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Chaitin Et Al
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic (Solomonoff–Kolmogorov–Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy. Mathematics and comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chicago
(''City in a Garden''); I Will , image_map = , map_caption = Interactive Map of Chicago , coordinates = , coordinates_footnotes = , subdivision_type = Country , subdivision_name = United States , subdivision_type1 = State , subdivision_type2 = Counties , subdivision_name1 = Illinois , subdivision_name2 = Cook and DuPage , established_title = Settled , established_date = , established_title2 = Incorporated (city) , established_date2 = , founder = Jean Baptiste Point du Sable , government_type = Mayor–council , governing_body = Chicago City Council , leader_title = Mayor , leader_name = Lori Lightfoot ( D) , leader_title1 = City Clerk , leader_name1 = Anna Valencia ( D) , unit_pref = Imperial , area_footnotes = , area_tot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. Biography Early life Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. His unmarried mother, Maria Y. Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do nobleman. Little is known about Andrey's father. He was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Kataev had been exiled from St. Petersburg to the Yaroslavl province after his participation in the revolutionary movem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definable Number
Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, \sqrt, can be defined as the unique positive solution to the equation x^2 = 2, and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable. Constructible numbers One way of specifying a real number uses geometric techniques. A real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−''n''. Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips ( binary) or rolls of a die (base 6). Even though there ''will'' be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
City College Of New York
The City College of the City University of New York (also known as the City College of New York, or simply City College or CCNY) is a public university within the City University of New York (CUNY) system in New York City. Founded in 1847, City College was the first free public institution of higher education in the United States. It is the oldest of CUNY's 25 institutions of higher learning, and is considered its flagship college. Located in Hamilton Heights overlooking Harlem in Manhattan, City College's 35-acre (14 ha) Collegiate Gothic campus spans Convent Avenue from 130th to 141st Streets. It was initially designed by renowned architect George B. Post, and many of its buildings have achieved landmark status. The college has graduated ten Nobel Prize winners, one Fields Medalist, one Turing Award winner, three Pulitzer Prize winners, and three Rhodes Scholars. Among these alumni, the latest is a Bronx native, John O'Keefe (2014 Nobel Prize in Medicine). City College' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bronx High School Of Science
The Bronx High School of Science, commonly called Bronx Science, is a public specialized high school in The Bronx in New York City. It is operated by the New York City Department of Education. Admission to Bronx Science involves passing the Specialized High Schools Admissions Test. Each November, about 30,000 eighth and ninth graders take the three-hour test for admittance to eight of the nine specialized high schools. The test is extremely competitive, with only 800 of the 30,000 applicants being accepted to Bronx Science each year. Founded in 1938 in the Bronx, New York City, Bronx Science is now situated in an educational area known as the Educational Mile in Bedford Park, a neighborhood in the northwest portion of the Bronx. Although originally known for its focus on mathematics and science, Bronx Science also emphasizes the humanities and social sciences and continually attracts students with a wide variety of interests beyond math and science. With eight Nobel Prize-wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering (field), information engineering, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a dice, die (with six equally likely outcomes). Some other important measures in information theory are mutual informat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonid Levin
Leonid Anatolievich Levin ( ; russian: Леони́д Анато́льевич Ле́вин; uk, Леоні́д Анато́лійович Ле́він; born November 2, 1948) is a Soviet-American mathematician and computer scientist. He is known for his work in randomness in computing, algorithmic complexity and intractability, average-case complexity, foundations of mathematics and computer science, algorithmic probability, theory of computation, and information theory. He obtained his master's degree at Moscow University in 1970 where he studied under Andrey Kolmogorov and completed the Candidate Degree academic requirements in 1972. He and Stephen Cook independently discovered the existence of NP-complete problems. This NP-completeness theorem, often called the Cook–Levin theorem, was a basis for one of the seven Millennium Prize Problems declared by the Clay Mathematics Institute with a $1,000,000 prize offered. The Cook–Levin theorem was a breakthrough in computer s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at Stockholm University.Member profile |
