He also introduced a multinumber: a function f(x) from a multiset to the natural numbers, giving the multiplicity of element x in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.
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One of the simplest and most natural examples is the multiset of prime factors of a number . Here the underlying set of elements is the set of prime divisors of . For example, the number 120 has the prime factorization :120 = 2^3 3^1 5^1 which gives the multiset . A related example is the multiset of solutions of an algebraic equation.
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That means the result set of SQL is a multiset. If it was a set, the repetitive records in the result set were eliminated. Another application of multiset is in modeling multigraphs. In multigraphs there can be multiple edges between any two given vertices.
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The work of Marius Nizolius (1498–1576) contains another early reference to the concept of multisets. Athanasius Kircher found the number of multiset permutations when one element can be repeated. Jean Prestet published a general rule for multiset permutations in 1675. John Wallis explained this rule in more detail in 1685.
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A merging operator can be expressed as a family of orderings over models, one for each possible multiset of knowledge bases to merge: the models of the result of merging a multiset of knowledge bases are the minimal models of the ordering associated to the multiset. A merging operator defined in this way satisfies the postulates for merging if and only if the family of orderings meets a given set of conditions. For the old definition of arbitration, the orderings are not on models but on pairs (or, in general, tuples) of models.
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For example, if we have a multiset, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case the multiset is being sharded. To finalize reduction properly, the "Shuffling" stage regroups the data among the nodes.
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In relational databases, a table can be a (mathematical) set or a multiset, depending on the presence of unicity constraints on some columns (which turns it into a candidate key). SQL allows the selection of rows from a relational table: this operation will in general yield a multiset, unless the keyword `DISTINCT` is used to force the rows to be all different, or the selection includes the primary (or a candidate) key. In ANSI SQL the `MULTISET` keyword can be used to transform a subquery into a collection expression: SELECT expression1, expression2... FROM table_name... is a general select that can be used as subquery expression of another more general query, while MULTISET(SELECT expression1, expression2... FROM table_name...) transforms the subquery into a collection expression that can be used in another query, or in assignment to a column of appropriate collection type.
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In a multiset (or bag), like in a set, the order of data items does not matter, but in this case duplicate data items are permitted. Examples of operations on multisets are the addition and removal of data items and determining how many duplicates of a particular data item are present in the multiset. Multisets can be transformed into lists by the action of sorting.
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In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity".
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Given a set A, the free commutative monoid on A is the set of all finite multisets with elements drawn from A, with the monoid operation being multiset sum and the monoid unit being the empty multiset. For example, if A = {a, b, c}, elements of the free commutative monoid on A are of the form :{ε, a, ab, a2b, ab3c4, ...}. The fundamental theorem of arithmetic states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers. The free commutative semigroup is the subset of the free commutative monoid which contains all multisets with elements drawn from A except the empty multiset.
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The 2007 HyperLogLog algorithm splits the multiset into subsets and estimates their cardinalities, then it uses the harmonic mean to combine them into an estimate for the original cardinality.
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Jenő Egerváry (1931) considered graphs in which each edge e has a non- negative integer weight we. The weight vector is denoted by w. The w-weight of a matching is the sum of weights of the edges participating in the matching. A w-vertex-cover is a multiset of vertices ("multiset" means that each vertex may appear several times), in which each edge e is adjacent to at least we vertices.
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When implemented in SQL standard the "default projection" returns a multiset instead a set, and the projection is obtained by the addition of the `DISTINCT` keyword to eliminate duplicate data.
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In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid.
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Y. Suzuki and H. Tanaka. Modeling p53 signaling pathways by using multiset processing. In G. Ciobanu, G. Pa ̆un, and M. J. Pérez- Jiménez, editors, Applications of Membrane Computing, Natural Computing Series, pages 203–214.
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If the cumulative reference list of an author's oeuvre is determined as the multiset union of the documents that the author has co-authored, then the author bibliographic coupling strength of two authors (or more precisely, of their oeuvres) is defined as the size of the multiset intersection of their cumulative reference lists, however. Bibliographic coupling can be useful in a wide variety of fields, since it helps researchers find related research done in the past. On the other hand, two documents are co-cited if they are both independently cited by one or more documents.
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If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
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A quadratic equation, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be , or it could be . In the latter case it has a solution of multiplicity 2.
