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Diophantine Set
In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x''''j'', ''y''1, ..., ''y''''k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., ''x''''j'' indicate parameters and ''y''1, ..., ''y''''k'' indicate unknowns. A Diophantine set is a subset ''S'' of \mathbb^j, the set of all ''j''-tuples of natural numbers, so that for some Diophantine equation ''P''(', ') = 0, :\bar \in S \iff (\exists \bar \in \mathbb^)(P(\bar,\bar)=0) . That is, a parameter value is in the Diophantine set ''S'' if and only if the associated Diophantine equation is satisfiable under that parameter value. The use of natural numbers both in ''S'' and the existential quantification merely reflects the usual applications in computability theory and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Recursive Set
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural number in a finite number of steps. A set is noncomputable (or undecidable) if it is not computable. Definition A subset S of the natural numbers is computable if there exists a total computable function f such that: :f(x)=1 if x\in S :f(x)=0 if x\notin S. In other words, the set S is computable if and only if the indicator function \mathbb_ is computable. Examples *Every recursive language is a computable. *Every finite or cofinite subset of the natural numbers is computable. **The empty set is computable. **The entire set of natural numbers is computable. **Every natural number is computable. *The subset of prime numbers is computable. *The set of Gödel numbers is computable. Non-examples *The set of Turing machines that halt is not computable. *The set of pairs of homeomorphic finite simplicial complexes is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Zhi Wei Sun
Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University. Biography Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall–Sun–Sun primes. Sun proved Sun's curious identity in 2002. In 2003, he presented a unified approach to three topics of Paul Erdős in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problems or EGZ Theorem. With Stephen Redmond, he posed the Redmond–Sun conjecture in 2006. In 2013, he published a paper containing many conjectures on primes, one of which states that for any positive integer m there are consecutive primes p_k,\ldots,p_n\ (k [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Recursive Language
In mathematics, logic and computer science, a recursive (or ''decidable'') language is a recursive subset of the Kleene closure of an alphabet. Equivalently, a formal language is recursive if there exists a Turing machine that decides the formal language. In theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply ''decidable''. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy. All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Recursively Enumerable Language
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable. The class of all recursively enumerable languages is called RE. Definitions There are three equivalent definitions of a recursively enumerable language: # A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language. # A recursively enumerable language is a formal language for which there exists a Turing mac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fibonacci Number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 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, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm can run forever, and no information is returned. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. Outside philosophy, Putnam contributed to mathematics and computer science. Together with Martin Davis (mathematician), Martin Davis he developed the Davis–Putnam algorithm for the Boolean satisfiability problem and he helped demonstrate the unsolvability of Hilbert's tenth problem. Putnam applied equal scrutiny to his own philosophical positions as to those of others, subjecting each position to rigorous analysis until he exposed its flaws. As a result, he acquired a reputation for frequently changing his positions. In philosophy of mind, Putnam argued against the type physicalism, type-identity of mental and physical states based on his hypothesis of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Martin Davis (mathematician)
Martin David Davis (March 8, 1928 – January 1, 2023) was an American mathematician and computer scientist who contributed to the fields of computability theory and mathematical logic. His work on Hilbert's tenth problem led to the MRDP theorem. He also advanced the Post–Turing model and co-developed the Davis–Putnam–Logemann–Loveland (DPLL) algorithm, which is foundational for Boolean satisfiability solvers. Davis won the Leroy P. Steele Prize, the Chauvenet Prize (with Reuben Hersh), and the Lester R. Ford Award. He was a fellow of the American Academy of Arts and Sciences and a fellow of the American Mathematical Society. Early life and education Davis's parents were Jewish immigrants to the United States from Łódź, Poland, and married after they met again in New York City. Davis was born in New York City on March 8, 1928. He grew up in the Bronx, where his parents encouraged him to obtain a full education.. He graduated from the prestigious Bronx High ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Julia Robinson
Julia Hall Bowman Robinson (December 8, 1919July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problems. Her work on Hilbert's tenth problem (now known as Matiyasevich's theorem or the MRDP theorem) played a crucial role in its ultimate resolution. Robinson was a 1983 MacArthur Fellow. Early years Robinson was born in St. Louis, Missouri, the daughter of Ralph Bowers Bowman and Helen (Hall) Bowman. Her father owned a machine equipment company while her mother was a school teacher before marriage. Her mother died when Robinson was 2 years old and her father remarried. Her older sister was the mathematical popularizer and biographer Constance Reid and her younger sister is Billie Comstock. When she was 9 years old, she was diagnosed with scarlet fever, which was shortly followed by rheumatic fever. This caused her to miss two years of school. When she was w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich (; born 2 March 1947 in Leningrad Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland ...) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem (Matiyasevich's theorem), which was presented in his doctoral thesis at LOMI (the Leningrad Department of the Steklov Institute of Mathematics). Biography Early years and education Yuri Matiyasevich was born in Leningrad on March 2, 1947. The first few classes he studied at school No. 255 with Sofia G. Generson, thanks to whom he became interested in mathematics. In 1961 he began to participate in all-Russian olympiads. From 1962 to 1963 he studied at Leningrad Saint Petersburg Lyceum 239, physical and mathematical school No. 239. Also from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
