奇点乘法中的不可避免无限分支
Inevitable Infinite Branching in the Multiplication of Singularities
摘要：
奇点出现在物理学中许多重要的数学模型中。在大多数这样的情况下，奇点都涉及到本质上非线性的背景。四十多年来，已经发展了足够普遍的非线性奇点理论。一个至关重要的相关特征是，在奇点的某些水平之上，乘法运算，以及一般来说，在这样的奇点上的非线性运算，不可避免地以无限多种方式分支，而不可能存在某种唯一的自然或规范方式来执行这样的非线性运算。因此，这种分支的选择必须来自外部的考虑。
英文摘要：
Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of singularities have been developed. A critically important related feature is that, above certain levels in singularities, the operation of multiplication, and in general, nonlinear operations on such singularities do inevitably branch in infinitely many ways, without the possibility for the existence of some unique natural or canonical way such nonlinear operations may be performed. Consequently, the choice in such branchings has to come from extraneous considerations.
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