1.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
2.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
3.
Diophantine equation
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In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
4.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
5.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
6.
Martin Davis
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Martin David Davis is an American mathematician, known for his work on Hilberts tenth problem. Daviss parents were Jewish immigrants to the US from Łódź, Poland, Davis grew up in the Bronx, where his parents encouraged him to obtain a full education. He received his Ph. D. from Princeton University in 1950 and he is Professor Emeritus at New York University. Davis is the co-inventor of the Davis–Putnam algorithm and the DPLL algorithms and he is also known for his model of Post–Turing machines. In 1975, Davis won the Leroy P. Steele Prize, the Chauvenet Prize and he became a fellow of the American Academy of Arts and Sciences in 1982, and in 2012, he was selected as one of the inaugural fellows of the American Mathematical Society. Davis, Martin, Weyuker, Elaine J. Sigal, Ron, computability, complexity, and languages, fundamentals of theoretical computer science. Engines of logic, mathematicians and the origin of the computer, review of Engines of logic, Wallace, Richard S. Mathematicians who forget the mistakes of history, a review of Engines of Logic by Martin Davis, ALICE A. I. Hardcover edition published as, The Universal Computer Articles Davis, Martin, Is mathematical insight algorithmic, Behavioral and Brain Sciences,13, criticism of non-standard analysis Halting problem Influence of non-standard analysis Martin Davis website
7.
Yuri Matiyasevich
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Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his solution of Hilberts tenth problem. As a winner of IMO Yuri Matiyasevich was accepted without exams to LSU, in 1966, he presented a talk at International Congress of Mathematicians held in Moscow. He was an undergraduate student at that time. In 1969–1970, he pursued Ph. D. studies at Leningrad Department of Steklov Institute of Mathematics under supervision of Sergey Maslov, in 1970, he received his Ph. D. degree at LOMI. In 1970–1974, he was a researcher at LOMI, in 1972, he obtained a second doctoral degree. In 1974–1980, he was a researcher at LOMI. Since 1980, Yuri Matiyasevich has been the head of Laboratory of mathematical logic at LOMI, since 1995, he has been a professor of Saint-Petersburg State University, initially at the chair of software engineering, later at the chair of algebra and number theory. In 1997, he was elected as a member of Russian Academy of Sciences. Since 1998, Yuri Matiyasevich has been a vice-president of St. Petersburg Mathematical Society, since 2002, he has been a head of St. Petersburg City Mathematical Olympiad. Since 2003, Matiyasevich has been a co-director of annual German–Russian student school JASS, in 2008, he was elected as a full member of Russian Academy of Sciences. 1964, Gold medal at the International Mathematical Olympiad held in Moscow,1970, Young mathematician prize of the Leningrad Mathematical Society. 1980, Markov Prize of Academy of Sciences of the USSR,1998, He received Humboldt Research Award to Outstanding Scholars. 2003, Honorary Degree, Université Pierre et Marie Curie,2007, Member of the Bayern Academy of Sciences. A polynomial related to the colorings of a triangulation of a sphere was named after Matiyasevich, see The Matiyasevich polynomial, four colour theorem, notable students include, Eldar Musayev, Maxim Vsemirnov, Alexei Pastor, Dmitri Karpov. Yuri Matiyasevich Hilberts 10th Problem, Foreword by Martin Davis and Hilary Putnam, real-time recognition of the inclusion relation. Reduction of an arbitrary Diophantine equation to one in 13 unknowns, decision Problems for Semi-Thue Systems with a Few Rules. Yuri Matiyasevich, Proof Procedures as Bases for Metamathematical Proofs in Discrete Mathematics, Yuri Matiyasevich, Elimination of bounded universal quantifiers standing in front of a quantifier-free arithmetical formula, Personal Journal of Yuri Matiyasevich
8.
