Word Problem (mathematics)
In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable. Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a ''solution to the word problem''. For example, imagine that x,y,z are symbols representing real numbers  then a relevant solution to the word problem would, given the input (x \cdot y)/z \mathrel (x/z)\cdot y, produce the output EQUAL, and similarly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Computational Mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06888 Computational Mathematics 2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Areas of computational mathematics Computational mathematics emerged as a distinct part of appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a wellknown example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Andrey Markov Jr
Andrey, Andrej or Andrei (in Cyrillic script: Андрей, Андреј or Андрэй) is a form of Andreas/ Ἀνδρέας in Slavic languages and Romanian. People with the name include: *Andrei of Polotsk ( – 1399), Lithuanian nobleman * Andrei Alexandrescu, Romanian computer programmer *Andrey Amador, Costa Rican cyclist *Andrei Arlovski, Belarusian mixed martial artist * Andrey Arshavin, Russian football player * Andrej Babiš, Czech prime minister *Andrey Belousov (born 1959), Russian politician *Andrey Bolotov, Russian agriculturalist and memoirist *Andrey Borodin, Russian financial expert and businessman *Andrei Chikatilo, prolific and cannibalistic Russian serial killer and rapist *Andrei Denisov (weightlifter) (born 1963), Israeli Olympic weightlifter *Andrey Ershov, Russian computer scientist *Andrey Esionov, Russian painter *Andrei Glavina, IstroRomanian writer and politician *Andrei Gromyko (1909–1989), Belarusian Soviet politician and diplomat * Andrey Ivanov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Governorate, Congress Poland, Russian Empire (now Poland) into a PolishJewish family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer, leading to his decision 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 in 1920 at Columbia University, supervised by Cassius Jackson Keyser, he did a postdoctorate at Princeton University in the 1920–1921 academic year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Systems Of Logic Based On Ordinals
''Systems of Logic Based on Ordinals'' was the PhD dissertation of the mathematician Alan Turing. Turing's thesis is not about a new type of formal logic, nor was he interested in socalled ‘ranked logic’ systems derived from ordinal or relative numbering, in which comparisons can be made between truthstates on the basis of relative veracity. Instead, Turing investigated the possibility of resolving the Godelian incompleteness condition using Cantor's method of infinites. This condition can be stated thus—in all systems with finite sets of axioms, an exclusiveor condition applies to expressive power and provability; i.e. one can have power and no proof, or proof and no power, but not both. The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed that for any formal system S powerful enough to represent arithmetic, there is a theorem G which is true but the system is unable to prove. G could be added as an additional axiom to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

On Formally Undecidable Propositions Of Principia Mathematica And Related Systems
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of ''Monatshefte für Mathematik.'' Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new techniques that Gödel invented to prove the incompleteness theorems. Outline and key results The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field of mathematical logic. These appear as theorems VI and XI, respectively, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

History Of The Church–Turing Thesis
The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing. The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious. This article provides detail of that debate and discovery from Peano's axioms in 1889 through recent discussion of the meaning of " axiom". Peano's nine axioms of arithmetic In 1889, Giuseppe Peano presented his ''The principles of arithmetic, presented by a new method'', based on the work of Dedekind. Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although :"Well before the nineteenth century mathematicians used the principle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without break ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Dehn's Algorithm
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem. History Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s. Dehn proved that fundamental groups of closed orientable surfaces of genus at least two have word problem solvable by what is now called Dehn's algorithm. His proof involved drawing the Cayley graph of such a group in the hyperbolic plane and performing curvature estimates via the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, JeanPierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States. Dehn was a student of David Hilbert, and in his habilitation in 1900 Dehn resolved Hilbert's third problem, making him the first to resolve one of Hilbert's wellknown 23 problems. Dehn's students include OttHeinrich Keller, Ruth Moufang, Wilhelm Magnus, and the artists Dorothea Rockburne and Ruth Asawa. Biography Dehn was born to a family of Jewish origin in Hamburg, Imperial Germany. He studied the foundations of geometry with Hilbert at Göttingen in 1899, and obtained a proof of the Jordan curve theorem for polygons. In 1900 he wrote his dissertation on the role of the Legendre angle sum theorem in axiomatic geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 