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Hofstadter Sequence
In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations. Sequences presented in ''Gödel, Escher, Bach: an Eternal Golden Braid'' The first Hofstadter sequences were described by Douglas Richard Hofstadter in his book ''Gödel, Escher, Bach''. In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are: Hofstadter Figure-Figure sequences The Hofstadter Figure-Figure (R and S) sequences are a pair of complementary integer sequences defined as follows : \begin R(1)&=1~ ;\ S(1)=2 \\ R(n)&=R(n-1)+S(n-1), \quad n>1. \end with the sequence S(n) defined as a strictly increasing series of positive integers not present in R(n). The first few terms of these sequences are :R: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, ... :S: 2, 4, 5, 6, 8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear System
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relations
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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. His 1979 book '' Gödel, Escher, Bach: An Eternal Golden Braid'' won both the Pulitzer Prize for general nonfiction"General Nonfiction" . ''Past winners and finalists by category''. The Pulitzer Prizes. Retrieved March 17, 2012. and a (at that time called The American Book Award) for Science. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel, Escher, Bach
''Gödel, Escher, Bach: an Eternal Golden Braid'', also known as ''GEB'', is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. It also discusses self-reference and formal rules, isomorphism, what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself. In response to confusion over the book's theme, Hofstadter emphasized that ''Gödel, Escher, Bach'' is not about the relationships of mathematics, art, and music—but rather about how cognition emerges from hidden neurological mechanisms. One point in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambek–Moser Theorem
The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two Complement (set theory), complementary sets. For instance, it applies to the partition of numbers into even number, even and odd number, odd, or into prime number, prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states that any two monotonic function, non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states that every complementary partition can be constructed in this way. When a formula is known for the natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the number not in the set. The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greg Huber
Greg is a masculine given name, and often a shortened form of the given name Gregory. Greg (more commonly spelled " Gregg") is also a surname. People with the name *Greg Abbott (other), multiple people *Greg Abel (born 1961/1962), Canadian businessman *Greg Adams (other), multiple people *Greg Allen (other), multiple people *Greg Anderson (other), multiple people *Greg Austin (other), multiple people *Greg Ball (other), multiple people *Greg Bell (other), multiple people *Greg Bennett (other), multiple people *Greg Berlanti (born 1972), American writer and producer *Greg Biffle (born 1969), American NASCAR driver *Greg Blankenship (born 1954), American football player *Greg Boyd (other), multiple people *Greg Boyer (other), multiple people *Greg Brady (broadcaster) (born 1971), Canadian sports radio host *Greg Brock (baseball) (born 1957), American baseball player *Greg Brooker (disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klaus Pinn
Klaus is a German, Dutch and Scandinavian given name and surname. It originated as a short form of Nikolaus, a German form of the Greek given name Nicholas. Notable persons whose family name is Klaus *Billy Klaus (1928–2006), American baseball player *Chris Klaus (born 1973), American entrepreneur *Frank Klaus (1887–1948), German-American boxer, 1913 Middleweight Champion *Fred Klaus (born 1967), German footballer *Josef Klaus (1910–2001), Chancellor of Austria 1966–1970 *Karl Ernst Claus (1796–1864), Russian chemist *Václav Klaus (born 1941), Czech politician, former President of the Czech Republic *Walter K. Klaus (1912–2012), American politician and farmer Notable persons whose given name is Klaus *Brother Klaus, Swiss patron saint *Klaus Augenthaler (born 1957), German football player and manager *Klaus Badelt (born 1967), German composer *Klaus Barbie (1913–1991), German SS-Hauptsturmführer and Holocaust Perpetrator *Klaus Bargsten (1911–2000), German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Of Convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
