wikipedia_math_articles.json

{"Linear algebra": {"categories": ["All articles to be expanded", "All pages needing cleanup", "Articles needing cleanup from August 2018", "Articles needing cleanup from September 2018", "Articles to be expanded from September 2018", "Articles using small message boxes", "Cleanup tagged articles with a reason field from August 2018", "Cleanup tagged articles with a reason field from September 2018", "Commons category link from Wikidata", "Linear algebra", "Numerical analysis", "Wikipedia articles with BNF identifiers", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NDL identifiers", "Wikipedia pages needing cleanup from August 2018", "Wikipedia pages needing cleanup from September 2018"], "title": "Linear algebra", "method": "Linear algebra", "url": "https://en.wikipedia.org/wiki/Linear_algebra", "summary": "Linear algebra is the branch of mathematics concerning linear equations such as \n\n  \n    \n      \n        \n          a\n          \n            1\n          \n        \n        \n          x\n          \n            1\n          \n        \n        +\n        \u22ef\n        +\n        \n          a\n          \n            n\n          \n        \n        \n          x\n          \n            n\n          \n        \n        =\n        b\n        ,\n      \n    \n    {\\displaystyle a_{1}x_{1}+\\cdots +a_{n}x_{n}=b,}\n  linear functions such as\n\n  \n    \n      \n        (\n        \n          x\n          \n            1\n          \n        \n        ,\n        \u2026\n        ,\n        \n          x\n          \n            n\n          \n        \n        )\n        \u21a6\n        \n          a\n          \n            1\n          \n        \n        \n          x\n          \n            1\n          \n        \n        +\n        \u2026\n        +\n        \n          a\n          \n            n\n          \n        \n        \n          x\n          \n            n\n          \n        \n        ,\n      \n    \n    {\\displaystyle (x_{1},\\ldots ,x_{n})\\mapsto a_{1}x_{1}+\\ldots +a_{n}x_{n},}\n  and their representations through matrices and vector spaces.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/2/2f/Linear_subspaces_with_shading.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", 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"http://mathworld.wolfram.com/topics/LinearAlgebra.html", "http://www.worldscientific.com/doi/suppl/10.1142/8261/suppl_file/8261_chap01.pdf", "http://www.worldscientific.com/worldscibooks/10.1142/8261", "http://www.math.brown.edu/~treil/papers/LADW/LADW.html", "http://www.math.miami.edu/~ec/book/", "http://joshua.smcvt.edu/linearalgebra/", "http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html", "http://linear.ups.edu/index.html", "http://data.bnf.fr/ark:/12148/cb11937509n", "http://www.math.technion.ac.il/iic/", "http://www.ams.org/mathscinet-getitem?mr=0276251", "http://www.numbertheory.org/book/", "http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm", "https://books.google.com/?id=GxmQxn2PF3IC&pg=PA18", "https://books.google.com/?id=yZ68h97pnAkC&pg=PA203", "https://books.google.com/books?id=De0GAAAAYAAJ&printsec=frontcover", "https://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221", "https://books.google.com/books?id=zLPm3xE2qWgC&printsec=frontcover", "https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9", "https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab", "https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/", "https://catalogue.bnf.fr/ark:/12148/cb11937509n", "https://id.loc.gov/authorities/subjects/sh85003441", "https://d-nb.info/gnd/4035811-2", "https://id.ndl.go.jp/auth/ndlna/00570681", "https://web.archive.org/web/20091031193126/http://matrixanalysis.com/DownloadChapters.html", "https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html", "https://arxiv.org/abs/math.HO/0405323", "https://www.encyclopediaofmath.org/index.php?title=p/l059040", "https://www.wikidata.org/wiki/Q82571"]}, "Multilinear_algebra": {"categories": ["All articles with style issues", "All articles with unsourced statements", "Articles with unsourced statements from April 2008", "Multilinear algebra", "Wikipedia articles needing clarification from July 2012", "Wikipedia articles with style issues from June 2012"], "title": "Multilinear algebra", "method": "Multilinear_algebra", "url": "https://en.wikipedia.org/wiki/Multilinear_algebra", "summary": "In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.", "images": ["https://upload.wikimedia.org/wikipedia/commons/2/2f/Linear_subspaces_with_shading.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", "https://upload.wikimedia.org/wikipedia/en/d/db/Symbol_list_class.svg"], "links": ["Abstract algebra", "Abstract index notation", "Academic Press", "Addison-Wesley", "Affine connection", "Albert 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cohomology", "Determinant", "Differential (infinitesimal)", "Differential equation", "Differential form", "Differential forms", "Differential geometry", "Digital object identifier", "Dimension", "Direct sum of modules", "Discrete geometry", "Discrete mathematics", "Dot product", "Dual space", "Dyadic tensor", "Dyadics", "Dynamical systems theory", "Eigenvalues and eigenvectors", "Einstein notation", "Einstein tensor", "Electromagnetic tensor", "Electromagnetism", "Elementary algebra", "Elwin Bruno Christoffel", "Engineering", "Euclidean geometry", "Euclidean space", "Euclidean vector", "Exterior algebra", "Exterior calculus", "Exterior covariant derivative", "Exterior derivative", "Exterior form", "Exterior product", "Fiber bundle", "Finite geometry", "Floating point", "Foundations of mathematics", "Free algebra", "Function space", "Functional analysis", "Game theory", "Gaussian elimination", "General covariance", "General relativity", "Geodesic", "Geometric algebra", "Geometry", "Glossary of tensor theory", "Gluon field strength tensor", "Gram\u2013Schmidt process", "Graph theory", "Grassmann algebra", "Gregorio Ricci-Curbastro", "Hermann Grassmann", "Hermann Weyl", "History of mathematics", "Hodge star operator", "Homological algebra", "Homology (mathematics)", "Hypercomplex number", "Index notation", "Information theory", "Inner product", "Inner product space", "International Standard Book Number", "International Standard Serial Number", "Invertible matrix", "Jacobian matrix", "Jan Arnoldus Schouten", "Kernel (linear algebra)", "Kronecker delta", "K\u00fcnneth theorem", "Leonhard Euler", "Levi-Civita connection", "Levi-Civita symbol", "Lie derivative", "Lie groups", "Linear algebra", "Linear combination", "Linear independence", "Linear map", "Linear span", "Linear subspace", "Linear transformation", "Lists of mathematics topics", "MATLAB", "Manifold", "Marcel Grossmann", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", 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"Pure mathematics", "Quaternion", "Quotient space (linear algebra)", "Raising and lowering indices", "Rank (linear algebra)", "Recreational mathematics", "Ricci calculus", "Ricci curvature", "Riemann curvature tensor", "Row space", "Scalar (mathematics)", "Scalar multiplication", "Set theory", "Seven-dimensional cross product", "Sparse matrix", "Spin tensor", "Spinor", "Springer Science+Business Media", "Statistics", "Stress\u2013energy tensor", "Symmetric algebra", "Symmetric power", "Symmetric tensor", "System of linear equations", "Tensor", "Tensor (intrinsic definition)", "Tensor algebra", "Tensor bundle", "Tensor calculus", "Tensor contraction", "Tensor density", "Tensor field", "Tensor operator", "Tensor product", "Tensors in curvilinear coordinates", "Tetrad (index notation)", "Theory of computation", "Topological space", "Topology", "Tor functor", "Torsion tensor", "Torus", "Transformation matrix", "Transport phenomena", "Transpose", "Trigonometry", "Triple product", "Tullio Levi-Civita", "Universal property", "Van der Waerden notation", "Vector (mathematics and physics)", "Vector algebra", "Vector projection", "Vector space", "Victor Schlegel", "Voigt notation", "Wedge product", "Weyl tensor", "Woldemar Voigt", "\u00c9lie Cartan"], "references": ["http://doi.org/10.1007%2FBF01454201", "http://www.worldcat.org/issn/1432-1807"]}, "Abstract algebra": {"categories": ["Abstract algebra", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Abstract algebra", "method": "Abstract algebra", "url": "https://en.wikipedia.org/wiki/Abstract_algebra", "summary": "In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.\nAlgebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified  way for expressing properties and constructions that are similar for various structures.\nUniversal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.", "images": ["https://upload.wikimedia.org/wikipedia/commons/a/a3/Arithmetic_symbols.svg", "https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/9/96/Rubik%27s_cube_v2.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abelian group", "Abstrakt Algebra", "Alexandre Vandermonde", "Algebra", "Algebra over a field", "Algebraic geometry", "Algebraic number theory", "Algebraic operation", "Algebraic structure", "Algebraic topology", "Analytic geometry", "Andrew Wiles", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Arthur Cayley", "Associative algebra", "Associative property", "Augustin Cauchy", "Axiomatic system", "Bartel van der Waerden", "Binary operation", "Boolean algebra", "Boson", "Calculus", "Camille Jordan", "Cancellation property", "Carl Friedrich Gauss", "Category (mathematics)", "Category theory", "Coding theory", "Commutative algebra", "Composition of binary quadratic forms", "Computational mathematics", "Computer science", "Control theory", "Coordinate vector", "Cyclic group", "David Hilbert", "Differential equation", "Differential geometry", "Diophantine equation", "Discrete geometry", "Discrete mathematics", "Dot product", "Dynamical systems theory", "Edmund F. Robertson", "Edward Waring", "Elementary algebra", "Emil Artin", "Emmy Noether", "Ernst Kummer", "Ernst Steinitz", "Euler's theorem", "Exterior algebra", "Fermat's Last Theorem", "Fermat's little theorem", "Field (mathematics)", "Finite geometry", "Force carrier", "Foundations of mathematics", "Functional analysis", "Fundamental group", "Fundamental theorem of symmetric polynomials", "Galois group", "Game theory", "Gauge theory", "Geometric algebra", "Geometry", "Georg Frobenius", "Glossary of field theory", "Glossary of linear algebra", "Glossary of order theory", "Glossary of ring theory", "Graduate Texts in Mathematics", "Graph theory", "Group (mathematics)", "Group theory", "Heinrich M. 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It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic.  Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.\nThe use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. 