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Rencontres Number
In combinatorics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set with specified numbers of fixed points: in other words, partial derangements. (''Rencontre'' is French for ''encounter''. By some accounts, the problem is named after a solitaire game.) For ''n'' ≥ 0 and 0 ≤ ''k'' ≤ ''n'', the rencontres number ''D''''n'', ''k'' is the number of permutations of that have exactly ''k'' fixed points. For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are ''D''7, 2 = 924 ways this could happen. Another often cited example is that of a dance school with 7 opposite-sex couples, where, after tea-break the participants are told to ''randomly'' find an opposite-sex partner to continue, then once more there are ''D''7, 2 = 924 possibilities that exactly 2 previous couples meet again by chance. Numerical va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution ( Hausdorff moment problem). The same is not true on unbounded intervals ( Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments of random variables. Significance of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Points (mathematics) .
{{disambiguation, math ...
Fixed point may refer to: * Fixed point (mathematics), a value that does not change under a given transformation * Fixed-point arithmetic, a manner of doing arithmetic on computers * Fixed point, a benchmark (surveying) used by geodesists * Fixed point join, also called a recursive join * Fixed point, in quantum field theory, a coupling where the beta function vanishes – see * Temperature reference point, usually defined by a phase change or triple point In thermodynamics, the triple point of a substance is the temperature and pressure at which the three Phase (matter), phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Distributions
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution, a random variable that can be counted *Discrete space, a simple example of a topological space *Discrete spline interpolation, the discrete analog of ordinary spline interpolation *Discrete time, non-continuous time, which results in discrete-time samples *Discrete variable In mathematics and statistics, a quantitative variable may be continuous or discrete. If it can take on two real values and all the values between them, the variable is con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutations
In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is &nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Riordan (mathematician)
John Francis Riordan (April 22, 1903 – August 27, 1988) was an American mathematician and the author of major early works in combinatorics, particularly ''Introduction to Combinatorial Analysis'' and ''Combinatorial Identities''. Biography Riordan was a graduate of Yale University. In his early life he wrote a number of poems and essays and a book of short-stories, ''On the Make'', published in 1929, and was Editor-in-Chief of ''Salient'' and ''The Figure in the Carpet'', literary magazines published by The New School for Social Research in New York. He married Mavis McIntosh, the well-known poet and literary agent and founder of McIntosh & Otis. The couple had two daughters: Sheila Riordan and Kathleen Riordan Speeth, and were long time residents of Hastings-on-Hudson, New York. Riordan's long professional career was at Bell Labs, which he joined in 1926 (a year after its foundation) and where he remained, publishing over a hundred scholarly papers on combinatorial analysis, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oberwolfach Problem
In mathematics, the Oberwolfach problem is an open problem that may be formulated either as a problem of scheduling seating assignments for diners, or more abstractly as a problem in graph theory, on the edge cycle covers of complete graphs. It is named after the Oberwolfach Research Institute for Mathematics, where the problem was posed in 1967 by Gerhard Ringel. It is known to be true for all sufficiently large complete graphs. Formulation In conferences held at Oberwolfach, it is the custom for the participants to dine together in a room with circular tables, not all the same size, and with assigned seating that rearranges the participants from meal to meal. The Oberwolfach problem asks how to make a seating chart for a given set of tables so that all tables are full at each meal and all pairs of conference participants are seated next to each other exactly once. An instance of the problem can be denoted as OP(x,y,z,\dots) where x,y,z,\dots are the given table sizes. Alternati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted B_n, where n is an integer greater than or equal to zero. Starting with B_0 = B_1 = 1, the first few Bell numbers are :1, 1, 2, 5, 15, 52, 203, 877, 4140, \dots . The Bell number B_n counts the different ways to partition a set that has exactly n elements, or equivalently, the equivalence relations on it. B_n also counts the different rhyme schemes for n -line poems. As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, B_n is the n -th moment of a Poisson distribution with mean 1. Counting Set partitions In general, B_n is the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Vadim Ponomarenko ( San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union (set theory), union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