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We now say that x∈V is an eigenvector of M if x is an eigenvector of T. Similarly, λ∈K is an eigenvalue of M if it is an eigenvalue of T, and with the same multiplicity, and the spectrum of M, written σM, is the multiset of all such eigenvalues.
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We can generalize the notion of a metric from a distance between two elements to a distance between two nonempty finite multisets of elements. A multiset is a generalization of the notion of a set such that an element can occur more than once. Define Z=XY if Z is the multiset consisting of the elements of the multisets X and Y, that is, if x occurs once in X and once in Y then it occurs twice in Z. A distance function d on the set of nonempty finite multisets is a metric if # d(X)=0 if all elements of X are equal and d(X) > 0 otherwise (positive definiteness), that is, (non-negativity plus identity of indiscernibles) # d(X) is invariant under all permutations of X (symmetry) # d(XY) \leq d(XZ)+d(ZY) (triangle inequality) Note that the familiar metric between two elements results if the multiset X has two elements in 1 and 2 and the multisets X,Y,Z have one element each in 3. For instance if X consists of two occurrences of x, then d(X)=0 according to 1.
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In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly, SQL operates on multisets and return identical records. For instance, consider "SELECT name from Student". In the case that there are multiple records with name "sara" in the student table, all of them are shown.
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Given S = {3,1,1,2,2,1}, a valid solution to the partition problem is the two sets S1 = {1,1,1,2} and S2 = {2,3}. Both sets sum to 5, and they partition S. Note that this solution is not unique. S1 = {3,1,1} and S2 = {2,2,1} is another solution. Not every multiset of positive integers has a partition into two subsets with equal sum.
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In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of the numbers in S2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization version of the partition problem, which is to partition the multiset S into two subsets S1, S2 such that the difference between the sum of elements in S1 and the sum of elements in S2 is minimized.
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Static sets allow only query operations on their elements — such as checking whether a given value is in the set, or enumerating the values in some arbitrary order. Other variants, called dynamic or mutable sets, allow also the insertion and deletion of elements from the set. A multiset is a special kind of set in which an element can figure several times.
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StoreSpy allowed viewing Items, Relationships, MultiSet, Nested Elements, Extensions and other types in the store along with its full metadata. It also presented a search interface to perform manual searches, and save them as virtual folders. The application also presented a graphical view of WinFS Rules. However, it did not allow editing of Items or their properties, though it was slated for inclusion in a future release.
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If the beads are not all distinct, having repeated colors, then there are fewer necklaces (and bracelets). The above necklace-counting polynomials give the number necklaces made from all possible multisets of beads. Polya's pattern inventory polynomial refines the counting polynomial, using variable for each bead color, so that the coefficient of each monomial counts the number of necklaces on a given multiset of beads.
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A complete intersection has a multidegree, written as the tuple (properly though a multiset) of the degrees of defining hypersurfaces. For example, taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position is an elliptic curve. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira.
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Regions communicate with each other via the transfer of objects. The computation by a membrane system starts with an initial configuration, where the number (multiplicity) of each object is set to some value for each region (multiset of objects). It proceeds by choosing, nondeterministically and in a maximally parallel manner, which rules are applied to which objects. The output of the computation is collected from an a priori determined output region.
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In computing, associative containers refer to a group of class templates in the standard library of the C++ programming language that implement ordered associative arrays. Being templates, they can be used to store arbitrary elements, such as integers or custom classes. The following containers are defined in the current revision of the C++ standard: `set`, `map`, `multiset`, `multimap`. Each of these containers differ only on constraints placed on their elements.
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In discrete mathematics, especially combinatorics and finite probability theory, -tuples arise in the context of various counting problems and are treated more informally as ordered lists of length . -tuples whose entries come from a set of elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of -tuples of an -set is . This follows from the combinatorial rule of product.
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Sorted arrays are the most space-efficient data structure with the best locality of reference for sequentially stored data. Elements within a sorted array are found using a binary search, in O(log n); thus sorted arrays are suited for cases when one needs to be able to look up elements quickly, e.g. as a set or multiset data structure. This complexity for lookups is the same as for self-balancing binary search trees.
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There are many ways to systematically generate all permutations of a given sequence. One classic, simple, and flexible algorithm is based upon finding the next permutation in lexicographic ordering, if it exists. It can handle repeated values, for which case it generates each distinct multiset permutation once. Even for ordinary permutations it is significantly more efficient than generating values for the Lehmer code in lexicographic order (possibly using the factorial number system) and converting those to permutations.