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Citizendium", "Wikipedia articles with BNE identifiers", "Wikipedia articles with BNF identifiers", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NDL identifiers", "Wikipedia articles with SUDOC identifiers"], "title": "Number theory", "method": "Number theory", "url": "https://en.wikipedia.org/wiki/Number_theory", "summary": "Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777\u20131855) said, \"Mathematics is the queen of the sciences \u2013 and number theory is the queen of mathematics.\" Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).\nIntegers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation).\nThe older term for number theory is arithmetic. By the early twentieth century, it had been superseded by \"number theory\". (The word \"arithmetic\" is used by the general public to mean \"elementary calculations\"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. 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dealing with limits\nand related theories, such as differentiation, integration, measure, infinite series, and analytic functions.These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.\nAnalysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects  (a metric space).", "images": ["https://upload.wikimedia.org/wikipedia/commons/c/c9/Archimedes_pi.svg", "https://upload.wikimedia.org/wikipedia/commons/9/95/Attracteur_%C3%A9trange_de_Lorenz.png", "https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e8/Nuvola_apps_kmplot.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["(\u03b5, \u03b4)-definition of limit", "Abstract algebra", "Algebra", "Algebraic geometry", "Algorithm", "Algorithm design", "Analysis of algorithms", "Analytic combinatorics", "Analytic function", "Analytic geometry", "Analytic number theory", "Applied mathematics", "Approximation", "Approximation theory", "Archimedes", "Area", "Areas of mathematics", "Arithmetic", "Arithmetization of analysis", "Augustin Louis Cauchy", "Axiom of choice", "Baire category theorem", "Bernard Bolzano", "Bernhard Riemann", "Bh\u0101skara II", "Biology", "Calculus", "Calculus of variations", "Camille Jordan", "Category theory", "Cauchy sequence", "Cavalieri's principle", "Celestial mechanics", "Chinese mathematics", "Classical mechanics", "Classification of discontinuities", "Clifford analysis", "Coding theory", "Combinatorics", "Complement (set theory)", "Complete metric space", "Complex analysis", "Complex number", "Complex numbers", "Complex plane", "Computability theory", "Computable analysis", "Computational mathematics", "Computational number theory", "Computer science", "Constructive analysis", "Continuous function", "Continuous probability", "Continuum (set theory)", "Control theory", "Convex analysis", "Countable", "Countably", "Counting measure", "Creative Commons", "Cryptography", "David Hilbert", "Decimal expansion", "Decision theory", "Dedekind cut", "Derivative", "Descartes", "Deterministic system (mathematics)", "Differentiable manifolds", "Differential entropy", "Differential equation", "Differential equations", "Differential game", "Differential geometry", "Differential topology", "Digital object identifier", "Discrete geometry", "Discrete mathematics", "Distance", "Dynamical systems", "Dynamical systems theory", "E. 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In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.\nIn pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions\u2014the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.\nIf a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/c/cd/Elmer-pump-heatequation.png", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Abstract differential equation", "Acta Eruditorum", "Albert Einstein", "Algebra", "Algebraic equations", "Algebraic geometry", "American Journal of Physics", "American Mathematical Society", "Analytic geometry", "Annalen der Physik", "Antiderivative", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Astronomy", "Asymptotic stability", "Augustin-Louis Cauchy", "Autonomous system (mathematics)", "Bernoulli differential equation", "Bibcode", "Biology", "Black\u2013Scholes", "Black\u2013Scholes equation", "Calculus", "Carl David Tolm\u00e9 Runge", "Category theory", "Cauchy\u2013Riemann equations", "Chaos theory", "Charles W. 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Freeman", "Wave equation", "Wave function", "Wiener process", "Wikisource", "Wronskian", "\u00c9mile Picard"], "references": ["http://www.mat.univie.ac.at/~gerald/ftp/book-ode/", "http://homes.chass.utoronto.ca/~cfraser/vibration.pdf", "http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx", "http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage=", "http://www.sosmath.com/diffeq/diffeq.html", "http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html", "http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode", "http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb", "http://adsabs.harvard.edu/abs/1916AnP...354..769E", "http://adsabs.harvard.edu/abs/1987AmJPh..55...33W", "http://tutorial.math.lamar.edu/classes/de/de.aspx", "http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/", "http://hti.umich.edu/u/umhistmath/", "http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001", "http://www.alberteinstein.info/gallery/gtext3.html", "http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm", "http://www.hedengren.net/research/models.htm", "http://doi.org/10.1002%2Fandp.19163540702", "http://doi.org/10.1119%2F1.15311", "http://goldbook.iupac.org/R05141.html", "http://www.jirka.org/diffyqs/", "http://www.khanacademy.org/math/differential-equations", "http://www.worldcat.org/oclc/2688081", "http://eqworld.ipmnet.ru/en/solutions/ode.htm", "https://books.google.com/books?id=", "https://books.google.com/books?id=9uf97reZZCUC&pg=PA191", "https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18", "https://d-nb.info/gnd/4012249-9", "https://id.ndl.go.jp/auth/ndlna/00560651", "https://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html", "https://web.archive.org/web/20130607120716/http://math.rareinfos.com/category/courses/solutions-differential-equations/homogeneous-linear-systems/", "https://www.wikidata.org/wiki/Q11214"]}, "Dynamical systems theory": {"categories": ["Complex systems theory", "Computational fields of study", "Dynamical systems", "Webarchive template wayback links"], "title": "Dynamical systems theory", "method": "Dynamical systems theory", "url": "https://en.wikipedia.org/wiki/Dynamical_systems_theory", "summary": "Dynamical systems theory is an area of mathematics used to describe the behavior of the  complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler\u2013Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set\u2014one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.\nThis theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.\nThis field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.", "images": ["https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5b/Lorenz_attractor_yb.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["A-not-B error", "AI", "Abstract algebra", "Algebra", "Algebraic geometry", "Ambient space", "Analytic geometry", "Andrey Kolmogorov", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Arithmetic dynamics", "Artificial life", "Baker's map", "Biological applications of bifurcation theory", "Biology", "Butterfly effect", "Calculus", "Calculus of variations", "Cantor set", "Category theory", "Cell (biology)", "Chaos theory", "Chaotic system", "Chemistry", "Classical mechanics", "Cognition", "Cognitive science", "Combinatorics and dynamical systems", "Complex plane", "Complex systems", "Complexity", "Computational mathematics", "Computer", "Computer science", "Continuous time", "Control theory", "David Luenberger", "Deterministic system (mathematics)", "Deterministic system (philosophy)", "Diane Larsen-Freeman", "Difference equations", "Differential-difference equations", "Differential equation", "Differential equations", "Differential geometry", "Digital object identifier", "Discrete geometry", "Discrete mathematics", "Discrete time", "Dmitri Anosov", "Dynamic approach to second language development", "Dynamic equations on time scales", "Dynamical system", "Dynamical system (definition)", "Dynamical systems", "Dynamicism", "Economics", "Electronic circuit", "Elementary algebra", "Elon Lindenstrauss", "Embodied Embedded Cognition", "Engineering", "Equations of motion", "Equilibrium point", "Ergodic theory", "Euler\u2013Lagrange equation", "Evolutionary computation", "Feedback", "Finite geometry", "Flow (mathematics)", "Formal system", "Foundations of mathematics", "Fourier transform", "Function (mathematics)", "Functional (mathematics)", "Functional analysis", "Functional space", "Game theory", "General topology", "Geometry", "Gingerbreadman map", "Graph dynamical system", "Graph theory", "Grigory Margulis", "Halo orbit", "Hillel Furstenberg", "History of mathematics", "Homogeneity (physics)", "Independent variable", "Information theory", "Integral equations", "International Standard Book Number", "Invariant measure", "Iterated function", "JSTOR", "J\u00fcrgen Moser", "Language attrition", "Linear algebra", "Linear system", "List of dynamical system topics", "List of types of systems theory", "Lists of mathematics topics", "Lorenz attractor", "Manifold", "Mathematical analysis", "Mathematical logic", "Mathematical model", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Mechanics", "Meteorology", "Michael A. Arbib", "Molecular biology", "Multilinear algebra", "Nature", "Neurodynamics", "Neurosciences", "Newtonian mechanics", "Nikolay Bogolyubov", "Nikolay Mitrofanovich Krylov", "Non-linear", "Nonlinear dynamical system", "Nonlinear system", "Nonlinearity", "Number theory", "Numerical analysis", "OCLC", "Operator (mathematics)", "Optimization (mathematics)", "Order theory", "Ordinary differential equations", "Oscillation", "Otomar H\u00e1jek", "Outline of mathematics", "People in systems and control", "Philosophy of mathematics", "Physics", "Planetary orbit", "Point (geometry)", "Polynomial", "Postcognitivism", "Principle of least action", "Probability theory", "Projected dynamical systems", "Psychology", "PubMed Identifier", "Pure mathematics", "Ralph Abraham (mathematician)", "Randomness", "Rational function", "Real line", "Real numbers", "Recreational mathematics", "Recurrent neural network", "Reductionist", "Science", "Science (journal)", "Scientific modelling", "Second language acquisition", "Self-organization", "Sequence", "Set (mathematics)", "Set theory", "Sharkovskii's theorem", "Shift operator", "Simulation", "Social sciences", "Society", "Sports biomechanics", "State space", "Statistical physics", "Statistics", "Stefan Banach", "Stephen Smale", "Steven Strogatz", "Stochastic differential equation", "Stock and flow", "Superposition