit proofs of our formulas from 2004. It is not, however, a plain check that our candidates satisfy Jean's recurrences. Under the way we retrieve and re-produce those surprising formulas.
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。
