Multisets have various applications. They are becoming fundamental in combinatorics. Multisets have become an important tool in the theory of relational databases, which often uses the synonym bag. For instance, multisets are often used to implement relations in database systems.
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Wayne Blizard traced multisets back to the very origin of numbers, arguing that “in ancient times, the number n was often represented by a collection of n strokes, tally marks, or units.” These and similar collections of objects are multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged. Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names.
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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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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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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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Let A and B be any multisets. If c is a mode of A \cup B and c otin A, then c is a mode of B.
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We also have associativity and permutation (or commutativity) for free as well, among other properties. In substructural logics, typically premises are not composed into sets, but rather they are composed into more fine-grained structures, such as trees or multisets (sets that distinguish multiple occurrences of elements) or sequences of formulae. For example, in linear logic, since contraction fails, the premises must be composed in something at least as fine-grained as multisets.
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In the 20th century, many generalizations of sets were invented, e.g., fuzzy sets (Zadeh, 1965), or rediscovered, e.g., multisets (Knuth, 1997). As a result, these generalizations created a unification problem in the foundation of mathematics.
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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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Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs have equal multisets of eigenvalues. enneahedra, the smallest possible cospectral polyhedral graphs Cospectral graphs need not be isomorphic, but isomorphic graphs are always cospectral.
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Horadam's research concerned generalised integers, formed from a sequence of real numbers greater than one (called generalised prime numbers) as the products of finite multisets of generalised primes. She was also the author of a textbook published by the University of New England, Principles of mathematics for economists (1982).
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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 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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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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Symbols represent chemicals which may react with other chemicals to form some product. In a P system each type of symbol is typically represented by a different letter. The symbol content of a membrane is therefore represented by a string of letters. Because the multiplicity of symbols in a region matters, multisets are commonly used to represent the symbol content of a region.
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Artificial Chemistries are often used in the study of protobiology, in trying to bridge the gap between chemistry and biology. A further motivation to study artificial chemistries is the interest in constructive dynamical systems. Yasuhiro Suzuki has modeled various systems such as membrane systems, signaling pathways (P53), ecosystems, and enzyme systems by using his method, abstract rewriting system on multisets (ARMS).
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In many cases, sequents are also assumed to consist of multisets or sets instead of sequences. Thus one disregards the order or even the numbers of occurrences of the formulae. For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises do not depend on these data. In substructural logic, however, this may become quite important.
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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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Membrane computing (or MC) is an area within computer science that seeks to discover new computational models from the study of biological cells, particularly of the cellular membranes. It is a sub-task of creating a cellular model. Membrane computing deals with distributed and parallel computing models, processing multisets of symbol objects in a localized manner. Thus, evolution rules allow for evolving objects to be encapsulated into compartments defined by membranes.
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Wendl examined a number of matching and covering problems in combinatorial probability, especially as these problems apply to molecular biology. He determined the distribution of match counts of pairs of integer multisets in terms of Bell polynomials,Wendl MC (2005) Probabilistic assessment of clone overlaps in DNA fingerprint mapping via a priori models, J. Comp. Biol. 12(3), 283-297. a problem directly relevant to physical mapping of DNA.
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Similar to set theory, named sets have axiomatic representations,Burgin (2011), p. 69–89 i.e., they are defined by systems of axioms and studied in axiomatic named set theory. Axiomatic definitions of named set theory show that in contrast to fuzzy sets and multisets, named set theory is completely independent of set theory or category theory while these theories are naturally conceived as sub- theories of named set theory.
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To keep track of this process, the nodes of a tableau itself are set out in the form of a tree and the branches of this tree are created and assessed in a systematic way. Such a systematic method for searching this tree gives rise to an algorithm for performing deduction and automated reasoning. Note that this larger tree is present regardless of whether the nodes contain sets, multisets, lists or trees.
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Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers X, Y and Z, each containing k elements, and a bound b. The goal is to select a subset M of X\times Y\times Z such that every integer in X, Y and Z occurs exactly once and that for every triple (x,y,z) in the subset x+y+z=b holds. This problem is labeled as [SP16] in.