Hilary Putnam
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Hilary Whitehall Putnam was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions to philosophy of mind, philosophy of language, philosophy of mathematics, at the time of his death, Putnam was Cogan University Professor Emeritus at Harvard University. As a result, he acquired a reputation for changing his own position. In the field of epistemology, he is known for his critique of the well known brain in a vat thought experiment and this thought experiment appears to provide a powerful argument for epistemological skepticism, but Putnam challenges its coherence. In the philosophy of perception Putnam came to endorse direct realism, in the past, he further held that there are no mental representations, sense data, or other intermediaries that stand between the mind and the world. Such transactions can further involve qualia, in his later work, Putnam became increasingly interested in American pragmatism, Jewish philosophy, and ethics, thus engaging with a wider array of philosophical traditions. He also displayed an interest in metaphilosophy, seeking to renew philosophy from what he identifies as narrow, outside philosophy, Putnam contributed to mathematics and computer science. Together with Martin Davis he developed the Davis–Putnam algorithm for the Boolean satisfiability problem and he was at times a politically controversial figure, especially for his involvement with the Progressive Labor Party in the late 1960s and early 1970s. Putnam was born in Chicago, Illinois, in 1926 and his father, Samuel Putnam, was a scholar of Romance languages, columnist, and translator who wrote for the Daily Worker, a publication of the American Communist Party, from 1936 to 1946. As a result of his fathers commitment to communism, Putnam had an upbringing, although his mother. The family lived in France until 1934, when returned to the United States. Putnam attended Central High School, there he met Noam Chomsky, the two had been friends—and often intellectual opponents—ever since. After teaching at Northwestern, Princeton, and MIT, he moved to Harvard in 1965 and his wife, Ruth Anna Jacobs took a teaching position in philosophy at Wellesley College. Hilary and Ruth Anna were married in 1962, the Putnams, rebelling against the antisemitism that they had experienced during their youth, decided to establish a traditional Jewish home for their children. Since they had no experience with the rituals of Judaism, they sought out invitations to other Jews homes for Seder and they had no idea how to do it, in the words of Ruth Anna. They therefore began to study Jewish ritual and Hebrew, and became more Jewishly interested, identified, in 1994, Hilary Putnam celebrated a belated Bar Mitzvah service. His wife had a Bat Mitzvah service four years later, Hilary was a popular teacher at Harvard. In keeping with the tradition, he was politically active
9.
Julia Robinson
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Julia Hall Bowman Robinson was an American mathematician best known for her work on decision problems and Hilberts tenth problem. Robinson was born in St. Louis, Missouri, the daughter of Ralph Bowers Bowman and her older sister was the mathematical popularizer and biographer Constance Reid. The family moved to Arizona and then to San Diego when the girls were a few years old and she entered San Diego State University in 1936 and transferred as a senior to University of California, Berkeley, in 1939. She received her BA degree in 1940 and continued in graduate studies and she received the Ph. D. degree in 1948 under Alfred Tarski with a dissertation on Definability and Decision Problems in Arithmetic. In 1975 she became a professor at Berkeley, teaching quarter-time because she still did not feel strong enough for a full-time job. Hilberts tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers, notices of the American Mathematical Society printed a film review and an interview with the director. College Mathematics Journal also published a film review and her Ph. D. thesis was on Definability and Decision Problems in Arithmetic. Robinsons work only strayed from decision problems twice, the first time was her first paper, published in 1948, on sequential analysis in statistics. The second was a 1951 paper in game theory where she proved that the fictitious play dynamics converges to the mixed strategy Nash equilibrium in two-player zero-sum games. This was posed as a problem at RAND with a $200 prize. Robinson was attracted to politics by the 1952 presidential campaign of Adlai Stevenson, in the 1950s Robinson was active in local Democratic party activities, and did less mathematics. She stuffed envelopes, rang doorbells, asked for votes, and she was Alan Cranstons campaign manager in Contra Costa County when he ran for his first political office, state controller. Robinsons heart had been damaged by rheumatic fever as a child and she was a self-reported late talker. She married Berkeley professor Raphael Robinson in 1941, in 1961, she underwent an operation to remove the scar tissue from her mitral valve. The operation was a success and she much more active physically. In 1984, she was diagnosed with leukemia and she underwent treatment and went into remission for a few months, but then the disease recurred and she died in Oakland, California, on July 30,1985. The collected works of Julia Robinson, Julia Bowman Robinson, Biographies of Women Mathematicians, Agnes Scott College OConnor, John J. Robertson, Edmund F. Julia Robinson, MacTutor History of Mathematics archive, University of St Andrews, Julia Robinson at the Mathematics Genealogy Project Julia Bowman Robinson on the Internet Trailer for Julia Robinson and Hilberts Tenth Problem on YouTube
10.