principle", "Symbolic dynamics", "Synergetics (Haken)", "System", "System dynamics", "Systemography", "The neo-Piagetian theories of cognitive development", "Theory of computation", "Tim van Gelder", "Time", "Topological dynamics", "Topology", "Trigonometry", "Vector space", "Vito Volterra", "Vladimir Arnold", "Wayback Machine", "Yakov G. Sinai"], "references": ["http://home.oise.utoronto.ca/~mlewis/Manuscripts/Promise.pdf", "http://geoffboeing.com/publications/nonlinear-chaos-fractals-prediction/", "http://mathworld.wolfram.com/DynamicalSystem.html", "http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html", "http://www.indiana.edu/~cogdev/labwork/dynamicsystem.pdf", "http://www.ncbi.nlm.nih.gov/pubmed/10836556", "http://www.ncbi.nlm.nih.gov/pubmed/12907229", "http://sysdyn.clexchange.org", "http://doi.org/10.1016%2FS1364-6613(03)00156-6", "http://doi.org/10.1111%2F1467-8624.00116", "http://doi.org/10.1126%2Fscience.238.4827.632", "http://doi.org/10.3390%2Fsystems4040037", "http://www.dynamicalsystems.org/", "http://www.jstor.org/stable/1700479", "http://www.sportsci.org/jour/03/psg.htm", "http://www.worldcat.org/oclc/24345312", "http://www.worldcat.org/oclc/343328", "http://www.worldcat.org/oclc/36713294", "http://www.worldcat.org/oclc/4195122", "http://www.worldcat.org/oclc/45873628", "http://www.worldcat.org/oclc/49839504", "http://www.worldcat.org/oclc/947600", "https://books.google.com/books?id=6UPvAAAAMAAJ", "https://books.google.com/books?id=MmoiJPsuFW8C", "https://books.google.com/books?id=ZPGBnZGV2_cC", "https://books.google.com/books?id=czhPAQAAIAAJ", "https://books.google.com/books?id=mvlQAAAAMAAJ", "https://books.google.com/books?id=oMbaAAAAMAAJ", "https://academic.oup.com/applij/article-abstract/18/2/141/134192", "https://web.archive.org/web/20080509163801/http://sysdyn.clexchange.org/", "https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html"]}, "Numerical analysis": {"categories": ["All articles lacking in-text citations", "Articles lacking in-text citations from November 2013", "Articles with German-language external links", "Computational science", "Mathematical physics", "Numerical analysis", "Use dmy dates from July 2012", "Webarchive template wayback links", "Wikipedia articles with GND identifiers"], "title": "Numerical analysis", "method": "Numerical analysis", "url": "https://en.wikipedia.org/wiki/Numerical_analysis", "summary": "Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.\nBefore the advent of modern computers, numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.\nOne of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of the square root of 2, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/2/28/LemonadeJuly2006.JPG", "https://upload.wikimedia.org/wikipedia/commons/1/12/Linear-regression.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/9/9d/Schumacher_%28Ferrari%29_in_practice_at_USGP_2005.jpg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/e/ef/Wind-particle.png", "https://upload.wikimedia.org/wikipedia/commons/0/0b/Ybc7289-bw.jpg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["ACM Computing Classification System", "Abramowitz and Stegun", "Abstract algebra", "Accelerator physics", "Acoustics", "Actuary", "Agrophysics", "Algebra", "Algebraic geometry", "Algorithm", "Algorithm design", "Algorithmic efficiency", "Analysis of algorithms", "Analytic geometry", "Application security", "Applied mathematics", "Applied physics", "Approximation", "Arbitrary precision arithmetic", "Areas of mathematics", "Arithmetic", "Artificial intelligence", "Astronomy", "Astroparticle physics", "Astrophysics", "Atmospheric physics", "Atomic, molecular, and optical physics", "Atomic physics", "Automata theory", "Automated planning and scheduling", "Babylonian mathematics", "Babylonian method", "Ballistics", "Biophysics", "Bisection method", "Branches of physics", "C (programming language)", "Calculus", "Carpentry", "Category theory", "Celestial mechanics", "Charles F. 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Trefethen", "Logic in computer science", "Loss of significance", "MATLAB", "Machine learning", "Magnetostatics", "Mahidol University", "Markov chain", "Massachusetts Institute of Technology", "Materials physics", "Mathematica", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical software", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Matrix decomposition", "Matrix splitting", "Mechanical calculator", "Medical physics", "Mesoscopic physics", "Michiel Hazewinkel", "Middleware", "Mixed reality", "Model of computation", "Modeling language", "Modern physics", "Molecular physics", "Monte Carlo integration", "Monte Carlo method", "Multi-task learning", "Multilinear algebra", "Multimedia database", "Multiprocessing", "Multithreading (computer architecture)", "NIST", "Natural language processing", "Netlib", "Network architecture", "Network performance", "Network protocol", "Network scheduler", "Network security", "Network service", "Networking hardware", "Neurophysics", "Newton's method", "Newton\u2013Cotes formulas", "Nicholas Higham", "Nobel Prize in Physics", "Nuclear astrophysics", "Nuclear physics", "NumPy", "Number theory", "Numerical Algorithms Group", "Numerical Recipes", "Numerical differentiation", "Numerical integration", "Numerical linear algebra", "Numerical method", "Numerical methods for linear least squares", "Numerical ordinary differential equations", "Numerical partial differential equations", "Numerical stability", "Numerical weather prediction", "Numerically stable", "Open-source software", "Operating system", "Operations research", "Optics", "Order theory", "Ordinary differential equation", "Outline of mathematics", "Oxford University", "Parallel computing", "Partial differential equation", "Particle physics", "Peripheral", "Phenomenology (physics)", "Philosophy of artificial intelligence", "Philosophy of mathematics", "Photo manipulation", 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criterion) from some set of available alternatives.In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding \"best available\" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/7/72/Max_paraboloid.svg", "https://upload.wikimedia.org/wikipedia/commons/3/33/Nelder-Mead_Simionescu.gif", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Active filter", "Aerospace engineering", "Affine scaling", "Agent (economics)", "Aharon Ben-Tal", "Albert W. 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branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.\nThe usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fr\u00e9chet and L\u00e9vy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.\nIn modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e9/Drum_vibration_mode12.gif", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Absolute value", "Absolutely convex set", "Absorbing set", "Abstract Wiener space", "Abstract algebra", "Alain Connes", "Algebra", "Algebraic geometry", "Algebraic interior", "Analytic geometry", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Avner Friedman", "Axiom of choice", "Baire category theorem", "Balanced set", "Banach-Steinhaus theorem", "Banach algebra", "Banach lattice", "Banach space", "Banach\u2013Alaoglu theorem", "Banach\u2013Mazur theorem", "Banach\u2013Saks theorem", "Banach\u2013Steinhaus theorem", "Barrelled space", "Barry Simon", "Bessel's inequality", "Biblioteca Nacional de Espa\u00f1a", "Biblioth\u00e8que nationale de France", "Bilinear form", "Bilinear map", "Bochner integral", "Bochner space", "Boolean prime ideal theorem", "Borel functional calculus", "Bornological space", "Bounded linear operator", "Bounded operator", "Bounded set (topological vector space)", "Bounding point", "Brauner space", "C*-algebra", "Calculus", "Calculus of variations", "Cardinal number", "Category theory", "Cauchy\u2013Schwarz inequality", "Closed graph theorem", "Closed linear operator", "Closed range theorem", "Combinatorial", "Compact operator", "Compact space", "Complete space", "Complex number", "Computational mathematics", "Computer science", "Cone (linear algebra)", "Continuous dual", "Continuous function", "Continuous function (topology)", "Continuous functional calculus", "Continuous linear operator", "Control theory", "Convex cone", "Convex hull", "Convex set", "Counting measure", "Densely defined", "Derivative", "Differential equation", "Differential equations", "Differential geometry", "Differentiation in Fr\u00e9chet spaces", "Digital object identifier", "Dimension (vector space)", "Discontinuous linear map", "Discrete geometry", "Discrete mathematics", "Dominate (mathematics)", "Drum head", "Dual norm", "Dual space", "Dual topology", "Dunford integral", "Dynamical systems theory", "Eberlein\u2013\u0160mulian theorem", "Ecole Centrale Paris", "Eigenfunction", "Elementary algebra", "Encyclopedia of Mathematics", "Ergodic theory", "Erwin Kreyszig", "Ess sup", "Euclidean space", "Extreme point", "F-space", "Finite geometry", "Foundations of mathematics", "Fourier transform", "Fredholm operator", "Freudenthal spectral theorem", "Frigyes Riesz", "Fr\u00e9chet derivative", "Fr\u00e9chet space", "Function space", "Functional (mathematics)", "Functional analysis (psychology)", "Functional calculus", "Functional derivative", "Game theory", "Gelfand\u2013Mazur theorem", "Gelfand\u2013Naimark theorem", "Geometry", "George Mackey", "Gerald Teschl", "Goldstine theorem", "Graph theory", "G\u00e2teaux derivative", "Hahn\u2013Banach theorem", "Hans Hahn (mathematician)", "Hausdorff space", "Ha\u00efm Brezis", "Hermitian adjoint", "Higher-order function", "Hilbert space", "Hilbert\u2013Schmidt operator", "History of mathematics", "Holomorphic functional calculus", "Hugo Steinhaus", "Hyperplane separation theorem", "Infinite-dimensional Lebesgue measure", "Infinite-dimensional holomorphy", "Information theory", "Inner product space", "Integer", "Integral", "Integral equations", "Integrated Authority File", "Interior (topology)", "International Standard Book Number", "Invariant subspace", "Invariant subspace problem", "Inverse function theorem", "Isometry", "Isomorphism", "Israel Gelfand", "Jacob T. Schwartz", "Jacques Hadamard", "James Munkres", "Jean Bourgain", "John B. Conway", "Juliusz Schauder", "Kakutani fixed-point theorem", "Kolmogorov", "Krein\u2013Milman theorem", "K\u014dsaku Yosida", "LF-space", "Law of large numbers", "Lebesgue-measurable function", "Library of Congress Control Number", "Linear algebra", "Linear form", "Linear functional", "Linear map", "Linear subspace", "Linear transformation", "List of functional analysis topics", "Lists of mathematics topics", "Locally convex topological vector space", "Lp space", "Lw\u00f3w School of Mathematics", "M. Riesz extension theorem", "Mackey space", "Mackey topology", "Mackey\u2013Arens theorem", "Mathematical analysis", "Mathematical formulation of quantum