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Some programming languages have sets among their builtin data structures. Such a data structure behaves as a finite set, that is, it consists of a finite number of data that are not specifically ordered, and may thus be considered as the elements of a set. In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. These programming languages have operators or functions for computing the complement and the set differences.
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Amicable numbers (m, n) satisfy \sigma(m)-m=n and \sigma(n)-n=m which can be written together as \sigma(m)=\sigma(n)=m+n. This can be generalized to larger tuples, say (n_1,n_2,\ldots,n_k), where we require :\sigma(n_1)=\sigma(n_2)= \dots =\sigma(n_k) = n_1+n_2+ \dots +n_k For example, (1980, 2016, 2556) is an amicable triple , and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple . Amicable multisets are defined analogously and generalizes this a bit further .
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In 1982 he received the venia legendi in applied computer science from the University of Karlsruhe with an inaugural dissertation (Habilitationsschrift) on Quicksort variants for multisets. Examiners were Thomas Ottmann, Wolfgang Janko and Jan van Leeuwen (Utrecht). In 1984 he was appointed professor at the Hochschule Fulda (Fulda University of Applied Sciences) and went from there in 1987 to the University of Kassel where he served as full professor and chairman of the database group since 1989 until his retirement in March 2015.
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The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums. As 19th-century mathematician Eugène Prouhet showed, the partition into evil and odious numbers of the numbers from 0 to 2^k-1, for any k, provides a solution to the Prouhet–Tarry–Escott problem of finding sets of numbers whose sums of powers are equal up to the kth power.
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Greenhill did her undergraduate studies at the University of Queensland, and remained there for a master's degree, working with Anne Penfold Street there. She earned her Ph.D. in 1996 at the University of Oxford, under the supervision of Peter M. Neumann. Her dissertation was From Multisets to Matrix Groups: Some Algorithms Related to the Exterior Square. After postdoctoral research with Martin Dyer at the University of Leeds and Nick Wormald at the University of Melbourne, Greenhill joined the University of New South Wales in 2003.
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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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For allowing loops, the above definition must be changed by defining edges as multisets of two vertices instead of two-sets. Such generalized graphs are called graphs with loops or simply graphs when it is clear from the context that loops are allowed. Generally, the set of vertices V is supposed to be finite; this implies that the set of edges is also finite. Infinite graphs are sometimes considered, but are more often viewed as a special kind of binary relation, as most results on finite graphs do not extend to the infinite case, or need a rather different proof.
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In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. The first odious numbers are: :1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38 ... These numbers give the positions of the nonzero values in the Thue–Morse sequence. Non-negative integers that are not odious are called evil numbers. The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums.
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In computer science, a collection or container is a grouping of some variable number of data items (possibly zero) that have some shared significance to the problem being solved and need to be operated upon together in some controlled fashion. Generally, the data items will be of the same type or, in languages supporting inheritance, derived from some common ancestor type. A collection is a concept applicable to abstract data types, and does not prescribe a specific implementation as a concrete data structure, though often there is a conventional choice (see Container for type theory discussion). Examples of collections include lists, sets, multisets, trees and graphs.
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There is some freedom of choice regarding the technical details of how sequents and structural rules are formalized. As long as every derivation in LK can be effectively transformed to a derivation using the new rules and vice versa, the modified rules may still be called LK. First of all, as mentioned above, the sequents can be viewed to consist of sets or multisets. In this case, the rules for permuting and (when using sets) contracting formulae are obsolete. The rule of weakening will become admissible, when the axiom (I) is changed, such that any sequent of the form \Gamma , A \vdash A , \Delta can be concluded.
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Integral cryptanalysis is a cryptanalytic attack that is particularly applicable to block ciphers based on substitution–permutation networks. Unlike differential cryptanalysis, which uses pairs of chosen plaintexts with a fixed XOR difference, integral cryptanalysis uses sets or even multisets of chosen plaintexts of which part is held constant and another part varies through all possibilities. For example, an attack might use 256 chosen plaintexts that have all but 8 of their bits the same, but all differ in those 8 bits. Such a set necessarily has an XOR sum of 0, and the XOR sums of the corresponding sets of ciphertexts provide information about the cipher's operation.
