Brouwer fixed-point theorem

Some simplifications are possible if equivalence classes of symmorphy transformations can be defined where operations S from the same class transform the space occupied by the object A the same way and differ only in parts of the space where A is not present. Furthermore, using the Brouwer fixed point theorem, a subgroup structure of symmorphy groups Gjph( ) provides a more detailed characterization of molecular shape. These aspects will not be reviewed here. 

Most symmorphy groups hp are rather complicated and their direct use for molecular shape characterization and shape similarity analysis is not a trivial task. Some simplifications are possible using a technique based on the Brouwer fixed point theorem, as described in reference [43]. 

Finally, the theorem asserts the existence of at least one periodic solution. After a bit of algebra, this follows directly from the Brouwer fixed point theorem (see Hastings and Murray). 

A famous result has been given by Brouwer. Consider a smooth map / that takes into itself / -> B. The Brouwer fixed-point theorem states... 

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