mechanics", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Maurice Ren\u00e9 Fr\u00e9chet", "Mazur's lemma", "Measure (mathematics)", "Measure space", "Michael C. Reed", "Michiel Hazewinkel", "Minkowski addition", "Minkowski functional", "Montel space", "Multilinear algebra", "Multiplication operator", "Nash\u2013Moser theorem", "National Diet Library", "Nelson Dunford", "Noncommutative geometry", "Norm (mathematics)", "Normal operator", "Normed space", "Normed vector space", "Nuclear operator", "Nuclear space", "Number theory", "Numerical analysis", "Open map", "Open mapping theorem (functional analysis)", "Open set", "Operator algebra", "Operator theory", "Operator topologies", "Order theory", "Orthonormal basis", "Outline of mathematics", "Paley\u2013Wiener integral", "Parseval's identity", "Paul L\u00e9vy (mathematician)", "Peter Lax", "Pettis integral", "Philosophy of mathematics", "Poland", "Polar decomposition", "Polar set", "Polarization identity", "Positive linear functional", "Probability", "Probability theory", "Projection-valued measure", "Pure mathematics", "Quantum mechanics", "Quasinorm", "Radial set", "Real number", "Recreational mathematics", "Reflexive space", "Regulated integral", "Representation theory", "Riesz representation theorem", "Riesz space", "Schauder basis", "Schauder fixed-point theorem", "Self-adjoint operator", "Separable space", "Sequence space", "Sergei Fomin", "Sesquilinear form", "Set theory", "Singular value decomposition", "Smith space", "Sobolev", "Spectral measure", "Spectral radius", "Spectral theorem", "Spectral theory", "Spectral theory of ordinary differential equations", "Spectrum (functional analysis)", "Spectrum of a C*-algebra", "Star domain", "Statistics", "Stefan Banach", "Stereotype space", "Strictly convex space", "Strictly singular operator", "Strong operator topology", "Strong topology", "Strong topology (polar topology)", "Sublinear function", "Surjective", "Symmetric set", "Tensor product of Hilbert spaces", "Theory of computation", "Topological group", "Topological ring", "Topological space", "Topological tensor product", "Topological vector space", "Topology", "Topology of uniform convergence", "Trace class", "Transpose of a linear map", "Trigonometry", "Ultrastrong topology", "Ultraweak topology", "Unbounded operator", "Uniform boundedness principle", "Unitary operator", "University of Colorado Colorado Springs", "Up to", "Upper Saddle River: Prentice Hall", "Vector measure", "Vector space", "Vector space basis", "Vito Volterra", "Walter Rudin", "Weak operator topology", "Weak topology", "Weak topology (polar topology)", "Weakly measurable function", "Webbed space", "Zonotope", "Zorn's lemma"], "references": ["http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html", "http://www.ebook3000.com/Theory-of-Linear-Operations--Volume-38--North-Holland-Mathematical-Library--by-S--Banach_134628.html", "http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf", "http://www.math.nyu.edu/phd_students/vilensky/Functional_Analysis.pdf", "http://www.uccs.edu/~gmorrow/", "http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX525044", "http://data.bnf.fr/ark:/12148/cb131627608", "http://www.coursera.org/", "http://www.coursera.org/course/functionalanalysis", "https://www.youtube.com/playlist?list=PLE1C83D79C93E2266", "https://catalogue.bnf.fr/ark:/12148/cb131627608", "https://id.loc.gov/authorities/subjects/sh85052312", "https://d-nb.info/gnd/4018916-8", "https://id.ndl.go.jp/auth/ndlna/00564962", "https://doi.org/10.1007%2F3-540-29587-9", "https://www.encyclopediaofmath.org/index.php?title=p/f042020", "https://www.wikidata.org/wiki/Q190549"]}, "Discrete geometry": {"categories": ["CS1 maint: Multiple names: authors list", "Discrete geometry"], "title": "Discrete geometry", "method": "Discrete geometry", "url": "https://en.wikipedia.org/wiki/Discrete_geometry", "summary": "Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.  Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth.  The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.\nDiscrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.", "images": ["https://upload.wikimedia.org/wikipedia/commons/3/32/Fano_plane_with_nimber_labels.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/3/36/Structural_rigidity_basic_examples.svg", "https://upload.wikimedia.org/wikipedia/commons/b/be/Unit_disk_graph.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["4-polytope", "Abstract algebra", "Abstract differential geometry", "Abstract polytope", "Abstract simplicial complex", "Affine plane (incidence geometry)", "Alexander Lubotzky", "Algebra", "Algebraic geometry", "Algebraic topology", "Analysis on fractals", "Analytic geometry", "Apeirotope", "Aperiodic tiling", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Armand Borel", "Arrangement of hyperplanes", "Arrangement of lines", "Augustin-Louis Cauchy", "Axel Thue", "Borsuk-Ulam theorem", "Building (mathematics)", "Calculus", "Category theory", "Cauchy's theorem (geometry)", "Circle", "Circle packing", "Combinatorial geometry", "Combinatorial optimization", "Combinatorial topology", "Combinatorics", "Computational geometry", "Computational mathematics", "Computer graphics", "Computer science", "Configuration (geometry)", "Control theory", "Convex geometry", "Convex lattice polytope", "Cycle graph", "Delaunay triangulation", "Differential equation", "Differential geometry", "Digital data", "Digital geometry", "Digitizing", "Dimension", "Directed graph", "Discrete Laplace operator", "Discrete Morse theory", "Discrete and Computational Geometry", "Discrete differential geometry", "Discrete exterior calculus", "Discrete group", "Discrete mathematics", "Discrete space", "Discrete subgroup", "Discrete topology", "Dynamical systems theory", "Edge (graph theory)", "Ehrhart polynomial", "Elementary algebra", "Ernst Steinitz", "Euclidean space", "Fair division", "Fano plane", "Field (mathematics)", "Finite geometry", "Finite set", "Finite subdivision rule", "Flexible polyhedron", "Foundations of mathematics", "Four colour theorem", "Functional analysis", "Game theory", "Geometric graph theory", "Geometry", "Geometry of numbers", "George Mostow", "Graph (discrete mathematics)", "Graph drawing", "Graph theory", "Grigory Margulis", "Group (mathematics)", "Hadwiger", "Harish-Chandra", "Harold Scott MacDonald Coxeter", "Hinge", "Hirsch conjecture", "History of mathematics", "Horst Martini", "Hyman Bass", "Hyperbolic space", "Hypersphere", "Image", "Image analysis", "Imre B\u00e1r\u00e1ny", "Incidence (geometry)", "Incidence structure", "Information theory", "Integer", "International Standard Book Number", "Intersection (set theory)", "Intersection graphs", "Invariant measure", "Jacob E. Goodman", "Johannes Kepler", "J\u00e1nos Pach", "Kepler conjecture", "Kneser graph", "K\u00e1roly Bezdek", "Lattice (discrete group)", "Lattice (group)", "Lie group", "Line (geometry)", "Line segment", "Linear algebra", "Linear independence", "Linear optimization", "Linkage (mechanical)", "Lists of mathematics topics", "Local field", "Locally compact", "L\u00e1szl\u00f3 Fejes T\u00f3th", "L\u00e1szl\u00f3 Lov\u00e1sz", "M. S. Raghunathan", "Manifold", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematical structure", "Mathematics", "Mathematics and art", "Mathematics education", "Matroid", "Mesh", "Metric space", "Multilinear algebra", "M\u00f6bius plane", "Nilpotent group", "Non-Euclidean geometry", "Number theory", "Numerical analysis", "Order theory", "Ordered field", "Ordered vector space", "Oriented matroid", "Outline of mathematics", "Packing problem", "Paul Erd\u0151s", "Periodic graph (geometry)", "Peter M. Gruber", "Philosophy of mathematics", "Pick's theorem", "Plane (geometry)", "Plane (mathematics)", "Point (geometry)", "Polygon", "Polyhedra", "Polyhedral combinatorics", "Polyhedral graph", "Polyhedron", "Polytope", "Probability theory", "Projective configuration", "Projective plane", "Pure mathematics", "Quasicrystal", "Quotient space (topology)", "Raster graphics", "Rational number", "Real numbers", "Recreational mathematics", "Reflection group", "Regular map (graph theory)", "Relative topology", "Rigid body", "Robert Zimmer (mathematician)", "Rockafellar", "Scale model", "Semisimple algebraic group", "Set (mathematics)", "Set theory", "Simple matroid", "Simplex", "Simplicial complex", "Simplicial complexes", "Simplicial set", "Skeleton (topology)", "Spectral shape analysis", "Sperner's lemma", "Sphere", "Sphere packing", "Statistics", "Structural rigidity", "Subgroup", "Tessellation", "Theory of computation", "Topological combinatorics", "Topological group", "Topological space", "Topology", "Toric geometry", "Triangle", "Triangle group", "Trigonometry", "Tsuneo Tamagawa", "Unit disk graph", "Vector space", "Vertex (graph theory)", "Visibility graphs", "Vladimir Boltyanski", "Voronoi diagram"], "references": ["http://www.renyi.hu/conferences/intuitiv_geometry/", "https://www.springer.com/us/book/9783319120980"]}, "Algebraic geometry": {"categories": ["Algebraic geometry", "All articles lacking in-text citations", "All articles with dead external links", "All articles with unsourced statements", "Articles lacking in-text citations from August 2016", "Articles with dead external links from September 2018", "Articles with permanently dead external links", "Articles with unsourced statements from November 2018", "Webarchive template wayback links", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Algebraic geometry", "method": "Algebraic geometry", "url": "https://en.wikipedia.org/wiki/Algebraic_geometry", "summary": "Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving  geometrical problems about these sets of zeros.\nThe fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.\nAlgebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.\nIn the 20th century, algebraic geometry split into several subareas.\n\nThe mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.\nThe study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.\nThe study of the real points of an algebraic variety is the subject of real algebraic geometry.\nA large part of singularity theory is devoted to the singularities of algebraic varieties.\nWith the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on \"intrinsic\" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e4/Crystal_Clear_app_3d.png", "https://upload.wikimedia.org/wikipedia/commons/d/d7/Parabola_%26_cubic_curve_in_projective_space.png", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/9/90/Slanted_circle.png", "https://upload.wikimedia.org/wikipedia/commons/8/88/Stereographic_projection_in_3D.