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In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
Input: a multiset S containing n positive integer elements. Conditions: S must be partitionable into m triplets, S 1, S 2, …, S m, where n = 3m. These triplets partition S in the sense that they are disjoint and they cover S. The target value T is computed by taking the sum of all elements in S, then divided by m.
In all versions of Python, boolean operators treat zero values or empty values such as "", 0, None, 0.0, [], and {} as false, while in general treating non-empty, non-zero values as true. The boolean values True and False were added to the language in Python 2.2.1 as constants (subclassed from 1 and 0 ) and were changed to be full blown ...
In number theory and computer science, the partition problem, or number partitioning, [1] is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S 1 and S 2 such that the sum of the numbers in S 1 equals the sum of the numbers in S 2.
The real-number Euclidean algorithm differs from its integer counterpart in two respects. First, the remainders r k are real numbers, although the quotients q k are integers as before. Second, the algorithm is not guaranteed to end in a finite number N of steps. If it does, the fraction a/b is a rational number, i.e., the ratio of two integers
There are two main reasons for using integer variables when modeling problems as a linear program: The integer variables represent quantities that can only be integer. For example, it is not possible to build 3.7 cars. The integer variables represent decisions (e.g. whether to include an edge in a graph) and so should only take on the value 0 or 1.
The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. If the space required to store the input numbers is not counted, it has a space requirement of O ( 1 ) {\displaystyle O(1)} , otherwise O ( n ) {\displaystyle O(n)} is required.
The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts. The number q(n) is also equal to the number of partitions of n in which only odd summands are permitted. [20]