11. The sum to compute their dot product is 1x0 + 0x1 = 0, in which no summand is negative. 12. It turns out that it is not always a summand , but it " is " a pure subgroup. 13. In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand . 14. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective. 15. Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot. 16. In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient. 17. Brown Peterson cohomology BP is a summand of MU ( " p " ), which is complex cobordism MU suspensions of BP. 18. For some re-indexing of the summand sets; this re-indexing depends on the particular point " x " being represented. 19. In the case of 0 } } for some, the value of the corresponding summand is taken to be, which is consistent with the limit: 20. :by replacing each product in the summand by the product of the " X " i in the order given by the permutation ?.