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e0/Togliatti_surface.png", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abelian integral", "Abelian variety", "Absolute geometry", "Abstract algebra", "Affine geometry", "Affine space", "Affine variety", "Ahmes", "Alexander Grothendieck", "Algebra", "Algebraic Geometry (book)", "Algebraic Geometry (journal)", "Algebraic geometric code", "Algebraic geometry of projective spaces", "Algebraic number theory", "Algebraic space", "Algebraic stack", "Algebraic statistics", "Algebraic surface", "Algebraic variety", "Algebraically closed field", "Algorithm", "Alhazen", "Alicia Dickenstein", "Allen Tannenbaum", "Altitude (triangle)", "American Mathematical Society", "Analytic function", "Analytic geometry", "Analytic variety", "Andr\u00e9 Weil", "Angle", "Apollonius of Perga", "Applied mathematics", "ArXiv", "Arakelov's geometry", "Archimedes", "Area", "Area of a circle", "Areas of mathematics", "Arithmetic", "Arithmetic geometry", "Arthur Cayley", "Aryabhata", "Atomic formula", "B. L. van der Waerden", "Barry A. Cipra", "Baudhayana", "Before Common Era", "Bernhard Riemann", "Bertrand To\u00ebn", "Birational isomorphism", "Birational transformation", "Blaise Pascal", "Brahmagupta", "Branches of geometry", "Bruno Buchberger", "B\u00e9zout's theorem", "Calculus", "Cambridge University Press", "Carl Friedrich Gauss", "Carlos Simpson", "Cassini oval", "Category theory", "Characteristic (algebra)", "Circle", "Circumference", "Closure operator", "Commutative algebra", "Compass-and-straightedge construction", "Complement (set theory)", "Complex analysis", "Complex geometry", "Complex manifold", "Computational complexity", "Computational complexity theory", "Computational geometry", "Computational mathematics", "Computational phylogenetics", "Computer algebra", "Computer science", "Congruence (geometry)", "Conic sections", "Continuous function", "Control theory", "Convex geometry", "Coordinate geometry", "Coordinate ring", "Critical point (mathematics)", "Cube", "Cubic curve", "Cubic equation", "Cuboid", "Curve", "Cusp (singularity)", "Cylinder (geometry)", "Cylindrical algebraic decomposition", "Daniel Lazard", "Daniel Pedoe", "David A. Cox", "David Eisenbud", "David Hilbert", "David Mumford", "Degree of an algebraic variety", "Delian problem", "Deligne-Mumford stack", "Derived affine scheme", "Derived algebraic geometry", "Diagonal", "Diameter", "Differentiable manifold", "Differential algebraic geometry", "Differential equation", "Differential geometry", "Differential graded commutative algebra", "Differential manifold", "Diffiety", "Digital geometry", "Digital object identifier", "Dimension", "Dimension of an algebraic variety", "Diophantine equations", "Discrete differential geometry", "Discrete geometry", "Discrete mathematics", "Domain of a function", "Dover", "Dual (category theory)", "Dynamical systems theory", "Econometrica", "Edmond Laguerre", "Elementary algebra", "Elimination theory", "Ellipse", "Elliptic-curve cryptography", "Elliptic curve", "Elliptic geometry", "Equation solving", "Equivalence of categories", "Euclid", "Euclidean geometry", "Euclidean space", "Faug\u00e8re's F4 and F5 algorithms", "Faug\u00e8re F5 algorithm", "Felix Klein", "Fermat's last theorem", "Field (mathematics)", "Field of fractions", "Finite geometry", "First-order logic", "Floating point", "Formal scheme", "Foundations of mathematics", "Four-dimensional space", "Fractal", "Francis Sowerby Macaulay", "Franciscus Vieta", "Function field of an algebraic variety", "Function inverse", "Functional analysis", "GAGA", "Gabrielle Vezzosi", "Galois connection", "Game theory", "Generic point", "Geometric algebra", "Geometric modelling", "Geometry", "George E. Collins", "Gerolamo Cardano", "Glossary of classical algebraic geometry", "Gottfried Wilhelm Leibniz", "Graph theory", "Grothendieck", "Grothendieck site", "Grothendieck topology", "Group (mathematics)", "Gr\u00f6bner basis", "G\u00e9rard Desargues", "Harold Scott MacDonald Coxeter", "Heisuke Hironaka", "Hellenistic Greece", "Henri Poincar\u00e9", "Hermann Minkowski", "Higher category theory", "Hilbert's Nullstellensatz", "Hilbert's basis theorem", "Hilbert's sixteenth problem", "Hilbert series", "History of geometry", "History of mathematics", "Homogeneous coordinate ring", "Homogeneous coordinates", "Homogeneous ideal", "Homogeneous polynomial", "Homological algebra", "Homological mirror symmetry", "Homotopy continuation", "Hyperbola", "Hyperbolic geometry", "Hypersphere", "Hypotenuse", "Ibn al-Haytham", "Ibn al-Yasamin", "Ideal (ring theory)", "Igor Shafarevich", "Image (mathematics)", "Ind-scheme", "Inflection point", "Information theory", "Integer programming", "Integral domain", "Integrated Authority File", "International Standard Book Number", "Intersection theory", "Irreducible component", "Isaac Newton", "Italian school of algebraic geometry", "JSTOR", "Jacob Lurie", "Jean-Pierre Serre", "Jean Dieudonn\u00e9", "Joe Harris (mathematician)", "Joseph Louis Lagrange", "Jye\u1e63\u1e6dhadeva", "J\u00e1nos Bolyai", "Kite (geometry)", "Kleinian geometry", "K\u0101ty\u0101yana", "Lemniscate of Bernoulli", "Length", "Leonhard Euler", "Library of Congress Control Number", "Line (geometry)", "Line at infinity", "Line segment", "Linear algebra", "List of algebraic surfaces", "List of geometers", "List of publications in mathematics", "Lists of mathematics topics", "Locus (mathematics)", "Manava", "Marseille", "Matching (graph theory)", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical singularity", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Mathematics in medieval Islam", "Maxim Kontsevich", "Maximal ideal", "Menaechmus", "Michael Atiyah", "Mikhail Leonidovich Gromov", "Miles Reid", "Minggatu", "Moduli space", "Monomial order", "Morphism", "Morphism of algebraic varieties", "Multilinear algebra", "Multivariate polynomial", "Multivariate resultant", "N-sphere", "Nasir al-Din al-Tusi", "National Diet Library", "Niccol\u00f2 Fontana Tartaglia", "Nikolai Durov", "Nikolai Lobachevsky", "Nisnevich topology", "Non-Euclidean geometry", "Noncommutative algebraic geometry", "Normal space", "Number field", "Number theory", "Numerical algebraic geometry", "Numerical analysis", "Omar Khayy\u00e1m", "One-dimensional space", "Order theory", "Ordered field", "Orthogonal", "Oscar Zariski", "Oswald Veblen", "Outline of geometry", "Outline of mathematics", "Parabola", "Parallel (geometry)", "Parallelogram", "Parameshvara", "Perpendicular", "Persian people", "Phillip Griffiths", "Philosophy of mathematics", "Pierre de Fermat", "Plane (geometry)", "Plane algebraic curve", "Point (geometry)", "Point at infinity", "Polygon", "Polynomial", "Polynomial equation", "Polynomial function", "Prime ideal", "Probability theory", "Projective geometry", "Projective plane", "Projective space", "Publications Math\u00e9matiques de l'IH\u00c9S", "Pure mathematics", "Pyramid (geometry)", "Pythagoras", "Pythagorean theorem", "Quadratic form", "Quadrilateral", "Quantifier elimination", "Quasicategory", "Quillen model category", "Radical of an ideal", "Rational function", "Rational mapping", "Rational number", "Rational variety", "Ray (geometry)", "Real algebraic geometry", "Recreational mathematics", "Rectangle", "Reduced ring", "Regular chain", "Regular function", "Regular point of an algebraic variety", "Renaissance", "Ren\u00e9 Descartes", "Resolution of singularities", "Resultant", "Rhomboid", "Rhombus", "Riemann-Roch theorem for algebraic curves", "Riemann surface", "Riemannian geometry", "Ring homomorphism", "Robin Hartshorne", "Robotics", "Sakabe K\u014dhan", "Scheme (mathematics)", "Scheme theory", "Set theory", "Sharaf al-Din al-Tusi", "Sheaf theory", "Sijzi", "Similarity (geometry)", "Singular point of a curve", "Singularity theory", "Smooth completion", "Smooth function", "Software", "Soliton", "Solution set", "Sphere", "Spherical geometry", "Springer Science+Business Media", "Square", "Statistics", "Symmetry", "Symplectic geometry", "Synthetic geometry", "Systems of polynomial equations", "Tarski\u2013Seidenberg theorem", "Tesseract", "Theory of computation", "Three-dimensional space", "Tietze extension theorem", "Togliatti surface", "Topological space", "Topology", "Transformation group", "Trapezoid", "Triangle", "Trigonometry", "Tropical geometry", "Two-dimensional space", "Universal algebraic geometry", "Valuation theory", "Van der Waerden", "Variety (universal algebra)", "Vertex (geometry)", "Virasena", "Volume", "W. V. D. Hodge", "Wayback Machine", "Whitney stratification", "Wiles's proof of Fermat's Last Theorem", "Wiley-Interscience", "Yang Hui", "Yasuaki Aida", "Zariski topology", "Zentralblatt MATH", "Zero of a function", "Zhang Heng", "\u00c9lie Cartan", "\u00c9l\u00e9ments de g\u00e9om\u00e9trie alg\u00e9brique", "\u00c9tale topology"], "references": ["http://stacks.math.columbia.edu/", "http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf", "http://econwpa.wustl.edu/econ-wp/game/papers/9309/9309001.pdf", "http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html", "http://lccn.loc.gov/2007938208", "http://neo-classical-physics.info/uploads/3/0/6/5/3065888/van_der_waerden_-_algebraic_geometry.pdf", "http://arxiv.org/abs/math-ph/0311005", "http://doi.org/10.2307%2F2317664", "http://www.jstor.org/stable/2317664", "http://www.jstor.org/stable/2951732", "http://planetmath.org/", "http://siam.org/pdf/news/1146.pdf", "http://zbmath.org/?format=complete&q=an:0118.36206", "http://zbmath.org/?format=complete&q=an:0203.23301", "http://zbmath.org/?format=complete&q=an:0367.14001", "http://zbmath.org/?format=complete&q=an:0701.14001", "http://zbmath.org/?format=complete&q=an:0779.14001", "http://zbmath.org/?format=complete&q=an:0796.14001", "http://zbmath.org/?format=complete&q=an:0796.14002", "http://zbmath.org/?format=complete&q=an:0796.14003", "http://zbmath.org/?format=complete&q=an:0797.14001", "http://zbmath.org/?format=complete&q=an:0797.14002", "http://zbmath.org/?format=complete&q=an:0821.14001", "http://zbmath.org/?format=complete&q=an:0836.14001", "http://zbmath.org/?format=complete&q=an:0861.13012", "http://zbmath.org/?format=complete&q=an:0945.14001", "http://zbmath.org/?format=complete&q=an:0960.14002", "http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html/", "https://books.google.com/?id=1wNGq87gWykC", "https://books.google.com/?id=9FljXoISr8AC", "https://books.google.com/?id=TytYUTy5V_IC", "https://books.google.com/?id=V--84aGmWh4C", "https://books.google.com/?id=Z8u3ngEACAAJ", "https://books.google.com/?id=eO_PAAAAIAAJ", "https://books.google.com/?id=fe0MJEPDwzAC", "https://books.google.com/?id=o2sA-wzDBLkC", "https://d-nb.info/gnd/4001161-6", "https://id.ndl.go.jp/auth/ndlna/00561224", "https://archive.org/stream/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/Roshdi%20Rashed%20(auth.)