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Room Square
A Room square, named after Thomas Gerald Room, is an ''n'' × ''n'' array filled with ''n'' + 1 different symbols in such a way that: # Each cell of the array is either empty or contains an unordered pair from the set of symbols # Each symbol occurs exactly once in each row and column of the array # Every unordered pair of symbols occurs in exactly one cell of the array. An example, a Room square of order seven, if the set of symbols is integers from 0 to 7: It is known that a Room square (or squares) exist if and only if ''n'' is odd but not 3 or 5. History The order-7 Room square was used by Robert Richard Anstice to provide additional solutions to Kirkman's schoolgirl problem in the mid-19th century, and Anstice also constructed an infinite family of Room squares, but his constructions did not attract attention. Thomas Gerald Room reinvented Room squares in a note published in 1955, and they came to be named after him. In his original paper on the subjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Gerald Room
Thomas Gerald Room Fellow of the Royal Society, FRS Fellow of the Australian Academy of Science, FAA (10 November 1902 – 2 April 1986) was an Australians, Australian mathematician who is best known for Room squares. He was a Foundation Fellow of the Australian Academy of Science. . Also published in ''Historical Records of Australian Science'' 7 (1): 109–122, . An abridged version ionline at the web site of the Australian Academy of Science First published in ''Australian Dictionary of Biography'', Volume 18, (MUP), 2012. Biography Thomas Room was born on 10 November 1902, near London, England. He studied mathematics in St John's College, Cambridge, and was a Wrangler (University of Cambridge), wrangler in 1923. He continued at Cambridge as a graduate student, and was elected as a fellow in 1925, but instead took a position at the University of Liverpool. He returned to Cambridge in 1927, at which time he completed his PhD, with a thesis supervised by H. F. Baker. Room remai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Richard Anstice
Robert Richard Anstice (1813–1853) was an English clergyman and mathematician who wrote two remarkable papers on combinatorics, published the same year he died in the Cambridge and Dublin mathematical journal. He pioneered the use of primitive roots in this field, anticipating the work of Eugen Netto on Steiner's triplets. Anstice studied at Christ Church, Oxford where he graduated in 1835, receiving a Master's in 1837. Nothing is known about his life in the next ten years. In 1846, he was ordained priest, and in the following year he became rector of Wigginton, Hertfordshire Wigginton (''Wigentone'' - 1086) is a large village and civil parish running north–south and perched at on the edge of the Chiltern Hills and aside the border with Buckinghamshire. It is part of Dacorum district in the county of Hertfords ...., MacTutor History of Mathematics. He died there in 1853 References Bibliography * * * External links * {{DEFAULTSORT:Anstice, Robert Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirkman's Schoolgirl Problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in ''The Lady's and Gentleman's Diary'' (pg.48). The problem states: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. Solutions A solution to this problem is an example of a ''Kirkman triple system'', which is a Steiner triple system having a ''parallelism'', that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks. Such Steiner systems that have a parallelism are also called ''resolvable''. There are exactly seven non-isomorphic solutions to the schoolgirl problem, as originally listed by Frank Nelson Cole in ''Kirkman Parades'' in 1922. The seven solutions are summarized in the table below, denoting the 15 girls with the letters A to O. From the numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duplicate Bridge
Duplicate bridge is a variation of contract bridge where the same set of bridge deals (i.e. the distribution of the 52 cards among the four hands) are played by different competitors, and scoring is based on relative performance. In this way, every hand, whether strong or weak, is played in competition with others playing identical cards, and the element of skill is heightened while that of chance is reduced. This stands in contrast to Bridge played without duplication, where each hand is freshly dealt and where scores may be more affected by chance in the short run. Four-way card holders known as Bridge boards are used to enable each player's hand to be preserved from table to table, and final scores are calculated by comparing each pair's result with others who played the same hand. In duplicate bridge, players normally play all the hands with the same partner, and compete either as a partnership (in a 'Pairs tournament') or on a team with one or more other partnerships ('Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have an identity element. A quasigroup with an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition. Algebra A quasigroup is a non-empty set ''Q'' with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), obeying the Latin square property. This states that, for each ''a'' and ''b'' in ''Q'', there exist uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin Square
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin square is The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3. Euler began the general theory of Latin squares. History The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years. Reduced form A Latin square is said to be ''reduced'' (also, ''normalized'' or ''in standard form'') if both its first row and its first column are in their natural order. For example, the La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Factorization
In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''- regular subgraph, and a ''k''-factorization partitions the edges of the graph into disjoint ''k''-factors. A graph ''G'' is said to be ''k''-factorable if it admits a ''k''-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a ''k''-regular graph is an edge coloring with ''k'' colors. A 2-factor is a collection of cycles that spans all vertices of the graph. 1-factorization If a graph is 1-factorable (ie, has a 1-factorization), then it has to be a regular graph. However, not all regular graphs are 1-factorable. A ''k''-regular graph is 1-factorable if it has chromatic index ''k''; examples of such graphs include: * Any regular bipartite graph. Hall's marriage theorem can be used to show that a ''k''-regular bipartite graph contains a perfect matching. One can then remove t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner System
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids. Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematical Statistics
The ''Annals of Mathematical Statistics'' was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the ''Annals of Statistics'' and the ''Annals of Probability''. In 1938, Samuel Wilks became editor-in-chief of the ''Annals'' and recruited a remarkable editorial staff: Fisher, Neyman, Cramér, Hotelling, Egon Pearson, Georges Darmois, Allen T. Craig, Deming, von Mises Mises or von Mises may refer to: * Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises ** Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von ..., H. L. Rietz, and Shewhart. References {{reflist External links Annals of Mathematical Statistics at Project Euclid Statistics journals Probability journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
