First to be considered are the stimulus-response pattern of these elementary automata. Top Terence Chi-Shen Tao (1975-) Australia,.S.A. Several theorems or concepts are named after Witten, including *alan turing thesis paper* Seiberg-Witten theory, the Weinberg-Witten theorem, the Gromov-Witten invariant, the Witten index, Witten conjecture, Witten-type Topological quantum field theory, etc. For his texts and theorems, he may be called the "Father of Trigonometry he was first to properly state and prove several theorems of planar and spherical trigonometry including the Law of Sines, and the (spherical) Law of Tangents. As an example of his careful rigor, he found a fundamental flaw in Steiner's Isoperimetric Theorem proof which no one else had noticed. (Although familiar with the utility of infinitesimals, he accepted the "Theorem of Eudoxus" which bans them to avoid Zeno's paradoxes.

#### Alan Turing, biography, Facts, Education

He may have been first to note that the square root of any integer, if not itself an integer, must be irrational. Galileo said of Cavalieri, "Few, if any, since Archimedes, have delved as far and as deep into the science of geometry." Top Pierre de Fermat (1601-1665) France Pierre de Fermat was the most brilliant mathematician of his era. He was one of the greatest mechanists ever, discovering Archimedes' Principle of Hydrostatics (a body partially or completely immersed in a fluid effectively loses weight equal to the weight of the fluid it displaces). He also worked with various spirals, paraboloids of revolution, etc. He discovered several trigonometric identities including a generalization of Ptolemy's Formula, the latter (then called prosthaphaeresis ) providing a calculation shortcut similar to logarithms in that multiplication is reduced to addition (or exponentiation reduced to multiplication). (Camille Jordan and.J.

He was far ahead of his time, but his writings eventually influenced Galileo, Leibniz, and another mathematician-priest more famous than himself: Giordano Bruno, who wrote "If Nicholas of Cusa had not been hindered by his priest's vestment, he would have. Newman strongly promoted Turing's principle of the stored-program computer, but unlike Turing, intended no personal involvement with engineering. Although Liu Hui mentions Chang's skill, it isn't clear Chang had the mathematical genius to qualify for this list, but he would still be a strong candidate due to his book's immense historical importance: It was the dominant Chinese mathematical. She was the daughter of the chief engineer of the Madras railways, who came from an Anglo-Irish family of somewhat similar social status. This is the long page, with list and biographies. He was first to prove that the Stirling and Euler generalizations of the factorial function are equivalent. After the war he developed his strength in cross-country running __alan turing thesis paper__ with frequent long-distance training and top-rank competition in amateur athletics. Many of the mathematical concepts of the early Greeks were discovered independently in early China. In 1938 Weyl called him "the greatest living master in differential geometry." Top Félix Édouard Justin Émile Borel (1871-1956) France Borel exhibited great talent while still in his teens, soon practically founded modern measure theory, and received several honors and prizes. Top Ghiyath al-Din Jamshid Mas'ud Al-Kashi (ca ) Iran, Transoxania (Uzbekistan) Al-Kashi was among the greatest calculators in the ancient world; wrote important texts applying arithmetic and algebra to problems in astronomy, mensuration and accounting; and developed trig tables far more accurate than earlier tables. He devised a range of code-breaking tools for cracking German ciphers, including an electromagnetic device called the.

He developed a very important result in analysis called the Selberg Integral. Madhava also did work with continued fractions, trigonometry, and geometry. Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers. He developed a theory of recursive functions which anticipated some computer science. In contrast, the more complex __alan turing thesis paper__ Enigma methods used in German Naval communications were generally regarded as unbreakable.

#### Alan Turing - a short biography

Serre has been much honored: he is the youngest ever to win a Fields Medal; 49 years after his Fields Medal he became the first recipient of the Abel Prize. In 2009, then British Prime Minister Gordon Brown offered an apology for the governments treatment of Turing, prompted by an online petition signed by 30,000 British citizens to clear Turing's name. State security also seems the likely cause of what he described as another intense crisis in March 1953, involving police searching for a visiting Norwegian who had come to see him. Dedekind was a key mentor for Georg Cantor: he introduced the notion that a bijection implied equinumerosity, used this to define infinitude (a set is infinite if equinumerous with its proper subset and was first to prove the Cantor-Bernstein-Schr?der Theorem. In addition to his own original research, his texts are noteworthy for preserving works of earlier mathematicians that would otherwise have been lost. He won the Order of Lenin three times and several prizes from Western countries. He was first to invent singular value decompositions. Methods for handling subroutines included a suggestion that the machine could expand its own programs from an abbreviated form, ideas well ahead of contemporary American plans. He developed new techniques, and principles of perspective geometry, for drawing, painting and sculpture; he was also an expert architect and engineer; and surely the most prolific inventor of all time. One key insight he had is that addends must be homogeneous (i.e., "apples shouldn't be added to oranges a seemingly trivial idea but which can aid intuition even today.

#### Computing Machinery and Intelligence, wikipedia

Go back to my home page. He developed rigorous definitions and axioms for set theory, as well as most of the notation of modern set theory. Several inventions are named after him,.g. Cardano's life had tragic elements. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle. But on seeing the solution Jacob Bernoulli immediately exclaimed "I **alan turing thesis paper** recognize the lion by his footprint." In 1687 Newton published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The coroner's verdict was suicide. Nonetheless, Michael Atiyah is still regarded as one of the very greatest mathematicians of the 20th century. He may have discovered the simple parametric form of primitive Pythagorean triplets (xx-yy, 2xy, xxyy), although the first explicit mention of this may be in Euclid's Elements. Eventually one of his papers was published in a journal; he was immediately given an honorary doctorate and was soon regarded as one of the best and most inspirational mathematicians in the world. Weierstrass demonstrated extreme brilliance as a youth, but during his college years he detoured into drinking and dueling and ended up as a degreeless secondary school teacher. He made much progress with the Prime Number Theorem, proving two distinct forms of that theorem, each incomplete but in a different way.