-The%20Development%20of%20Arabic%20Mathematics_%20Between%20Arithmetic%20and%20Algebra-Springer%20Netherlands%20(1994)#page/n111/mode/1up", "https://web.archive.org/web/20040415021548/http://planetmath.org/encyclopedia/AlgebraicGeometry.html", "https://web.archive.org/web/20131113165913/https://pantherfile.uwm.edu/stevesch/Math%20Lectures/Jean%20Dieudonne%20EIAJ.mov", "https://web.archive.org/web/20160303230428/http://siam.org/pdf/news/1146.pdf#", "https://web.archive.org/web/20171112123436/http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html#", "https://www.wikidata.org/wiki/Q180969"]}, "Analytic geometry": {"categories": ["Analytic geometry", "Wikipedia articles with BNF identifiers", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NDL identifiers"], "title": "Analytic geometry", "method": "Analytic geometry", "url": "https://en.wikipedia.org/wiki/Analytic_geometry", "summary": "In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.\nAnalytic geometry is widely used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.\nUsually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor\u2013Dedekind axiom.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/0/0e/Cartesian-coordinate-system.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e4/Crystal_Clear_app_3d.png", "https://upload.wikimedia.org/wikipedia/commons/6/67/Distance_Formula.svg", "https://upload.wikimedia.org/wikipedia/commons/b/ba/FourGeometryTransformations.svg", "https://upload.wikimedia.org/wikipedia/commons/8/88/Stereographic_projection_in_3D.svg"], "links": ["Affine coordinates", "Affine geometry", "Affine transformations", "Ahmes", "Algebraic equation", "Algebraic geometry", "Alhazen", "Altitude (triangle)", "Ancient Greece", "Angle", "Apollonius of Perga", "Archimedes", "Area", "Area of a circle", "Aryabhata", "Aviation", "Baudhayana", "Before Common Era", "Bernhard Riemann", "Biblioth\u00e8que nationale de France", "Blaise Pascal", "Brahmagupta", "Branches of geometry", "CRC Press", "Calculus", "Cantor\u2013Dedekind axiom", "Carl Benjamin Boyer", "Carl Friedrich Gauss", "Cartesian coordinate system", "Cartesian coordinates", "Cartesian plane", "Circle", "Circumference", "Compass-and-straightedge construction", "Computational geometry", "Cones", "Congruence (geometry)", "Conic section", "Convex geometry", "Coordinate system", "Coordinate systems", "Cross product", "Cube", "Cubic equation", "Cuboid", "Curve", "Cylinder (geometry)", "Cylindrical coordinates", "David Hilbert", "Derivative", "Descartes", "Diagonal", "Diameter", "Differential geometry", "Digital geometry", "Digital object identifier", "Dimension", "Discourse on Method", "Discrete differential geometry", "Discrete geometry", "Discriminant", "Distance", "Dot product", "Ellipse", "Ellipsoid", "Elliptic geometry", "Engineering", "Equation", "Euclid", "Euclidean geometry", "Euclidean plane", "Euclidean space", "Felix Klein", "Finite geometry", "Force", "Formula", "Four-dimensional space", "Fractal", "Frans van Schooten", "French language", "Geometry", "Graph of a function", "Harold Scott MacDonald Coxeter", "Henri Poincar\u00e9", "Hermann Minkowski", "History of geometry", "Hyperbola", "Hyperbolic geometry", "Hyperboloid", "Hypersphere", "Hypotenuse", "Ibn al-Yasamin", "Independent variable", "Infinitesimal", "Integrated Authority File", "International Standard Book Number", "Internet Archive", "Intersection (geometry)", "Intersection (set theory)", "James Stewart (mathematician)", "John Casey (mathematician)", "John Stillwell", "Jye\u1e63\u1e6dhadeva", "J\u00e1nos Bolyai", "Kite (geometry)", "K\u0101ty\u0101yana", "La Geometrie", "Latin", "Length", "Leonhard Euler", "Library of Congress Control Number", "Line (geometry)", "Line segment", "Linear equation", "List of geometers", "Locus (mathematics)", "Manava", "Menaechmus", "Michael Atiyah", "Mikhail Leonidovich Gromov", "Minggatu", "Nasir al-Din al-Tusi", "National Diet Library", "Nikolai Lobachevsky", "Non-Euclidean geometry", "Normal (geometry)", "Normal vector", "Omar Khayy\u00e1m", "One-dimensional space", "Ordered pair", "Orthogonal", "Orthogonality", "Oswald Veblen", "Outline of geometry", "Parabola", "Paraboloid", "Parallel (geometry)", "Parallelogram", "Parameshvara", "Parametric equation", "Perpendicular", "Persian people", "Physics", "Pierre de Fermat", "Plane (geometry)", "Plane (mathematics)", "Point (geometry)", "Polar coordinates", "Polygon", "Projective geometry", "Projective space", "Pyramid (geometry)", "Pythagoras", "Pythagorean theorem", "Quadratic equation", "Quadratic polynomial", "Quadric surface", "Quadrilateral", "Radius", "Ray (geometry)", "Real number", "Real numbers", "Rectangle", "Ren\u00e9 Descartes", "Rhomboid", "Rhombus", "Riemannian geometry", "Rocket", "Roger Cooke", "Root of a function", "Sakabe K\u014dhan", "Sijzi", "Similarity (geometry)", "Slope", "Slope-intercept form", "Solution set", "Space curve", "Space science", "Spaceflight", "Sphere", "Spherical coordinates", "Spherical geometry", "Square", "Square (geometry)", "Straight line", "Subset", "Surface (mathematics)", "Symmetry", "Symplectic geometry", "Synthetic geometry", "Tangent", "Tangent line", "Tangent space", "Tesseract", "The Geometry Center", "Three-dimensional space", "Trapezoid", "Triangle", "Tuple", "Two-dimensional space", "University of Minnesota", "Vector (geometric)", "Vector (geometry)", "Vector space", "Vertex (geometry)", "Virasena", "Volume", "X-intercept", "Y-intercept", "Yang Hui", "Yasuaki Aida", "Zhang Heng", "\u00c9lie Cartan"], "references": ["http://www.mathopenref.com/tocs/coordpointstoc.html", "http://mathworld.wolfram.com/Plane.html", "http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html", "http://data.bnf.fr/ark:/12148/cb11938440r", "http://gallica.bnf.fr/ark:/12148/bpt6k56523g/f73.image.langEN", "http://gallica.bnf.fr/ark:/12148/bpt6k6213144d/f147.image.langEN", "http://doi.org/10.2307%2F2305740", "https://books.google.com/books?id=75mAJPcAWT8C", "https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45", "https://catalogue.bnf.fr/ark:/12148/cb11938440r", "https://id.loc.gov/authorities/subjects/sh85054141", "https://d-nb.info/gnd/4001867-2", "https://id.ndl.go.jp/auth/ndlna/00564622", "https://archive.org/details/cu31924001520455", "https://www.wikidata.org/wiki/Q134787"]}, "Differential geometry": {"categories": ["Articles that may contain original research from February 2017", "Differential geometry", "Geometry processing", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Differential geometry", "method": "Differential geometry", "url": "https://en.wikipedia.org/wiki/Differential_geometry", "summary": "Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.\nSince the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.", "images": ["https://upload.wikimedia.org/wikipedia/commons/1/1b/Ambox_question.svg", "https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e4/Crystal_Clear_app_3d.png", "https://upload.wikimedia.org/wikipedia/commons/d/d1/GPB_circling_earth.jpg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Hyperbolic_triangle.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/88/Stereographic_projection_in_3D.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Abstract differential geometry", "Abstract index notation", "Affine connection", "Affine differential geometry", "Affine geometry", "Ahmes", "Albert Einstein", "Algebra", "Algebraic geometry", "Alhazen", "Almost complex manifold", "Almost complex structure", "Altitude (triangle)", "Ambient space", "Analysis on fractals", "Analytic geometry", "Analytical mechanics", "Angle", "Angular momentum", "Annulus (mathematics)", "Antisymmetric tensor", "Apollonius of Perga", "Applied mathematics", "Arc length", "Archimedes", "Area", "Area of a circle", "Areas of mathematics", "Arithmetic", "Aryabhata", "Atlas (topology)", "Augustin-Louis Cauchy", "Banach norm", "Basic introduction to the mathematics of curved spacetime", "Basis (linear algebra)", "Baudhayana", "Beamforming", "Before Common Era", "Bernhard Riemann", "Bilinear form", "Biophysics", "Black holes", "Blaise Pascal", "Brahmagupta", "Branches of geometry", "CR structure", "Calculus", "Carl Friedrich Gauss", "Carl Gustav Jacobi", "Cartan formalism (physics)", "Category theory", "Cauchy stress tensor", "Chemistry", "Christoffel symbols", "Circle", "Circumference", "Compass-and-straightedge construction", "Complex manifold", "Complex manifolds", "Computational geometry", "Computational mathematics", "Computer-aided geometric design", "Computer graphics", "Computer science", "Computer vision", "Congruence (geometry)", "Connection (mathematics)", "Connection form", "Contact geometry", "Continuum mechanics", "Control theory", "Convex geometry", "Coordinate system", "Courant algebroid", "Covariance and contravariance of vectors", "Covariant derivative", "Cube", "Cuboid", "Curvature", "Curve", "Curved space", "Cylinder (geometry)", "Darboux's theorem", "David Hilbert", "Definite bilinear form", "Derivations of the Lorentz transformations", "Diagonal", "Diameter", "Diffeomorphism", "Differentiable manifold", "Differential calculus", "Differential equation", "Differential form", "Differential forms", "Differential geometry of curves", "Differential geometry of surfaces", "Differential topology", "Digital geometry", "Digital object identifier", "Digital signal processing", "Dimension", "Directional derivative", "Discrete differential geometry", "Discrete geometry", "Discrete mathematics", "Doubly special relativity", "Dyadics", "Dynamical systems theory", "Econometrics", "Economics", "Einstein", "Einstein field equations", "Einstein notation", "Einstein tensor", "Electromagnetic tensor", "Electromagnetism", "Elementary algebra", "Elliptic geometry", "Elwin Bruno Christoffel", "Encyclopedia of Mathematics", "Engineering", "Equivalence principle", "Euclid", "Euclidean geometry", "Euclidean space", "Exterior calculus", "Exterior covariant derivative", "Exterior derivative", "Exterior form", "Exterior product", "Felix Klein", "Fiber bundle", "Finite geometry", "Finsler manifold", "First order of approximation", "Fisher information metric", "Foundations of mathematics", "Four-dimensional space", "Four-vector", "Fractal", "Functional analysis", "G.D. Birkhoff", "Game theory", "Gaspard Monge", "Gauss", "Gaussian curvature", "General relativity", "General theory of relativity", "Geodesic", "Geometric calculus", "Geometric control", "Geometric modeling", "Geometrothermodynamics", "Geometry", "Glossary of differential geometry and topology", "Glossary of tensor theory", "Gluon field strength tensor", "Graph theory", "Grassmannian", "Gravitational lensing", "Gravity", "Gregorio Ricci-Curbastro", "Grigori Perelman", "Group (mathematics)", "Hamiltonian mechanics", "Hamiltonian system", "Harold Scott MacDonald Coxeter", "Henri Poincar\u00e9", "Hermann Grassmann", "Hermann Minkowski", "Hermann Weyl", "Hermitian manifold", "History of geometry", "History of mathematics", "Hodge manifold", "Hodge star operator", "Holomorphic function", "Hyperbolic geometry", "Hyperbolic paraboloid", "Hypersphere", "Hypotenuse", "Ibn al-Yasamin", "Image processing", "Index notation", "Information geometry", "Information theory", "Integral calculus", "Integral