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In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
In its simplest form, it relates the expectation of a sum of randomly many finite-mean, independent and identically distributed random variables to the expected number of terms in the sum and the random variables' common expectation under the condition that the number of terms in the sum is independent of the summands.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
Symptoms frequently get better after a few days. [3] A skin tag may remain after the healing of an external hemorrhoid. [4] While the exact cause of hemorrhoids remains unknown, a number of factors that increase pressure in the abdomen are believed to be involved. [4] This may include constipation, diarrhea, and sitting on the toilet for long ...
This example is an instance of the Few Sums, Many Products [6] version of the sum-product problem of György Elekes and Imre Z. Ruzsa. A consequence of their result is that any set with small additive doubling (such as an arithmetic progression ) has the lower bound on the product set | AA | = Ω(| A | 2 log −1 (| A |)) .
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. For this reason, many special cases and generalizations have been examined.
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The algorithm performs summation with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around. Thus the summation proceeds with "guard digits" in c , which is better than not having any, but is not as good as performing the calculations with double the ...
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