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Max-sum MSSP is a special case of MKP in which the value of each item equals its weight. The knapsack problem is a special case of MKP in which m=1. The subset-sum problem is a special case of MKP in which both the value of each item equals its weight, and m=1. The MKP has a Polynomial-time approximation scheme. [6]
Maximum subarray problems arise in many fields, such as genomic sequence analysis and computer vision.. Genomic sequence analysis employs maximum subarray algorithms to identify important biological segments of protein sequences that have unusual properties, by assigning scores to points within the sequence that are positive when a motif to be recognized is present, and negative when it is not ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The problem is NP-hard even when all input integers are positive (and the target-sum T is a part of the input). This can be proved by a direct reduction from 3SAT. [2] It can also be proved by reduction from 3-dimensional matching (3DM): [3] We are given an instance of 3DM, where the vertex sets are W, X, Y. Each set has n vertices.
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of VaR models which do not rely on the assumption of normality of risk factors.
The value of this limit, should it exist, is the (C, α) sum of the integral. Analogously to the case of the sum of a series, if α = 0, the result is convergence of the improper integral. In the case α = 1, (C, 1) convergence is equivalent to the existence of the limit
If every x in M can be written in exactly one way as a sum of finitely many elements of the M i, then we say that M is the internal direct sum of the submodules M i (Halmos 1974, §18). In this case, M is naturally isomorphic to the (external) direct sum of the M i as defined above ( Adamson 1972 , p.61).