Greatest common divisor
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In mathematics, the greatest common divisor of two or more integers, when at least one of them is not zero, is the largest positive integer that is a divisor of both numbers. For example, the GCD of 8 and 12 is 4, the greatest common divisor is also known as the greatest common factor, highest common factor, greatest common measure, or highest common divisor. This notion can be extended to polynomials and other commutative rings, in this article we will denote the greatest common divisor of two integers a and b as gcd. What is the greatest common divisor of 54 and 24, the number 54 can be expressed as a product of two integers in several different ways,54 ×1 =27 ×2 =18 ×3 =9 ×6. Thus the divisors of 54 are,1,2,3,6,9,18,27,54, similarly, the divisors of 24 are,1,2,3,4,6,8,12,24. The numbers that these two share in common are the common divisors of 54 and 24,1,2,3,6. The greatest of these is 6 and that is, the greatest common divisor of 54 and 24. The greatest common divisor is useful for reducing fractions to be in lowest terms, for example, gcd =14, therefore,4256 =3 ⋅144 ⋅14 =34. Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1, for example,9 and 28 are relatively prime. For example, a 24-by-60 rectangular area can be divided into a grid of, 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, therefore,12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, in practice, this method is only feasible for small numbers, computing prime factorizations in general takes far too long. Here is another example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180, first, find the prime factorizations of the two numbers,48 =2 ×2 ×2 ×2 ×3,180 =2 ×2 ×3 ×3 ×5. What they share in common is two 2s and a 3, Least common multiple =2 ×2 × ×3 ×5 =720 Greatest common divisor =2 ×2 ×3 =12. To compute gcd, divide 48 by 18 to get a quotient of 2, then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as, gcd = a gcd = gcd, in this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a problem is NC-equivalent to the problem of integer linear programming with two variables, if either problem is in NC or is P-complete, the other is as well
11.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
12.
Emil Leon Post
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Emil Leon Post was a Polish-born American mathematician and logician. He is best known for his work in the field eventually became known as computability theory. Post was born in Augustów, Suwałki Governorate, Russian Empire into a Polish-Jewish family that immigrated to New York City in May 1904 and his parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his arm in a car accident. This loss was a significant obstacle to being a professional astronomer and he decided to pursue mathematics, rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B. S. in Mathematics. After completing his Ph. D. in mathematics at Columbia University, supervised by Cassius Jackson Keyser, Post then became a high school mathematics teacher in New York City. Post married Gertrude Singer in 1929, with whom he had a daughter, Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton. In 1936, he was appointed to the department at the City College of New York. He died in 1954 of an attack following electroshock treatment for depression. Post also devised truth tables independently of Wittgenstein and C. S. Peirce, jean Van Heijenoorts well-known source book on mathematical logic reprinted Posts classic article setting out these results. While at Princeton, Post came very close to discovering the incompleteness of Principia Mathematica, Post initially failed to publish his ideas as he believed he needed a complete analysis for them to be accepted. In 1936, Post developed, independently of Alan Turing, a model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, Post devised a method of auxiliary symbols by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages and this question, which became known as Posts problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the priority method in recursion theory. Post made a fundamental and still-influential contribution to the theory of polyadic, or n-ary and his major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n −1. He also demonstrated that a group operation on a set can be expressed in terms of a group operation on the same set
13.
