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Necklace Problem
The necklace problem is a problem in recreational mathematics concerning the reconstruction of necklaces (cyclic arrangements of binary values) from partial information. Formulation The necklace problem involves the reconstruction of a necklace of n beads, each of which is either black or white, from partial information. The information specifies how many copies the necklace contains of each possible arrangement of k black beads. For instance, for k=2, the specified information gives the number of pairs of black beads that are separated by i positions, for i=0,\dots \lfloor n/2-1 \rfloor . This can be made formal by defining a k-configuration to be a necklace of k black beads and n-k white beads, and counting the number of ways of rotating a k-configuration so that each of its black beads coincides with one of the black beads of the given necklace. The necklace problem asks: if n is given, and the numbers of copies of each k-configuration are known up to some threshold k\le K, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics (especially social choice theory), dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods. The archetypal fair division algorithm is divide and choose. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necklace (combinatorics)
In combinatorics, a ''k''-ary necklace of length ''n'' is an equivalence class of ''n''-character strings over an alphabet of size ''k'', taking all rotations as equivalent. It represents a structure with ''n'' circularly connected beads which have ''k'' available colors. A ''k''-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other, they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace. Formally, one may represent a necklace as an orbit of the cyclic group acting on ''n''-character strings over an alphabet of size ''k'', and a bracelet as an orbit of the dihedral group. One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem. Equivalence classes Number of necklaces There are :N_k(n)=\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noga Alon
Noga Alon ( he, נוגה אלון; born 17 February 1956) is an Israeli mathematician and a professor of mathematics at Princeton University noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers. Academic background Alon is a Professor of Mathematics at Princeton University and a Baumritter Professor Emeritus of Mathematics and Computer Science at Tel Aviv University, Israel. He graduated from the Hebrew Reali School in 1974 and received his Ph.D. in Mathematics at the Hebrew University of Jerusalem in 1983 and had visiting positions in various research institutes including MIT, The Institute for Advanced Study in Princeton, IBM Almaden Research Center, Bell Labs, Bellcore and Microsoft Research. He serves on the editorial boards of more than a dozen international journals; since 2008 he is the editor-in-chief of ''Random Structures and Algorithms''. He has given lectures in many conferences, including plenary addresses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yair Caro , a location in Scotland
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Yair may refer to: *A spelling variant of the Jewish name Jair or Ya'ir *Yair (name), list of people with the name Yair *Yair, Scottish Borders Yair, also known as The Yair, is an estate in the Scottish Borders. It stands by the River Tweed in the former county of Selkirkshire, north-west of Selkirk, and south of Edinburgh. The name comes from the old Scots word for a fish trap. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ilia Krasikov
Ilia may refer to: Science and medicine *''Apatura ilia'' or lesser purple emperor, a butterfly *Ilium (bone) (plural: "ilia"), pelvic bone People * Ilia (name), numerous **Ilia II, the current Catholicos-Patriarch of All Georgia Places *Ilia, Hunedoara, Romania *Elis (regional unit), Greece *Elis Province, Greece Arts and literature *Ilia, a character in ''Idomeneo'', an opera by Mozart *Ilia (The Legend of Zelda), a character in the video game ''The Legend of Zelda: Twilight Princess'' * Ilia (''Star Trek''), a character in ''Star Trek: The Motion Picture'' *Ilia, a nation of the continent Elibe from the ''Fire Emblem'' series *Ilia the Righteous, a prominent figure of new Georgian literature *Rhea Silvia or Ilia, the mother of Romulus and Remus in Roman mythology Other *Illinois Institute of Art – Chicago, a nonprofit institution *Ilia (band), a rock band *Arturo Umberto Illia (1900–1983), former president of Argentina See also *Elia (other) *Ilija (disambigua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yehuda Roditty
