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In One Line and Anywhere. Permutations as Linear Orders. Inversions.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
For our purposes, a permutation of a multiset is just a way of listing all its elements. It is straightforward to see, and is proved in most undergraduate textbooks on enumerative combinatorics, that the number of all permutations of the multiset K={1a1,2a2,⋯,kak} is n!a1!a2!⋯ak!,
Structured Text Representations
Published in Jan Žižka, František Dařena, Arnošt Svoboda, Text Mining with Machine Learning, 2019
Jan Žižka, František Dařena, Arnošt Svoboda
Documents are therefore often treated as sets of independent words or expressions. One word might appear multiple times in one document. The number of occurrences can also play an important role. The more times a word appears, the more important it probably is. Sets generally do not allow multiple occur-rences of items. An exception is so called multisets or bags that allow multiple instances of their elements. In the text mining domain, the later term is commonly used and the bag-of-words model is a model commonly used in natural language processing. It become very popular because of its simplicity and the straightforward process of creation while providing satisfactory results [135].
Quantifying errors in the aerosol mixing-state index based on limited particle sample size
Published in Aerosol Science and Technology, 2020
J. T. Gasparik, Q. Ye, J. H. Curtis, A. A. Presto, N. M. Donahue, R. C. Sullivan, M. West, N. Riemer
In this article we consider particle populations to be sets, which means that all particles in the population must have at least one species with a different mass. In particular, this excludes monodisperse populations. To overcome this limitation we could consider populations to be multisets in the sense of (Knuth 1998, p. 473). Roughly speaking, a multiset is an extension of a set to allow elements to appear multiple times and for which the set union and set union operators have been appropriately extended. For multisets, the equivalent to the Iverson bracket is the multiplicity operator which gives the integer number of occurrences of any particle in the population. All the theoretical results in this article carry through for multisets, but we restrict ourselves to regular sets for convenience.
Dynamic distributed clustering in wireless sensor networks via Voronoi tessellation control
Published in International Journal of Control, 2019
Given the position sets and , each CH node computes which are its associated sensor nodes (i.e. the nodes whose distance to j is smaller than the distances to the other CH nodes) and collects them in the neighbour set . Each CH node stores the past and the current neighbour sets in the multiset, with . is a multiset in the sense that each node can appear in more than once; ⊎ denotes the operation of multiset union3. The number of times a node i appears in is called the multiplicity of the node, and is denoted in the following with . The multiplicity of a node is in practice equal to the number of times the node was associated to the CH node s up to time tk. With little abuse of notation, let tn(i, j) denote the time instant when node i was associated to for the nth time, i.e. raising the multiplicity of node i in to n. As already specified, each time an element is associated to , the CH node also keeps trace of the node transmission rate and position. In the following, and , will denote, respectively, the position and the transmission rate of the node i when multiplicity n was gained in .