Chinese remainder theorem
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This theorem has this name because it is a theorem about remainders and was first discovered in the 3rd century AD by the Chinese mathematician Sunzi in Sunzi Suanjing. The Chinese remainder theorem is true over every principal ideal domain and it has been generalized to any commutative ring, with a formulation involving ideals. What amounts to an algorithm for solving this problem was described by Aryabhata, special cases of the Chinese remainder theorem were also known to Brahmagupta, and appear in Fibonaccis Liber Abaci. The result was later generalized with a solution called Dayanshu in Qin Jiushaos 1247 Mathematical Treatise in Nine Sections. The notion of congruences was first introduced and used by Gauss in his Disquisitiones Arithmeticae of 1801, Gauss introduces a procedure for solving the problem that had already been used by Euler but was in fact an ancient method that had appeared several times. Nk be integers greater than 1, which are often called moduli or divisors, Let us denote by N the product of the ni. The Chinese remainder theorem asserts that if the ni are pairwise coprime and this may be restated as follows in term of congruences, If the ni are pairwise coprime, and if a1. Ak are any integers, then there exists an x such that x ≡ a 1 ⋮ x ≡ a k. This means that for doing a sequence of operations in Z / N Z, one may do the same computation independently in each Z / n i Z. This may be faster than the direct computation if N. This is widely used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers, the theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family. The existence and the uniqueness of the solution may be proven independently, however, the first proof of existence, given below, uses this uniqueness. Suppose that x and y are both solutions to all the congruences, as x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N divides also x − y, If x and y are supposed to be non negative and less than N, then their difference may be a multiple of N only if x = y. The map x ↦ maps congruence classes modulo N to sequences of congruence classes modulo ni, the proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the number of elements, the map is also surjective. This proof is simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to situations where the following proof can
14.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
15.
Turing machine
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Despite the models simplicity, given any computer algorithm, a Turing machine can be constructed that is capable of simulating that algorithms logic. The machine operates on an infinite memory tape divided into discrete cells, the machine positions its head over a cell and reads the symbol there. The Turing machine was invented in 1936 by Alan Turing, who called it an a-machine, thus, Turing machines prove fundamental limitations on the power of mechanical computation. Turing completeness is the ability for a system of instructions to simulate a Turing machine, a Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More specifically, it is a capable of enumerating some arbitrary subset of valid strings of an alphabet. Assuming a black box, the Turing machine cannot know whether it will eventually enumerate any one specific string of the subset with a given program and this is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus, a Turing machine that is able to simulate any other Turing machine is called a universal Turing machine. The thesis states that Turing machines indeed capture the notion of effective methods in logic and mathematics. Studying their abstract properties yields many insights into computer science and complexity theory, at any moment there is one symbol in the machine, it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, however, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings, the Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, in the original article, Turing imagines not a mechanism, but a person whom he calls the computer, who executes these deterministic mechanical rules slavishly. If δ is not defined on the current state and the current tape symbol, Q0 ∈ Q is the initial state F ⊆ Q is the set of final or accepting states. The initial tape contents is said to be accepted by M if it eventually halts in a state from F, Anything that operates according to these specifications is a Turing machine. The 7-tuple for the 3-state busy beaver looks like this, Q = Γ = b =0 Σ = q 0 = A F = δ = see state-table below Initially all tape cells are marked with 0. In the words of van Emde Boas, p.6, The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like. For instance, There will need to be many decisions on what the symbols actually look like, and a failproof way of reading and writing symbols indefinitely
16.
Goldbach's conjecture
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Goldbachs conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states, Every even integer greater than 2 can be expressed as the sum of two primes, the conjecture has been shown to hold up through 4 ×1018, but remains unproven despite considerable effort. A Goldbach number is an integer that can be expressed as the sum of two odd primes. The expression of an even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers,6 =3 +38 =3 +510 =3 +7 =5 +512 =7 +5. 100 =3 +97 =11 +89 =17 +83 =29 +71 =41 +59 =47 +53. He then proposed a second conjecture in the margin of his letter and he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time, a modern version of Goldbachs marginal conjecture is, Every integer greater than 5 can be written as the sum of three primes. In the letter dated 30 June 1742, Euler stated, Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, Goldbachs third version is the form in which the conjecture is usually expressed today. It is also known as the strong, even, or binary Goldbach conjecture, while the weak Goldbach conjecture appears to have been finally proved in 2013, the strong conjecture has remained unsolved. For small values of n, the strong Goldbach conjecture can be verified directly, for instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤105. With the advent of computers, many more values of n have been checked, one record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781. A very crude version of the heuristic argument is as follows. The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is an even integer and m is a number between 3 and n/2, then one might expect the probability of m and n − m simultaneously being prime to be 1 /. Since this quantity goes to infinity as n increases, we expect that every even integer has not just one representation as the sum of two primes, but in fact has very many such representations. This heuristic argument is somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd then n − m is odd, and if m is even, then n − m is even
17.
