Double Factorial
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Double Factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the double factorial is :n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, and for odd it is :n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of t ...
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Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function ...
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Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\righ ...
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Euler Tour Technique
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree. The ETT allows for efficient, parallel computation of solutions to common problems in algorithmic graph theory. It was introduced by Tarjan and Vishkin in 1984. Construction Given an undirected tree presented as a set of edges, the Euler tour representation (ETR) can be constructed in parallel as follows: * We construct a symmetric list of directed edges: ** For each undirected edge in the tree, insert (''u'',''v'') and (''v'',''u'') in the edge list. * Sort the edge list lexicographically. (Here we assume that the nodes of the tree are ordered, and that the root is the first element in this order.) * Construct adjacency lists for each ...
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Heap (data Structure)
In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a ''max heap'', for any given node C, if P is a parent node of C, then the ''key'' (the ''value'') of P is greater than or equal to the key of C. In a ''min heap'', the key of P is less than or equal to the key of C. The node at the "top" of the heap (with no parents) is called the ''root'' node. The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspe ...
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Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is so ...
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Stirling Permutation
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 (2''k'' − 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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Journal Of Computational Biology
The ''Journal of Computational Biology'' is a monthly peer-reviewed scientific journal covering computational biology and bioinformatics. It was established in 1994 and is published by Mary Ann Liebert, Inc. The editors-in-chief are Sorin Istrail (Brown University) and Michael S. Waterman (University of Southern California). According to the ''Journal Citation Reports'', the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 0.879. Since 1997, authors of accepted proceedings papers at the Research in Computational Molecular Biology (RECOMB) conference have been invited to submit a revised version to a special issue of the journal. References External links * Mary Ann Liebert academic journals Publications established in 1994 English ...
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Telephone Number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on vertices, the number of permutations on elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from ) Applications John Riordan provides the following explanation for these numbers: suppose that people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different pa ...
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Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory. Education and career Alfred Brauer was Richard's brother and seven years older. They were born to a Jewish family. Both were interested in science and mathematics, but Alfred was injured in combat in World War I. As a boy, Richard dreamt of becoming an inventor, and in February 1919 enrolled in Technische Hochschule Berlin-Charlottenburg. He soon transferred to University of Berlin. Except for the summer of 1920 when he studied at University of Freiburg, he studied in Berlin, being awarded his PhD on 16 March 1926. Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result. The problem also was solved by Heinz Hopf at the same time. Richard wrote his thesis ...
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Chord Diagram (mathematics)
In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle. The number of different chord diagrams that may be given for a set of 2n cyclically ordered objects is the double factorial (2n-1)!!. There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross. In knot theory, a chord diagram can be used to described the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, t ...
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Permutations
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of s ...
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Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...; and reciprocal ciphers such as the ROT13 transformation and the Beaufort cipher, Beaufort polyalphabetic cipher. The Function composition, composition of two invol ...
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