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A (general) block design is a set together with a family of subsets of (repeated subsets are allowed). Normally a block design is required to satisfy numerical regularity conditions. As an incidence structure, is the set of points and is the set of lines, usually called blocks in this context (repeated blocks must have distinct names, so is actually a set and not a multiset). If all the subsets in have the same size, the block design is called uniform.
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For instance, they were important in early AI languages, such as QA4, where they were referred to as bags, a term attributed to Peter Deutsch. A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set). Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets.
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Query Rewriting is a typically automatic transformation that takes a set of database tables views and/or queries, usually indices, often gathered data and query statistics, and other metadata, and yields a set of different queries, which produce the same results but execute with better performance (for example, faster, or with lower memory use). Query rewriting can be based on relational algebra or an extension thereof (e.g. multiset relational algebra with sorting, aggregation and three-valued predicates i.e. NULLs as in the case of SQL).
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The selective communication is where the computing comes in. Graphical representations may have numerous elements, according to the variation of the model that is being studied. For example, a rule may produce the special symbol δ, in which case the membrane that contains it is dissolved and all its contents move up in the region hierarchy. The variety of suggestions from biology and the range of possibilities to define the architecture and the functioning of a membrane-based multiset processing device are practically endless.
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Combinatorics. 97 rules and 45 examples. Generating permutations (including of a multiset), combinations, partitions of a number, binomial coefficients, generalized Fibonacci numbers. Narayana Pandita noted the equivalence of the figurate numbers and the formulae for the number of combinations of different things taken so many at a time. The book contains a rule to determine the number of permutations of n objects and a classical algorithm for finding the next permutation in lexicographic ordering though computational methods have advanced well beyond that ancient algorithm.
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Multisets appeared explicitly in the work of Richard Dedekind. Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described generalized sets ("sets" whose characteristic functions may take any integer value - positive, negative or zero). Monro (1987) investigated the category Mul of multisets and their morphisms, defining a multiset as a set with an equivalence relation between elements "of the same sort", and a morphism between multisets as a function which respects sorts.
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The median can be used as a measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewed, extreme values are not known, or outliers are untrustworthy, i.e., may be measurement/transcription errors. For example, consider the multiset : 1, 2, 2, 2, 3, 14. The median is 2 in this case, (as is the mode), and it might be seen as a better indication of the center than the arithmetic mean of 4, which is larger than all-but-one of the values.
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The Spectrum Exchangeability Principle, Sx. The probability w(\Theta) of a state description \Theta depends only on the spectrum of \Theta, that is, on the multiset of sizes of equivalence classes with respect to the equivalence relation \sim_\Theta. Li with Sx. As the Language Invariance Principle but all the probability functions in the family also satisfy Spectrum Exchangeability. The Principle of Induction, PI. Let \Theta be a state description and a_k a constant not appearing in \Theta. Let \Phi, \Psi be state descriptions extending \Theta to include (just) a_k.
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The bag-of-words model is a simplifying representation used in natural language processing and information retrieval (IR). In this model, a text (such as a sentence or a document) is represented as the bag (multiset) of its words, disregarding grammar and even word order but keeping multiplicity. The bag-of-words model has also been used for computer vision. The bag-of-words model is commonly used in methods of document classification where the (frequency of) occurrence of each word is used as a feature for training a classifier.
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A specific choice of columns which uniquely identify rows is called the primary key. "Table" is another term for "relation"; although there is the difference in that a table is usually a multiset (bag) of rows where a relation is a set and does not allow duplicates. Besides the actual data rows, tables generally have associated with them some metadata, such as constraints on the table or on the values within particular columns. The data in a table does not have to be physically stored in the database.
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Designs without repeated blocks are called simple, in which case the "family" of blocks is a set rather than a multiset. In statistics, the concept of a block design may be extended to non- binary block designs, in which blocks may contain multiple copies of an element (see blocking (statistics)). There, a design in which each element occurs the same total number of times is called equireplicate, which implies a regular design only when the design is also binary. The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
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The rules typically are expressed in terms of finite sets of formulae, although there are logics for which we must use more complicated data structures, such as multisets, lists, or even trees of formulas. Henceforth, "set" denotes any of {set, multiset, list, tree}. If there is such a rule for every logical connective then the procedure will eventually produce a set which consists only of atomic formulae and their negations, which cannot be broken down any further. Such a set is easily recognizable as satisfiable or unsatisfiable with respect to the semantics of the logic in question.
