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