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Also, a monomial is a multiset of indeterminates; for example, the monomial x 3 y 2 corresponds to the multiset {x, x, x, y, y}. A multiset corresponds to an ordinary set if the multiplicity of every element is 1. An indexed family (a i) i∈I, where i varies over some index set I, may define a multiset, sometimes written {a i}.
In some cases a multiset in this counting sense may be generalized to allow negative values, as in Python. C++'s Standard Template Library implements both sorted and unsorted multisets. It provides the multiset class for the sorted multiset, as a kind of associative container, which implements this multiset using a self-balancing binary search ...
Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the leftmost set, will be the middle set, and
Sometimes the items derive from a common type; even deriving from the most general type of a programming language such as object or variant. Although easily confused with implementations in programming languages, collection, as an abstract concept, refers to mathematical concepts which can be misunderstood when the focus is on an implementation.
The 3-partition problem remains NP-complete even when the integers in S are bounded above by a polynomial in n.In other words, the problem remains NP-complete even when representing the numbers in the input instance in unary. i.e., 3-partition is NP-complete in the strong sense or strongly NP-complete.
For example, the prime factorization of the integer 60 is 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors.
In mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna.It is often used in context of termination of programs or term rewriting systems.
In C++, associative containers are a group of class templates in the standard library of the C++ programming language that implement ordered associative arrays. [1] Being templates , they can be used to store arbitrary elements, such as integers or custom classes.
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