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In the context of this article, all graphs will be simple and undirected, unless stated otherwise. This means that the edges of the graph form a set (and not a multiset) and each edge is a pair of distinct vertices. Graphs are assumed to have an implicit representation in which each vertex has a unique identifier or label and in which it is possible to test the adjacency of any two vertices, but for which adjacency testing is the only allowed primitive operation. Informally, a graph property is a property of a graph that is independent of labeling.
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In computing it may be required to generate permutations of a given sequence of values. The methods best adapted to do this depend on whether one wants some randomly chosen permutations, or all permutations, and in the latter case if a specific ordering is required. Another question is whether possible equality among entries in the given sequence is to be taken into account; if so, one should only generate distinct multiset permutations of the sequence. An obvious way to generate permutations of n is to generate values for the Lehmer code (possibly using the factorial number system representation of integers up to n!), and convert those into the corresponding permutations.
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For generating random permutations of a given sequence of n values, it makes no difference whether one applies a randomly selected permutation of n to the sequence, or chooses a random element from the set of distinct (multiset) permutations of the sequence. This is because, even though in case of repeated values there can be many distinct permutations of n that result in the same permuted sequence, the number of such permutations is the same for each possible result. Unlike for systematic generation, which becomes unfeasible for large n due to the growth of the number n!, there is no reason to assume that n will be small for random generation.
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Runtime verification, if used in combination with provably correct recovery code, can provide an invaluable infrastructure for program verification, which can significantly lower the latter's complexity. For example, formally verifying heap-sort algorithm is very challenging. One less challenging technique to verify it is to monitor its output to be sorted (a linear complexity monitor) and, if not sorted, then sort it using some easily verifiable procedure, say insertion sort. The resulting sorting program is now more easily verifiable, the only thing being required from heap-sort is that it does not destroy the original elements regarded as a multiset, which is much easier to prove.
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Cypher uses compact fixed- and variable-length patterns which combine visual representations of node and relationship (edge) topologies, with label existence and property value predicates. (These patterns are usually referred to as "ASCII art" patterns, and arose originally as a way of commenting programs which used a lower-level graph API.) By matching such a pattern against graph data elements, a query can extract references to nodes, relationships and paths of interest. Those references are emitted as a "binding table" where column names are bound to a multiset of graph elements. The name of a column becomes the name of a "binding variable", whose value is a specific graph element reference for each row of the table.
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Rel is an implementation of Tutorial D. Even the query language of SQL is loosely based on a relational algebra, though the operands in SQL (tables) are not exactly relations and several useful theorems about the relational algebra do not hold in the SQL counterpart (arguably to the detriment of optimisers and/or users). The SQL table model is a bag (multiset), rather than a set. For example, the expression (R \cup S) \setminus T = (R \setminus T) \cup (S \setminus T) is a theorem for relational algebra on sets, but not for relational algebra on bags; for a treatment of relational algebra on bags see chapter 5 of the "Complete" textbook by Garcia-Molina, Ullman and Widom.
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A superkey or super-key is defined in the relational model of database organization as a set of attributes of a relation variable for which it holds that in all relations assigned to that variable, there are no two distinct tuples (rows) that have the same values for the attributes in this set. It can be defined as a set of attributes of a relation schema upon which all attributes of the schema are functionally dependent. The set of all attributes is a trivial superkey, because in relational algebra duplicate rows are not permitted: rows are a set (no duplicates), not a multiset (duplicates allowed). The superkey is also known as superset key.
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The current version of IBM Informix is 14.10. The major enhancements made over previous releases were adding built-in index compression, integration of JSON collections with support for MongoDB JSON drivers into the server, and an enhancement permitting database objects to be partitioned across multiple servers in a cluster or grid (aka sharding). Queries can optionally return data from the locally connected server instance or from an entire grid with the same SQL. Informix version 14.10 introduced support for partial indexing where only a subset of the rows in a table are indexed and for multi-valued key indexes which support indexing the elements within multi-valued data types such as LIST, SET, MULTISET, and BSON array fields.
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More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree always form a multiset of cardinality . A special case of the above are the eigenvalues of a matrix, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial, and the geometric multiplicity, which is defined as the dimension of the kernel of (where is an eigenvalue of the matrix ). These three multiplicities define three multisets of eigenvalues, which may be all different: Let be a matrix in Jordan normal form that has a single eigenvalue.