Fermat's Last Theorem
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In number theory, Fermats Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n =1 and n =2 have been known to have many solutions since antiquity. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. The Pythagorean equation, x2 + y2 = z2, has an number of positive integer solutions for x, y, and z. Around 1637, Fermat wrote in the margin of a book that the general equation an + bn = cn had no solutions in positive integers. Although he claimed to have a proof of his conjecture, Fermat left no details of his proof. His claim was discovered some 30 years later, after his death and this claim, which came to be known as Fermats Last Theorem, stood unsolved in mathematics for the following three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics, attempts to prove it prompted substantial development in number theory, and over time Fermats Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n =4 proved, it suffices to prove the theorem for n that are prime numbers. Over the next two centuries, the conjecture was proved for only the primes 3,5, and 7, in the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and as the modularity theorem, it stood on its own and it was widely seen as significant and important in its own right, but was widely considered completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermats Last Theorem and this potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermats Last Theorem that depended on the modularity theorem. On hearing this, English mathematician Andrew Wiles, who had a fascination with Fermats Last Theorem. In 1993, after six years working secretly on the problem, Wiless paper was massive in size and scope. A flaw was discovered in one part of his paper during peer review and required a further year and collaboration with a past student, Richard Taylor. As a result, the proof in 1995 was accompanied by a second smaller joint paper to that effect
18.
Riemann hypothesis
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In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann, after whom it is named, the name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers, along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann zeta function ζ is a function whose argument s may be any complex number other than 1 and it has zeros at the negative even integers, that is, ζ =0 when s is one of −2, −4, −6. These are called its trivial zeros, However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros, the Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that, The real part of every non-trivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the non-trivial zeros lie on the line consisting of the complex numbers 1/2 + i t. There are several books on the Riemann hypothesis, such as Derbyshire, Rockmore. The books Edwards, Patterson, Borwein et al. and Mazur & Stein give mathematical introductions, while Titchmarsh, Ivić, furthermore, the book Open Problems in Mathematics, edited by John Forbes Nash Jr. and Michael Th. Rassias, features an essay on the Riemann hypothesis by Alain Connes. The convergence of the Euler product shows that ζ has no zeros in this region, the Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to continue the function to give it a definition that is valid for all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If the real part of s is greater than one, then the function satisfies ζ = ∑ n =1 ∞ n +1 n s =11 s −12 s +13 s − ⋯. However, the series on the right converges not just when the part of s is greater than one. Thus, this alternative series extends the function from Re >1 to the larger domain Re >0. The zeta function can be extended to these values, as well, by taking limits, in the strip 0 < Re <1 the zeta function satisfies the functional equation ζ =2 s π s −1 sin Γ ζ. If s is an even integer then ζ =0 because the factor sin vanishes
19.
Four color theorem
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Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers, according to an article by the math historian Kenneth May, “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. Martin Gardner wrote an account of what was known at the time about the four color theorem in his September 1960 Mathematical Games column in Scientific American magazine. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fools hoax column of April 1975, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer, Appel and Hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. To dispel remaining doubt about the Appel–Haken proof, a proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour. Additionally, in 2005, the theorem was proved by Georges Gonthier with general purpose theorem proving software, the intuitive statement of the four color theorem, i. e. First, all corners, points that belong to three or more countries, must be ignored. In addition, bizarre maps can require more than four colors, second, for the purpose of the theorem, every country has to be a connected region, or contiguous. In the real world, this is not true, because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, In this map and this map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a color is used for all bodies of water. For maps in which more than one country may have multiple disconnected regions, a simpler statement of the theorem uses graph theory
20.
Thoralf Skolem
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Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolems father was a school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology, in 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light, in 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science, Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and his notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold, Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. This senior post allowed Skolem to conduct research free of administrative, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the courses in algebra and number theory. Skolems Ph. D. student Øystein Ore went on to a career in the USA, Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift for many years. He was also the editor of Mathematica Scandinavica. After his 1957 retirement, he made trips to the United States. He remained intellectually active until his sudden and unexpected death, for more on Skolems academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory and he mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras, Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices, in 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every lattice is distributive and, as a partial converse
21.
Ring (mathematics)
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In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity
22.