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A. B. Kempe (1877) How to draw a straight line; a lecture on linkages, London: Macmillan and Co. In 1879 Kempe wrote his famous "proof" of the four colour theorem, shown incorrect by Percy Heawood in 1890. Much later, his work led to fundamental concepts such as the Kempe chain and unavoidable sets. Kempe (1886) revealed a rather marked philosophical bent, and much influenced Charles Sanders Peirce. Kempe also discovered what are now called multisets, although this fact was not noted until long after his death.A. B. Kempe, (1886) "A memoir on the theory of mathematical form," Philosophical Transactions of the Royal Society of London 177: 1–70Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870–1940.
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Forms of integral cryptanalysis have since been applied to a variety of ciphers, including Hierocrypt, IDEA, Camellia, Skipjack, MISTY1, MISTY2, SAFER++, KHAZAD, and FOX (now called IDEA NXT). Unlike differential cryptanalysis, which uses pairs of chosen plaintexts with a fixed XOR difference, integral cryptanalysis uses sets or even multisets of chosen plaintexts of which part is held constant and another part varies through all possibilities. For example, an attack might use 256 chosen plaintexts that have all but 8 of their bits the same, but all differ in those 8 bits. Such a set necessarily has an XOR sum of 0, and the XOR sums of the corresponding sets of ciphertexts provide information about the cipher's operation.
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In this case, the representation of a tuple as a sequence of stars and bars, with the bars dividing the stars into bins, is unchanged. The weakened restriction of nonnegativity (instead of positivity) means that one may place multiple bars between two stars, as well as placing bars before the first star or after the last star. Thus, for example, when n = 7 and k = 5, the tuple (4, 0, 1, 2, 0) may be represented by the following diagram. This establishes a one-to-one correspondence between tuples of the desired form and selections with replacement of gaps from the available gaps, or equivalently ()-element multisets drawn from a set of size .
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In a set-theoretical definition,Burgin (2011), p. 89-96 named sets are built using sets similar to constructions of fuzzy sets or multisets. Namely, a set-theoretical named set is a triad X = (X, f, I), in which X and I are two sets and f is a set-theoretical correspondence (binary relation) between X and I. Note that not all named sets are set-theoretical. The most transparent example of non-set-theoretical named sets is given by algorithmic named sets, which have the form X = (X, A, I), in which X and I are two constructive objects, for example, sets of words, and A is an algorithm that transforms X into I.
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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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Chapter 11 connects the low- dimensional faces together into the skeleton of a polytope, and proves the van Kampen–Flores theorem about non-embeddability of skeletons into lower- dimensional spaces. Chapter 12 studies the question of when a skeleton uniquely determines the higher-dimensional combinatorial structure of its polytope. Chapter 13 provides a complete answer to this theorem for three- dimensional convex polytopes via Steinitz's theorem, which characterizes the graphs of convex polyhedra combinatorially and can be used to show that they can only be realized as a convex polyhedron in one way. It also touches on the multisets of face sizes that can be realized as polyhedra (Eberhard's theorem) and on the combinatorial types of polyhedra that can have inscribed spheres or circumscribed spheres.
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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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In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved.
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An integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrange's four- square theorem shows that w^2+x^2+y^2+z^2 is universal. Ramanujan generalized this to aw^2+bx^2+cy^2+dz^2 and found 54 multisets that can each generate all positive integers, namely, :{1, 1, 1, d}, 1 ≤ d ≤ 7 :{1, 1, 2, d}, 2 ≤ d ≤ 14 :{1, 1, 3, d}, 3 ≤ d ≤ 6 :{1, 2, 2, d}, 2 ≤ d ≤ 7 :{1, 2, 3, d}, 3 ≤ d ≤ 10 :{1, 2, 4, d}, 4 ≤ d ≤ 14 :{1, 2, 5, d}, 6 ≤ d ≤ 10 There are also forms whose image consists of all but one of the positive integers. For example, {1,2,5,5} has 15 as the exception.
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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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