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Somos Sequence
In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but surprisingly, a few Somos sequences have the property that all of their members are integers. Recurrence equations For an integer number ''k'' larger than 1, the Somos-''k'' sequence (a_0, a_1, a_2, \ldots ) is defined by the equation :a_n a_ = a_ a_ + a_ a_ + \cdots + a_ a_ when ''k'' is odd, or by the analogous equation :a_n a_ = a_ a_ + a_ a_ + \cdots + (a_)^2 when ''k'' is even, together with the initial values : ''a''''i'' = 1 for ''i'' < ''k''. For ''k'' = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, ''k'' = 4, the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Somos
Michael Somos is an American mathematician, who was a visiting scholar in the Georgetown University Mathematics and Statistics department for four years and is a visiting scholar at The Catholic University of America. In the late eighties he proposed a conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers. Somos' quadratic recurrence constant is also named after him. Notes References * Michael Somos and Robert Haas, "A Linked Pair of Sequences Implies the Primes Are Infinite", ''The American Mathematical Monthly'', volume 110, number 6 (June – July, 2003), pp. 539–540 External links Michael Somos's homepage '' |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Algebra
Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy various conditions. Definitions Suppose that ''F'' is an integral domain, such as the field Q(''x''1,...,''x''''n'') of rational functions in ''n'' variables over the rational numbers Q. A cluster of rank ''n'' consists of a set of ''n'' elements of ''F'', usually assumed to be an algebraically independent set of generators of a field extension ''F''. A seed consists of a cluster of ''F'', together with an exchange matrix ''B'' with integer entries ''b''''x'',''y'' indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be skew-symmetric, so that ''b''''x'',''y'' = –''b''''y'',''x'' for all ''x'' and ''y''. More generally the matrix might be skew-symmetrizable, meaning there are positive integers '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Applied Mathematics
''Advances in Applied Mathematics'' is a peer-reviewed mathematics journal publishing research on applied mathematics. Its founding editor was Gian-Carlo Rota (Massachusetts Institute of Technology); from 1980 to 1999, Joseph P. S. Kung (University of North Texas) served as managing editor. It is currently published by Elsevier with eight issues per year and edited by Hal Schenck (Auburn University) and Catherine Yan (Texas A&M University). Abstracting and indexing The journal is abstracted and indexed by: * ACM Guide to Computing Literature * CompuMath Citation Index * Current Contents/Physics, Chemical, & Earth Sciences * ''Mathematical Reviews'' * Science Citation Index * Scopus According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Journal Of Combinatorics
The ''Electronic Journal of Combinatorics'' is a peer-reviewed open access scientific journal covering research in combinatorial mathematics. The journal was established in 1994 by Herbert Wilf (University of Pennsylvania) and Neil Calkin (Georgia Institute of Technology). The Electronic Journal of Combinatorics is a founding member of the Free Journal Network. According to the ''Journal Citation Reports'', the journal had a 2017 impact factor of 0.762. Editors-in-chief Current The current editors-in-chief at ''Electronic Journal of Combinatorics'' are: * Maria Axenovich, Karlsruhe Institute of Technology, Germany * Miklós Bóna, University of Florida, United States * Julia Böttcher, London School of Economics, United Kingdom * Richard A. Brualdi, University of Wisconsin, Madison, United States * Zdeněk Dvořák, Charles University, Czech Republic * Nikolaos Fountoulakis, University of Birmingham, United Kingdom * Eric Fusy, CNRS/LIX, École Polytechnique, France * Feli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Göbel's Sequence
In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation :x_n = \frac,\!\, with starting value :x_0 = x_1 = 1. Göbel's sequence starts with : 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... The first non-integral value is ''x''43. History This sequence was developed by the German mathematician Fritz Göbel in the 1970s. In 1975, the Dutch mathematician Hendrik Lenstra showed that the 43rd term is not an integer. Generalization Göbel's sequence can be generalized to ''k''th powers by :x_n = \frac. The least indices at which the ''k''-Göbel sequences assume a non-integral value are :43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ... Regardless of the value chosen for ''k'', the initial 19 terms are always integers. See also * Somos sequence In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numberphile
''Numberphile'' is an Educational entertainment, educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis and Kruskal's tree theorem. The videos are produced by Brady Haran, a former BBC video journalist and creator of Periodic Videos, Sixty Symbols, and several other YouTube channels. Videos on the channel feature several university professors, maths communicators and famous mathematicians. In 2018, Haran released a spin-off (media), spin-off audio podcast titled ''The Numberphile Podcast''. YouTube channel The ''Numberphile'' YouTube channel was started on 15 September 2011. Most videos consist of Haran interviewing an expert on a number, mathematical theorem or other mathematical concept. The expert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable if there exists an algorithm that, given '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