Countable set
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In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a set or a countably infinite set. Some authors use countable set to mean countably infinite alone, to avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable, today, countable sets form the foundation of a branch of mathematics called discrete mathematics. A set S is countable if there exists a function f from S to the natural numbers N =. If such an f can be found that is also surjective, in other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N. As noted above, this terminology is not universal, some authors use countable to mean what is here called countably infinite, and do not include finite sets. Alternative formulations of the definition in terms of a function or a surjective function can also be given. In 1874, in his first set theory article, Cantor proved that the set of numbers is uncountable. In 1878, he used one-to-one correspondences to define and compare cardinalities, in 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. A set is a collection of elements, and may be described in many ways, one way is simply to list all of its elements, for example, the set consisting of the integers 3,4, and 5 may be denoted. This is only effective for small sets, however, for larger sets, even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite, these sets have more than n elements for any integer n, for example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it out that infinite sets do have a well-defined notion of size. To understand what this means, we first examine what it does not mean, for example, there are infinitely many odd integers, infinitely many even integers, and infinitely many integers overall. However, it out that the number of even integers. This is because we arrange things such that for every integer, or, more generally, n→2n, see picture. However, not all sets have the same cardinality
23.
Algebraic number field
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In mathematics, an algebraic number field F is a finite degree field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a space over Q. The study of number fields, and, more generally. The notion of algebraic number field relies on the concept of a field, a field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of numbers, commonly denoted Q. Another notion needed to define algebraic number fields is vector spaces, to the extent needed here, vector spaces can be thought of as consisting of sequences whose entries are elements of a fixed field, such as the field Q. Any two such sequences can be added by adding the one per one. Furthermore, any sequence can be multiplied by an element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly, vector spaces are allowed to be infinite-dimensional, that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector consists of finite sequences. An algebraic number field is a finite field extension of the field of rational numbers. Here degree means the dimension of the field as a space over Q. The smallest and most basic number field is the field Q of rational numbers, many properties of general number fields are modelled after the properties of Q. The Gaussian rationals, denoted Q, form the first nontrivial example of a number field and its elements are expressions of the form a+bi where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the rules of arithmetic. Explicitly, + = + i, = + i, non-zero Gaussian rational numbers are invertible, which can be seen from the identity = a 2 + b 2 =1. It follows that the Gaussian rationals form a field which is two-dimensional as a vector space over Q. More generally, for any square-free integer d, the quadratic field Q is a number field obtained by adjoining the square root of d to the field of rational numbers
24.
Jan Denef
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Jan Denef is a Belgian mathematician. He is Professor of Mathematics at the Katholieke Universiteit Leuven and he is a specialist of model theory, number theory and algebraic geometry. He is well known for his work on Hilberts tenth problem. He has also worked on number theory. Recently he proved a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem and he was an invited speaker at ICM2002. Jan Denef at the Mathematics Genealogy Project Denefs home page One of his key publications on motivic integration
25.
Abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers and they are named after Niels Henrik Abel. The concept of a group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules. The theory of groups is generally simpler than that of their non-abelian counterparts. On the other hand, the theory of abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b, the symbol • is a general placeholder for a concretely given operation. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, commutativity For all a, b in A, a • b = b • a. A group in which the operation is not commutative is called a non-abelian group or non-commutative group. There are two main conventions for abelian groups – additive and multiplicative. Generally, the notation is the usual notation for groups, while the additive notation is the usual notation for modules. To verify that a group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal and this is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the th entry of the table equals the th entry, every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form a group under addition, as do the integers modulo n. Every ring is a group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group, in particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication
26.