geometry", "Integrated Authority File", "International Standard Book Number", "Introduction to general relativity", "Introduction to the mathematics of general relativity", "Isometry", "Jan Arnoldus Schouten", "Joseph Louis Lagrange", "Jye\u1e63\u1e6dhadeva", "J\u00e1nos Bolyai", "Kite (geometry)", "Kronecker delta", "K\u00e4hler manifold", "K\u0101ty\u0101yana", "Lagrangian mechanics", "Length", "Leonhard Euler", "Levi-Civita connection", "Levi-Civita symbol", "Lie algebroid", "Lie derivative", "Lie group", "Line (geometry)", "Line segment", "Linear algebra", "Linear transformation", "List of differential geometry topics", "List of geometers", "List of publications in mathematics", "Lists of mathematics topics", "Lorentz transformation", "Lorentzian manifold", "MIMO", "Manava", "Manfredo do Carmo", "Manifold", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematician", "Mathematics", "Mathematics and art", "Mathematics education", "Mathematics of general relativity", "Matrix (mathematics)", "Metric tensor", "Metric tensor (general relativity)", "Michael Atiyah", "Michael Spivak", "Michiel Hazewinkel", "Mikhail Leonidovich Gromov", "Minggatu", "Minkowski diagram", "Minkowski space", "Mixed tensor", "Moment of inertia", "Multi-index notation", "Multilinear algebra", "Multivariable calculus", "Multivector", "Nash embedding theorem", "Nasir al-Din al-Tusi", "National Diet Library", "Newton's law of universal gravitation", "Nijenhuis tensor", "Nikolai Lobachevsky", "Non-Euclidean geometry", "Non-degenerate", "Noncommutative geometry", "Nonmetricity tensor", "Number theory", "Numerical analysis", "Omar Khayy\u00e1m", "One-dimensional space", "Operation (mathematics)", "Order theory", "Orthogonal", "Oswald Veblen", "Outline of geometry", "Outline of mathematics", "Parallel (geometry)", "Parallel transport", "Parallelogram", "Parameshvara", "Penrose graphical notation", "Perpendicular", "Phase space", "Philosophy of mathematics", "Physics", "Plane (geometry)", "Poincar\u00e9 conjecture", "Poincar\u00e9\u2013Birkhoff theorem", "Point (geometry)", "Polygon", "Positive definite bilinear form", "Principal bundle", "Probability", "Probability theory", "Projective differential geometry", "Projective geometry", "Pseudo-Riemannian manifold", "Pure mathematics", "Pyramid (geometry)", "Pythagoras", "Pythagorean theorem", "Quadrilateral", "Raising and lowering indices", "Ray (geometry)", "Recreational mathematics", "Rectangle", "Ren\u00e9 Descartes", "Representation of a Lie group", "Rhomboid", "Rhombus", "Ricci calculus", "Ricci curvature", "Ricci flow", "Riemann curvature tensor", "Riemannian geometry", "Riemannian manifold", "Riemannian metric", "Riemannian symmetric space", "Sakabe K\u014dhan", "Satellites", "Set theory", "Shape operator", "Sijzi", "Similarity (geometry)", "Skew-symmetric matrix", "Smooth function", "Smooth manifold", "Space-time", "Spacetime", "Spacetime topology", "Special relativity", "Sphere", "Spherical geometry", "Spin tensor", "Spinor", "Square", "Statistics", "Stress\u2013energy tensor", "Structural geology", "Surface (topology)", "Symmetric bilinear form", "Symmetric tensor", "Symmetry", "Symplectic geometry", "Symplectic manifold", "Symplectomorphism", "Synthetic differential geometry", "Synthetic geometry", "Systolic geometry", "Tangent bundle", "Tensor", "Tensor (intrinsic definition)", "Tensor algebra", "Tensor bundle", "Tensor calculus", "Tensor contraction", "Tensor density", "Tensor field", "Tensor operator", "Tensor product", "Tensors in curvilinear coordinates", "Tesseract", "Tetrad (index notation)", "Theodore Frankel", "Theorema Egregium", "Theorema egregium", "Theory of computation", "Thermodynamics", "Three-dimensional space", "Topology", "Torsion tensor", "Transport phenomena", "Transpose", "Trapezoid", "Triangle", "Trigonometry", "Tullio Levi-Civita", "Two-dimensional space", "Van der Waerden notation", "Vector (mathematics and physics)", "Vector bundle", "Vector field", "Vector space", "Vertex (geometry)", "Virasena", "Voigt notation", "Volume", "Volume form", "Weyl tensor", "William Rowan Hamilton", "Wireless", "Woldemar Voigt", "Yang Hui", "Yasuaki Aida", "Zhang Heng", "\u00c9lie Cartan"], "references": ["http://www.maths.adelaide.edu.au/michael.murray/teaching_old.html", "http://geocalc.clas.asu.edu/pdf/Shape%20in%20GC-2012.pdf", "http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/", "http://math.stanford.edu/~conrad/diffgeomPage/", "http://users.loni.ucla.edu/~ajoshi/final_thesis.pdf", "http://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf", "http://quod.lib.umich.edu/u/umhistmath/abr1255.0001.001", "http://users.ece.utexas.edu/~rheath/papers/2002/grassbeam/paper.pdf", "http://www.cs.elte.hu/geometry/csikos/dif/dif.html", "http://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf", "http://staff.science.uva.nl/~leo/agacse2010/talks_world/Hestenes.pdf", "http://doi.org/10.1109%2FICASSP.2005.1416480", "http://doi.org/10.1109%2FTIT.2003.817466", "http://www.encyclopediaofmath.org/index.php/Differential_geometry", "http://VirtualMathMuseum.org/", "http://VirtualMathMuseum.org/Surface/a/bk/curves_surfaces_palais.pdf", "https://d-nb.info/gnd/4012248-7", "https://id.ndl.go.jp/auth/ndlna/00560656", "https://web.archive.org/web/20110604092900/http://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf", "https://www.encyclopediaofmath.org/index.php?title=p/d032170", "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1416480", "https://www.wikidata.org/wiki/Q188444"]}, "Finite geometry": {"categories": ["CS1 errors: external links", "Combinatorics", "Finite geometry", "Webarchive template wayback links"], "title": "Finite geometry", "method": "Finite geometry", "url": "https://en.wikipedia.org/wiki/Finite_geometry", "summary": "A finite geometry is any geometric system that has only a finite number of points.\nThe familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.  Other significant types of finite geometry are finite M\u00f6bius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.\nFinite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries.  Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes.  Similar results hold for other kinds of finite geometries.", "images": ["https://upload.wikimedia.org/wikipedia/commons/0/0a/150614-PG-3-2-schoolgirls-arrangement.png", "https://upload.wikimedia.org/wikipedia/commons/e/e4/Crystal_Clear_app_3d.png", "https://upload.wikimedia.org/wikipedia/commons/5/5b/Fano3space.png", "https://upload.wikimedia.org/wikipedia/commons/a/af/Fano_plane.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a5/Fano_plane_Hasse_diagram.svg", "https://upload.wikimedia.org/wikipedia/commons/e/eb/Hesse_configuration.svg", "https://upload.wikimedia.org/wikipedia/commons/0/02/Order_2_affine_plane.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/88/Stereographic_projection_in_3D.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Affine geometry", "Affine plane (incidence geometry)", "Affine space", "Ahmes", "Algebra", "Algebraic geometry", "Alhazen", "Altitude (triangle)", "American Mathematical Monthly", "Analytic geometry", "Angle", "Apollonius of Perga", "Applied mathematics", "ArXiv", "Archimedes", "Area", "Area of a circle", "Areas of mathematics", "Arithmetic", "Aryabhata", "Axiom", "Axiomatic projective space", "Baudhayana", "Before Common Era", "Beniamino Segre", "Benz plane", "Bernhard Riemann", "Binomial coefficient", "Blaise Pascal", "Block design", "Brahmagupta", "Branches of geometry", "Bruck\u2013Ryser theorem", "Burkard Polster", "Calculus", "Cambridge University Press", "Carl Friedrich Gauss", "Category theory", "Circle", "Circumference", "Collineation", "Collineation group", "Compass-and-straightedge construction", "Computational geometry", "Computational mathematics", "Computer-assisted proof", "Computer science", "Congruence (geometry)", "Control theory", "Convex geometry", "Cube", "Cuboid", "Curve", "Cylinder (geometry)", "David Hilbert", "Desargues's theorem", "Desarguesian plane", "Diagonal", "Diameter", "Differential equation", "Differential geometry", "Digital geometry", "Digital object identifier", "Dimension", "Discrete differential geometry", "Discrete geometry", "Discrete mathematics", "Division ring", "Duality (mathematics)", "Dynamical systems theory", "Elementary algebra", "Elliptic geometry", "Empty set", "Ergebnisse der Mathematik und ihrer Grenzgebiete", "Eric W. Weisstein", "Euclid", "Euclidean geometry", "Exponent", "Fano plane", "Felix Klein", "Finite field", "Finite set", "Foundations of mathematics", "Four-dimensional space", "Fractal", "Functional analysis", "GF(2)", "Galois field", "Galois geometry", "Game theory", "Gaussian binomial coefficient", "General linear group", "Generalized polygon", "Geometry", "Gino Fano", "Graph theory", "Group (mathematics)", "Harold Scott MacDonald Coxeter", "Henri Poincar\u00e9", "Hermann Minkowski", "Hesse configuration", "History of geometry", "History of mathematics", "Homogeneous coordinates", "Hyperbolic geometry", "Hypersphere", "Hypotenuse", "Ibn al-Yasamin", "Incidence (geometry)", "Incidence geometry", "Incidence relation", "Incidence structure", "Information theory", "Integer", "International Standard Book Number", "International Standard Serial Number", "Inversive geometry", "Isomorphism", "Italians", "JSTOR", "Jean-Pierre Serre", "Jye\u1e63\u1e6dhadeva", "J\u00e1nos Bolyai", "Karl Georg Christian von Staudt", "Kirkman's schoolgirl problem", "Kite (geometry)", "K\u0101ty\u0101yana", "Laguerre plane", "Length", "Leonhard Euler", "Line (geometry)", "Line segment", "Linear algebra", "Linear space (geometry)", "List of geometers", "Lists of mathematics topics", "Lynn Batten", "Manava", "MathWorld", "Mathematical Reviews", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Michael Atiyah", "Mikhail Leonidovich Gromov", "Minggatu", "Multilinear algebra", "M\u00f6bius plane", "Nasir al-Din al-Tusi", "Near polygon", "Nikolai Lobachevsky", "Non-Desarguesian plane", "Non-Euclidean geometry", "Number theory", "Numerical analysis", "Omar Khayy\u00e1m", "One-dimensional space", "Order theory", "Orthogonal", "Oswald Veblen", "Outline of geometry", "Outline of mathematics", "PSL(2,7)", "Parallel (geometry)", "Parallelogram", "Parameshvara", "Partial geometry", "Pasch's axiom", "Perpendicular", "Philosophy of mathematics", "Pixel", "Plane (geometry)", "Playfair's axiom", "Point (geometry)", "Polar space", "Polygon", "Positive integer", "Positive number", "Prime number", "Prime power", "Probability theory", "Projective geometry", "Projective plane", "Projective space", "Pure mathematics", "Pyramid (geometry)", "Pythagoras", "Pythagorean theorem", "Quadrilateral", "Ray (geometry)", "Recreational mathematics", "Rectangle", "Ren\u00e9 Descartes", "Rhomboid", "Rhombus", "Riemannian geometry", "Sakabe K\u014dhan", "Set theory", "Sijzi", "Similarity (geometry)", "Sphere", "Spherical