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Let \pi be a permutation. If i < j and \pi(i) > \pi(j), either the pair of places (i, j) or the pair of elements \bigl(\pi(i), \pi(j)\bigr) is called an inversion of \pi. The inversion is usually defined for permutations, but may also be defined for sequences: Let S be a sequence (or multiset permutation). If i < j and S(i) > S(j), either the pair of places (i, j) or the pair of elements \bigl(S(i), S(j)\bigr) is called an inversion of S. For sequences inversions according to the element-based definition are not unique, because different pairs of places may have the same pair of values.
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In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non- adjacent vertices. It is also called the Seidel matrix or--its original name-- the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and J. J. Seidel in 1966 and extensively exploited by Seidel and coauthors.
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In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, for each value i appearing in the permutation, the values between the two copies of i are larger than i. For instance, the 15 Stirling permutations of order three are :1,1,2,2,3,3; 1,2,2,1,3,3; 2,2,1,1,3,3; :1,1,2,3,3,2; 1,2,2,3,3,1; 2,2,1,3,3,1; :1,1,3,3,2,2; 1,2,3,3,2,1; 2,2,3,3,1,1; :1,3,3,1,2,2; 1,3,3,2,2,1; 2,3,3,2,1,1; :3,3,1,1,2,2; 3,3,1,2,2,1; 3,3,2,2,1,1. The number of Stirling permutations of order k is given by the double factorial (2k − 1)!!. Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative.
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1, G. Rozenberg and A. Salomaa (Eds.), Springer 1997, S. 210-217. For his second thesis he developed variants of Quicksort suitable for multiset and proved that they achieved the lower bound for quicksort algorithms previously given by Robert Sedgewick. Following a sabbatical stay at the IBM Scientific Center Heidelberg he took an interest in the Non-First Normal-Form data model, also known as nested relational model, and designed a graphical editor which also served as base for research on synchronous groupware. In 1986 he authored an E-learning course "Introduction to Unix", which originally was a contribution to Hermann Maurer's COSTOC-Project, and with several portings was in active use until 2015, thus constituting one of the longest running examples of courseware.
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These include formal sums over B, which are expressions of the form \sum a_i b_i where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. Every set B has a free abelian group with B as its basis.
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Relational completeness clearly does not imply that any interesting database query can be expressed in relationally complete languages. Well-known examples of inexpressible queries include simple aggregations (counting tuples, or summing up values occurring in tuples, which are operations expressible in SQL but not in relational algebra) and computing the transitive closure of a graph given by its binary edge relation (see also expressive power). Codd's theorem also doesn't consider SQL nulls and the three-valued logic they entail; the logical treatment of nulls remains mired in controversy.For recent work extending Codd's theorem in this direction see Additionally, SQL allows duplicate rows (has multiset semantics.) Nevertheless, relational completeness constitutes an important yardstick by which the expressive power of query languages can be compared.
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Let V be a finite- dimensional vector space over some field K and suppose T: V → V is a linear map. The spectrum of T, denoted σT, is the multiset of roots of the characteristic polynomial of T. Thus the elements of the spectrum are precisely the eigenvalues of T, and the multiplicity of an eigenvalue λ in the spectrum equals the dimension of the generalized eigenspace of T for λ (also called the algebraic multiplicity of λ). Now, fix a basis B of V over K and suppose M∈MatK(V) is a matrix. Define the linear map T: V→V point-wise by Tx=Mx, where on the right-hand side x is interpreted as a column vector and M acts on x by matrix multiplication.
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The problem of finding a single simple cycle that covers each vertex exactly once, rather than covering the edges, is much harder. Such a cycle is known as a Hamiltonian cycle, and determining whether it exists is NP-complete.. Much research has been published concerning classes of graphs that can be guaranteed to contain Hamiltonian cycles; one example is Ore's theorem that a Hamiltonian cycle can always be found in a graph for which every non-adjacent pair of vertices have degrees summing to at least the total number of vertices in the graph.. The cycle double cover conjecture states that, for every bridgeless graph, there exists a multiset of simple cycles that covers each edge of the graph exactly twice. Proving that this is true (or finding a counterexample) remains an open problem...