Judah or Yehuda is the name of a biblical patriarch, Judah (son of Jacob). It may also refer to: Historical ethnic, political and geographic terms * Tribe of Judah, one of the twelve Tribes of Israel; their allotment corresponds to Judah or Judaea * Judea, the name of part of the Land of Israel ** Kingdom of Judah, an Iron Age kingdom of the Southern Levant *** History of ancient Israel and Judah ** Yehud (Persian province), a name introduced in the Babylonian period ** Judaea (Roman province) People * Judah (given name), or Yehudah, including a list of people with the name * Judah (surname) Other uses * Judah, Indiana, a small town in the United States * N Judah, a light trail line in San Francisco, U.S. * Yehuda Matzos, an Israeli matzo company See also * Juda (other) * Judas (other) * Jude (other) * Jews, an ethnoreligious group and nation originating from the Israelites and Hebrews of historical Israel and Judah * Judas Iscariot Judas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion–exclusion Principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \cup B, = , A, + , B, - , A \cap B, where ''A'' and ''B'' are two finite sets and , ''S'', indicates the cardinality of a set ''S'' (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets ''A'', ''B'' and ''C'' is given by :, A \cup B \cup C, = , A, + , B, + , C, - , A \cap B, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jamie Radcliffe
Jamie is a unisex name. It is a diminutive form of James or, more rarely, other names. It is also given as a name in its own right. People Female * Jamie Anne Allman (born 1977), American actress * Jamie Babbit (born 1970), American film and television director * Jamie Belsito (born 1973), American politician * Jamie Bernadette, American actress and occasional producer * Jamie Bochert (born 1978), American fashion model and musician * Jamie Brewer, American actress and model * Jamie Broumas (born 1959), American jazz singer * Jamie Chadwick (born 1998), British racing driver * Jamie Chung (born 1983), American actress * Jamie Clayton (born 1978), American actress and model * Jamie Lee Curtis (born 1958), American actress and author * Jamie Dantzscher (born 1982), American artistic gymnast * Jamie Finn (born 1998, Irish footballer * Jamie Gauthier, American Democratic politician * Jamie Ginn (born 1982), American beauty queen * Jamie Gorelick (born 1950), American lawyer * Jamie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bracelet (combinatorics)
In combinatorics, a ''k''-ary necklace of length ''n'' is an equivalence class of ''n''-character strings over an alphabet of size ''k'', taking all rotations as equivalent. It represents a structure with ''n'' circularly connected beads which have ''k'' available colors. A ''k''-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other, they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace. Formally, one may represent a necklace as an orbit of the cyclic group acting on ''n''-character strings over an alphabet of size ''k'', and a bracelet as an orbit of the dihedral group. One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem. Equivalence classes Number of necklaces There are :N_k(n)=\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moreau's Necklace-counting Function
In combinatorial mathematics, the necklace polynomial, or Moreau's necklace-counting function, introduced by , counts the number of distinct necklaces of ''n'' colored beads chosen out of α available colors. The necklaces are assumed to be aperiodic (not consisting of repeated subsequences), and counted up to rotation (rotating the beads around the necklace counts as the same necklace), but without flipping over (reversing the order of the beads counts as a different necklace). This counting function also describes, among other things, the dimensions in a free Lie algebra and the number of irreducible polynomials over a finite field. Definition The necklace polynomials are a family of polynomials M(\alpha,n) in the variable \alpha such that :\alpha^n \ =\ \sum_ d \, M(\alpha, d). By Möbius inversion they are given by : M(\alpha,n) \ =\ \sum_\mu\!\left(\right)\alpha^d, where \mu is the classic Möbius function. A closely related family, called the general necklace polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necklace Splitting Problem
Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon and Douglas B. West. The basic setting involves a necklace with beads of different colors. The necklace should be divided between several partners (e.g. thieves), such that each partner receives the same amount of every color. Moreover, the number of ''cuts'' should be as small as possible (in order to waste as little as possible of the metal in the links between the beads). Variants The following variants of the problem have been solved in the original paper: #Discrete splitting: The necklace has k\cdot n beads. The beads come in t different colors. There are k\cdot a_i beads of each color i, where a_i is a positive integer. Partition the necklace into k parts (not necessarily contiguous), each of which has exactly a_i beads of color ''i''. Use at most (k-1)t cuts. Note that if the beads of each color are c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
