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The following very simple algorithm has an approximation ratio of 1/2: [17] Order the inputs by descending value; Put the next-largest input into the subset, as long as it fits there. When this algorithm terminates, either all inputs are in the subset (which is obviously optimal), or there is an input that does not fit.
Therefore the POF is 1/(3e), which is unbounded. Alice has two items with values 1 and e, for some small e>0. George has two items with value e. The capacity is 1. The maximum sum is 1, attained by giving Alice the item with value 1 and George nothing. But the max-min allocation gives both agents value e. Therefore the POF is 1/(2e), which is ...
Numeric literals in Python are of the normal sort, e.g. 0, -1, 3.4, 3.5e-8. Python has arbitrary-length integers and automatically increases their storage size as necessary. Prior to Python 3, there were two kinds of integral numbers: traditional fixed size integers and "long" integers of arbitrary size.
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
Fig. 3 Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4. Standard examples of posets arising in mathematics include:
8.3.1 Counter-examples: images of operations not distributing 8.3.2 Conditions guaranteeing that images distribute over set operations 8.3.2.1 Conditions for f(L⋂R) = f(L)⋂f(R)
The first four partial sums of 1 + 2 + 4 + 8 + ⋯. In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity.
The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.