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If ƒ must be injective, then the selection must involve n distinct elements of X, so it is a subset of X of size n, also called an n-combination. Without the requirement, a same element of X may occur multiple times in the selection, and the result is a multiset of size n of elements from X, also called an n-multicombination or n-combination with repetition. In these cases the requirement of a surjective ƒ means that every label is to be used at least once, respectively that every element of X be included in the selection at least once. Such a requirement is less natural to handle mathematically, and indeed the former case is more easily viewed first as a grouping of elements of N, with in addition a labelling of the groups by the elements of X.
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All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13 ... A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k (-1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n.
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This construction has been generalized, in algebraic geometry, to the notion of a divisor. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of a system of polynomial equations. In the case where the system of equations has one degree of freedom (its solutions form an algebraic curve or Riemann surface), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors can again be represented as elements of a free abelian group, with multiplication or division of functions corresponding to addition or subtraction of group elements.
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The assumption implicit in the revision operator is that the new piece of information P is always to be considered more reliable than the old knowledge base K. This is formalized by the second of the AGM postulates: P is always believed after revising K with P. More generally, one can consider the process of merging several pieces of information (rather than just two) that might or might not have the same reliability. Revision becomes the particular instance of this process when a less reliable piece of information K is merged with a more reliable P. While the input to the revision process is a pair of formulae K and P, the input to merging is a multiset of formulae K, T, etc. The use of multisets is necessary as two sources to the merging process might be identical. When merging a number of knowledge bases with the same degree of plausibility, a distinction is made between arbitration and majority.
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Powers of two are usually ignored, because of the way the human ear perceives octaves as equivalent. An x-dimensional tone-dimension contains x factors. "An Euler-Fokker genus with two dimensions may be represented in a two-dimensional (rectangular) tone- grid, one with three dimensions in a three-dimensional (block-shaped) tone- lattice. Euler-Fokker genera are characterized by a listing of the number of steps in each dimension. The number of steps is represented by a repeated mention of the dimension, so that there arise descriptions such as [3 3 5 5], [3 5 7], [3 3 5 5 7 7 11 11], etc." For example, the multiset {3, 3, 7} yields the Euler–Fokker genus [3, 3, 7], which contains these pitches: 1 3 =3 7=7 3×3 =9 3×7=21 3×3×7=63 Normalized to fall within an octave, these become: 1/1, 9/8, 21/16, 3/2, 7/4, 63/32.
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The nonnegative integer r is called the free rank or Betti number of the module M. The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the Chinese remainder theorem for R. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
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As mentioned above, weak orders have applications in utility theory. In linear programming and other types of combinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real- valued objective function; the phenomenon of ties in these orderings is called "degeneracy", and several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent problems caused by degeneracy.. Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographic breadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of a graph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a total order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is the output of the algorithm.. In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering.
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They chose the name because of a connection to certain polynomials defined from the Stirling numbers, which are in turn named after 18th-century Scottish mathematician James Stirling.. Euler tour of a plane tree with its edges labeled by construction order Stirling permutations may be used to describe the sequences by which it is possible to construct a rooted plane tree with k edges by adding leaves one by one to the tree. For, if the edges are numbered by the order in which they were inserted, then the sequence of numbers in an Euler tour of the tree (formed by doubling the edges of the tree and traversing the children of each node in left to right order) is a Stirling permutation. Conversely every Stirling permutation describes a tree construction sequence, in which the next edge closer to the root from an edge labeled i is the one whose pair of values most closely surrounds the pair of i values in the permutation.. Stirling permutations have been generalized to the permutations of a multiset with more than two copies of each value.. Researchers have also studied the number of Stirling permutations that avoid certain patterns..
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The elements of any distributive lattice form a median graph, a structure in which any three elements P, Q, and R (here, stable matchings) have a unique median element m(P,Q,R) that lies on a shortest path between any two of them. It can be defined as: :m(P,Q,R)=(P\wedge Q)\vee(P\wedge R)\vee(Q\wedge R)=(P\vee Q)\wedge(P\vee R)\wedge(Q\vee R). For the lattice of stable matchings, this median can instead be taken element- wise, by assigning each doctor the median in the doctor's preferences of the three hospitals matched to that doctor in P, Q, and R and similarly by assigning each hospital the median of the three doctors matched to it. More generally, any set of an odd number of elements of any distributive lattice (or median graph) has a median, a unique element minimizing its sum of distances to the given set. For the median of an odd number of stable matchings, each participant is matched to the median element of the multiset of their matches from the given matchings.
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