#### Helped Win wwii And Was Thanked With

One of his award citations states "Thurston has fantastic geometric insight and vision: his ideas have completely revolutionized the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry." Top Edward Witten (1951-).S.A. Top Pappus of Alexandria (ca 300) Egypt, Greece Pappus, along with Diophantus, may have been one of the two greatest Western mathematicians during the 13 centuries that separated Hipparchus and Fibonacci. His approximation 223/71 22/7 was the best of his day. His generalized notions of distance and curvature described new possibilities for the geometry of space itself. He made revolutionary advances in number theory, algebra and analysis, and was also a composer of music. Newton contributed little to number theory, and satisfies the "great breadth" requirement due to his huge contributions to physics. Conway's great creativity and breadth certainly make him one of the greatest living mathematicians. Although others also developed the techniques independently, Newton is regarded as the "Father of Calculus" (which he called "fluxions he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). His long and productive career earned the Abel Prize for his "vast and lasting impact on the theory of numbers and his incisive contributions and illuminating insights. Original work *alan turing thesis paper* on this aspect of automata theory was done by Warren.

Legendre had spent much of his life studying elliptic integrals, but Abel inverted these to get elliptic functions, and was first **alan turing thesis paper** to observe (but in a manuscript mislaid by Cauchy) that they were doubly periodic. He was a founder of fields like metamathematics and modern logic. Top Christiaan Huygens (1629-1695) Holland, France Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist and theoretical physicist of his era; he may have helped inspire Newton. The Unit Theorem is unusually difficult to prove; it is said that Dirichlet discovered the proof while listening to music in the Sistine Chapel. (The equation e x xk / k! Any physical neuron can be sufficiently excited by an oncoming impulse to fire another impulse into the network of which it forms a part, or else the threshold will not be reached because the stimulus is absent or inadequate. Hamilton's Principle of Least Action, and its associated equations and concept of configuration space, led to a revolution in mathematical physics. He was first to publish general solutions to cubic and quartic equations, and first to publish the use of complex numbers in calculations. Qin's textbook discusses various algebraic procedures, includes word problems requiring quartic or quintic equations, explains a version of Horner's Method for finding solutions to such equations, includes Heron's Formula for a triangle's area, and introduces the zero symbol and decimal fractions. He was first to notice that the set of Platonic regular solids was incomplete if concave solids are admitted, and first to prove that there were only 13 Archimedean solids.

1600, John Machin 1706. He did more important work in geometry and topology; for example, he proved theorems about embedding (or "unknotting manifolds in Euclidean space. This is the source of __alan turing thesis paper__ their witchery." Top Julius Pl?cker (1801-1868) Germany Pl?cker was one of the most innovative geometers, inventing line geometry (extending the atoms of geometry beyond just points enumerative geometry (which considered such questions. Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different. It seems unlikely that Diophantus actually had proofs for such "lemmas. Top Christian Felix Klein (1849-1925) Germany Klein's key contribution was an application of invariant theory to unify geometry with group theory. Noether was an unusual and inspiring teacher; her successful students included Emil Artin, Max Deuring, Jacob Levitzki, etc. Kepler developed a rudimentary notion of universal gravitation, and used it to produce the best explanation for tides before Newton; however he seems not to have noticed that his empirical laws implied inverse-square gravitation. He also introduced the notions of definable number and oracle (important in modern computer science and was an early pioneer in the study of neural networks. While studying lens refraction, he invented the Ovals of Descartes. Jean le Rond, named after the Parisian church where he was abandoned as a baby, played a very key role in that development. His excellent approximation to 3 indicates that he'd partially anticipated the method of continued fractions. If, in this sense of comparison, the functional response of the automaton is identical to the functional value of the logical statement (polynomial the automaton is then said to compute the statement (polynomial) or the statement is said to be computable.

Top Giordano Bruno (1548-1600) Italy, England Bruno wrote on mnemonics, philosophy, cosmology and more, and further developed Nicholas of Cusa's pandeism (though it wasn't then known by that name). He solved Alhazen's Billiard Problem (originally posed as a problem in mirror design a difficult construction which continued to intrigue several great mathematicians including Huygens. With his "question mark function" and "sausage he was also a pioneer in the study of fractals. Top André Weil (1906-1998) France,.S.A. Which is now linked with the name Fibonacci. Top Adrien Marie Legendre (1752-1833) France Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. Combining Hamilton's quaternions, Grassmann's exterior algebra, and his own geometric intuition and understanding of physics, he developed biquaternions, and generalized this to geometric algebra, which paralleled work by Klein. Al-Farisi was another ancient mathematician who noted FLT4, although attempting no proof.) Another of Leonardo's noteworthy achievements was proving that the roots of a certain cubic equation could not have any of the constructible forms Euclid had outlined in Book 10 of his Elements. In the meanwhile the idea lived only in his mind.

#### Artificial Intelligence, internet Encyclopedia of Philosophy

Many theorems and inventions are named __alan turing thesis paper__ after him, for example Fréchet Distance, which has many applications in applied math,.g. Eratosthenes had the nickname Beta ; he was a master of several fields, but was only second-best of his time. As an undergraduate at King's College, Cambridge from 1931, he entered a world more encouraging to free-ranging thought. (Ptolemy's model predicted phases, but timed quite differently from Galileo's observations.) Since the planets move without friction, their motions offer a pure view of the Laws of Motion; this is one reason that the heliocentric breakthroughs of Copernicus, Kepler and. If his writings had survived he'd surely be considered one of the most brilliant and innovative geometers of antiquity.