Barry Mazur
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Barry Charles Mazur is an American mathematician and a Gerhard Gade University Professor at Harvard University. Born in New York City, Mazur attended the Bronx High School of Science and MIT, regardless, he was accepted for graduate school and received his Ph. D. from Princeton University in 1959, becoming a Junior Fellow at Harvard from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard and his early work was in geometric topology. In an elementary fashion, he proved the generalized Schoenflies conjecture, both Brown and Mazur received the Veblen Prize for this achievement. He also discovered the Mazur manifold and the Mazur swindle and his observations in the 1960s on analogies between primes and knots were taken up by others in the 1990s giving rise to the field of arithmetic topology. Coming under the influence of Alexander Grothendiecks approach to algebraic geometry, Mazurs first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. This proof was carried in his seminal paper Modular curves and the Eisenstein ideal, the ideas of this paper and Mazurs notion of Galois deformations, were among the key ingredients in Wiless proof of Fermats Last Theorem. Mazur and Wiles had earlier worked together on the conjecture of Iwasawa theory. Number theory swarms with bugs, waiting to bite the tempted flower-lovers who and he expanded his thoughts in the 2003 book Imagining Numbers and Circles Disturbed, a collection of essays on mathematics and narrative that he edited with writer Apostolos Doxiadis. In 1982 he was elected a member of the National Academy of Sciences, in early 2013, he was presented with one of the 2011 National Medals of Science by President Barack Obama. Prime numbers and the Riemann hypothesis, New York, NY, Cambridge University Press. Collected works of John Tate, parts i and ii, fontaine–Mazur conjecture Mazurs control theorem Homepage of Barry Mazur OConnor, John J. Robertson, Edmund F. Barry Mazur, MacTutor History of Mathematics archive, University of St Andrews. Video of Mazur talking about his work, from the National Science & Technology Medals Foundation Barry Mazur on MathSciNet
27.
Algebraic variety
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, a variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. For example, some definitions provide that algebraic variety is irreducible, under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility, the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that a variety may have singular points. Generalizing this result, Hilberts Nullstellensatz provides a correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry, an affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way, the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k can be viewed as k-valued functions on An by evaluating f at the points in An, i. e. by choosing values in k for each xi. For each set S of polynomials in k, define the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. This topology is called the Zariski topology.2 Given a subset V of An, let f in k be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates, however, because f is homogeneous, f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish, Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the ring of V is the quotient of the polynomial ring by this ideal.10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties and it might not have an embedding into projective space
28.
S. Barry Cooper
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S. Barry Cooper was a British mathematician and computability theorist. He was a Professor of Pure Mathematics at the University of Leeds, Cooper grew up in Bognor Regis and attended Chichester High School for Boys, during which time he played scrum-half for the under-15s England rugby team. Cooper graduated from Jesus College, Oxford in 1966, and in 1970 received his PhD from University of Leicester under the supervision of Reuben Goodstein, yates, with a thesis entitled Degrees of Unsolvability. In the 1970s, he was also a figure in the Chile Solidarity Campaign. Cooper was appointed Lecturer in the School of Mathematics at the University of Leeds in 1969 and he was promoted to Reader in Mathematical Logic in 1991 and to Professor of Pure Mathematics in 1996. In 2011, he was awarded a doctorate at the University of Sofia St. Kliment Ohridski. His book Computability Theory made the technical research area accessible to a new generation of students and he was a leading mover of the return to basic questions of the kind considered by Alan Turing, and of interdisciplinary developments related to computability. He was President of the Association Computability in Europe, and Chair of the Turing Centenary Advisory Committee which co-ordinated the Alan Turing Year. The book Alan Turing, His Work and Impact, edited by Cooper and Jan van Leeuwen and he was a keen long-distance runner, and was also interested in jazz and improvised music, founding Leeds Jazz and being involved in the Termite Club. Cooper died on 26 October 2015 after a short illness, Cooper was a member of the editorial board for The Rutherford Journal. Alan Turing – His Work and Impact, New York, Elsevier, New Computational Paradigms - Changing Conceptions of What is Computable, Springer. Harrington, L. Lachlan, A. H. Lempp, S. Soare, the d. r. e. degrees are not dense. Annals of Pure and Applied Logic, British Computer Society, Electronic Workshops in Computing. Invited paper from Alan Mathison Turing 2004, A celebration of his life and achievements, Computability and Models, Perspectives East and West. Plenum Publishers, New York, Boston, Dordrecht, London, cS1 maint, Uses editors parameter Cooper, S. B. London Mathematical Society Lecture Notes Series 259, Cambridge University Press, Cambridge, New York, cS1 maint, Uses editors parameter S. Barry Cooper homepage S. Barry Coopers Mathematics Genealogy Page Computability in Europe homepage The Alan Turing Centenary homepage
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Proceedings of Symposia in Pure Mathematics
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. Mary W. Gray challenged that situation by sitting in on the Council meeting in Atlantic City, when she was told she had to leave, she refused saying she would wait until the police came. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L