geometry", "Springer-Verlag", "Square", "Square (algebra)", "Statistics", "Symmetry", "Symplectic geometry", "Synthetic geometry", "Terence Tao", "Tesseract", "Theory of computation", "Thomas Penyngton Kirkman", "Three-dimensional space", "Topology", "Transactions of the American Mathematical Society", "Trapezoid", "Triangle", "Trigonometry", "Two-dimensional space", "Veblen\u2013Young theorem", "Vector space", "Vertex (geometry)", "Virasena", "Volume", "Wayback Machine", "Wedderburn's little theorem", "Yang Hui", "Yasuaki Aida", "Zhang Heng", "\u00c9lie Cartan"], "references": ["http://cage.ugent.be/geometry/links.php", "http://cage.ugent.be/~fdc/courses/GGaGP.php", "http://www.cecm.sfu.ca/organics/papers/lam/", "http://mathworld.wolfram.com/FiniteGeometry.html", "http://sbseminar.wordpress.com/", "http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/", "http://fog.ccsf.edu/~mgreenbe/FiniteGeometries.pdf", "http://www.math.mtu.edu/~jbierbra/HOMEZEUGS/finitegeom04.ps", "http://www.ams.org/featurecolumn/archive/finitegeometries.html", "http://www.ams.org/journals/tran/1906-007-02/S0002-9947-1906-1500747-6/S0002-9947-1906-1500747-6.pdf", "http://www.ams.org/mathscinet-getitem?mr=0008892", "http://www.ams.org/mathscinet-getitem?mr=0233275", "http://www.ams.org/mathscinet-getitem?mr=1629468", "http://arxiv.org/abs/1311.7177", "http://www.cambridge.org/hr/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/finite-geometry-and-combinatorial-applications", "http://doi.org/10.1007%2F978-3-642-15627-4", "http://doi.org/10.1007%2FBF03024845", "http://doi.org/10.2307%2F1990331", "http://doi.org/10.2307%2F2323798", "http://www.jstor.org/stable/1990331", "http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0488.0499.ocr.pdf", "http://www.worldcat.org/issn/0002-9947", "http://www.maths.sussex.ac.uk/Staff/JWPH/", "http://www.maths.sussex.ac.uk/Staff/JWPH/RESEARCH/index.html", "https://mathoverflow.net/q/38632/34461", "https://web.archive.org/web/20100817074231/http://home.wlu.edu/~mcraea/Finite_Geometry/Applications/Prob31SchoolGirl/problem31.html", "https://web.archive.org/web/20110927121417/http://home.wlu.edu/~mcraea/finite_geometry/Applications/Prob31SchoolGirl/problem31.html", "https://web.archive.org/web/20131029221809/http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf", "https://web.archive.org/web/20150330135512/http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0488.0499.ocr.pdf", "https://arxiv.org/abs/1310.6482"]}, "Topology": {"categories": ["All articles with unsourced statements", "Articles with Curlie links", "Articles with inconsistent citation formats", "Articles with short description", "Articles with unsourced statements from July 2009", "Mathematical structures", "Topology", "Use dmy dates from July 2012", "Wikipedia articles needing clarification from December 2017", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Topology", "method": "Topology", "url": "https://en.wikipedia.org/wiki/Topology", "summary": "In mathematics, topology (from the Greek \u03c4\u03cc\u03c0\u03bf\u03c2, place, and \u03bb\u03cc\u03b3\u03bf\u03c2, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. \nAn n-dimensional topological space is a space (not necessarily Euclidean) with certain properties of connectedness and compactness.The space may be continuous (like all points on a rubber sheet), or discrete (like the set of integers). It can be open (like the set of points inside a circle) or closed (like the set of points inside a circle, together with the points on the circle).\nTopology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for \"geometry of place\") and analysis situs (Greek-Latin for \"picking apart of place\").  Leonhard Euler's Seven Bridges of K\u00f6nigsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.  By the middle of the 20th century, topology had become a major branch of mathematics.\n\n", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/74/Alphabet_homeo.png", "https://upload.wikimedia.org/wikipedia/commons/7/7c/Alphabet_homotopy.png", "https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/2/2d/Klein_bottle_translucent.png", "https://upload.wikimedia.org/wikipedia/commons/b/b9/Kleinsche_Flasche.png", "https://upload.wikimedia.org/wikipedia/commons/d/d9/M%C3%B6bius_strip.jpg", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/4/46/Trefoil_knot_arb.png", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5d/Konigsberg_bridges.png", "https://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", "https://upload.wikimedia.org/wikipedia/en/d/db/Symbol_list_class.svg"], "links": ["Abstract algebra", "Algebra", "Algebraic geometry", "Algebraic topology", "Analysis Situs (paper)", "Analytic geometry", "Applied mathematics", "ArXiv", "Areas of mathematics", "Arithmetic", "Augustin-Louis Cauchy", "Bernhard Riemann", "Betti number", "Bibcode", "Braggadocio (typeface)", "CW complex", "Calabi-Yau manifold", "Calculus", "Cantor set", "Category theory", "Cesare Arzel\u00e0", "Characteristic classes", "Circle", "CiteSeerX", "Classification theorem", "Clifford A. Pickover", "Clopen set", "Closed set", "Cohomology", "Colin Adams (mathematician)", "Combinatorial topology", "Compact (topology)", "Compact space", "Complex geometry", "Complex plane", "Compressive strength", "Computational mathematics", "Computer science", "Condensed matter physics", "Configuration space (physics)", "Conformal geometry", "Connectedness", "Contact mechanics", "Continuity (topology)", "Continuous function", "Continuous function (topology)", "Continuous map", "Continuum (topology)", "Control theory", "Covering space", "Cowlick", "Cross-cap", "Crumpling", "Curlie", "Curvature", "Deformation theory", "Differentiable function", "Differentiable manifold", "Differential equation", "Differential geometry", "Differential topology", "Digital object identifier", "Dimension", "Discrete geometry", "Discrete mathematics", "Disentanglement puzzle", "Dynamical systems theory", "Edward Witten", "Electrophoresis", "Elementary algebra", "Encyclopedia of Mathematics", "English alphabet", "Enrico Betti", "Equivariant topology", "Euclidean space", "Euler's polyhedron formula", "Euler characteristic", "Eulerian path", "Evolutionary biology", "Exercise (mathematics)", "Family of sets", "Felix Hausdorff", "Fields Medal", "Finite geometry", "Font", "Foundations of mathematics", "Four-manifold", "Fourier series", "Fractal dimension", "Free group", "Function (mathematics)", "Functional analysis", "Fundamental group", "Game theory", "General topology", "Genotype", "Genus (topology)", "Geometric topology", "Geometrization conjecture", "Geometry", "Georg Cantor", "Giulio Ascoli", "Gottfried Leibniz", "Graph theory", "Greek language", "Grothendieck topology", "Hairy ball theorem", "Handle decomposition", "Hassler Whitney", "Hausdorff space", "Henri Poincar\u00e9", "History of mathematics", "Homeomorphic", "Homeomorphism", "Homology (mathematics)", "Homotopy", "Homotopy equivalence", "Homotopy group", "Information theory", "Injective function", "Integer", "Integrated Authority File", "International Standard Book Number", "Invariant (mathematics)", "JSTOR", "Jacques Hadamard", "Johann Benedict Listing", "John L. Kelley", "Joint", "Jordan-Sch\u00f6nflies theorem", "Kaliningrad", "Kazimierz Kuratowski", "Klein bottle", "Knot theory", "Lattice (order)", "Lawvere\u2013Tierney topology", "Lemniscate", "Leonhard Euler", "Limit of a sequence", "Line (geometry)", "Linear algebra", "List of algebraic topology topics", "List of examples in general topology", "List of general topology topics", "List of geometric topology topics", "List of important publications in mathematics", "List of publications in mathematics", "List of topology topics", "Lists of mathematics topics", "Local flatness", "Low-dimensional topology", "Ludwig Schl\u00e4fli", "Manifold", "Materials science", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Maurice Fr\u00e9chet", "Maxim Kontsevich", "Mechanical engineering", "Metric (mathematics)", "Metric space", "Michael Artin", "Michiel Hazewinkel", "Moduli spaces", "Molecules", "Motion planning", "Multilinear algebra", "Myriad (font)", "M\u00f6bius strip", "National Diet Library", "Nature (journal)", "Neighborhood (mathematics)", "Neighbourhood (mathematics)", "Nicolas Bourbaki", "Number theory", "Numerical analysis", "OCLC", "Open set", "Orbit space", "Order theory", "Orientability", "Orientable manifold", "Outline of mathematics", "Persistent homology", "Peter Johnstone (mathematician)", "Phenotype", "Philosophy of mathematics", "Physical cosmology", "Plane (geometry)", "Point-set topology", "Point (geometry)", "Pointless topology", "Polyhedron", "Probability theory", "Projective plane", "PubMed Central", "PubMed Identifier", "Pure mathematics", "Quantum field theory", "Quotient space (topology)", "Real line", "Real number", "Real projective plane", "Recreational mathematics", "Riemannian curvature", "Robot", "Ronald Brown (mathematician)", "Ryszard Engelking", "Sans-serif", "Set-theoretic topology", "Set theory", "Seven Bridges of K\u00f6nigsberg", "Sheaf (mathematics)", "Simon Donaldson", "Simplicial complex", "Smooth structure", "Space (mathematics)", "Spacetime topology", "Sphere", "Springer Science+Business Media", "Standard Topology", "Statistics", "Stencil", "Stretching", "String theory", "Surface (topology)", "Surgery theory", "Surjective function", "Tearing", "The Geometry Center", "The Spectator", "Theory of computation", "Topography", "Topoisomer", "Topological data analysis", "Topological geometry", "Topological invariant", "Topological order", "Topological quantum field theory", "Topological space", "Topology (disambiguation)", "Topology glossary", "Torus", "Trefoil knot", "Trigonometry", "Trivial knot", "Typography", "Uniform space", "Uniformization theorem", "Up to", "Van Kampen's theorem", "Vaughan Jones", "Vector field", "Vito Volterra", "Wac\u0142aw Sierpi\u0144ski", "Zentralblatt MATH"], "references": ["http://at.yorku.ca/i/a/a/b/23.htm", "http://at.yorku.ca/topology/", "http://sma.epfl.ch/~ojangure/Notes.pdf", "http://dictionary.reference.com/browse/topology", "http://meta.math.stackexchange.com/questions/2840/what-is-geometric-topology", "http://www.math.cornell.edu/~hatcher/AT/ATpage.html", "http://adsabs.harvard.edu/abs/2011PNAS..10814741C", "http://adsabs.harvard.edu/abs/2017NatSR...741621S", "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.63.7808", "http://www.geom.uiuc.edu/zoo/", "http://www.math.wayne.edu/~rrb/topology.html", "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3169141", "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5290745", "http://www.ncbi.nlm.nih.gov/pubmed/11894994", "http://www.ncbi.nlm.nih.gov/pubmed/21873249", "http://www.ncbi.nlm.nih.gov/pubmed/28155863", "http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf", "http://www.ams.org/online_bks/hmath1/hmath1-whitney10.pdf", "http://arxiv.org/abs/1203.5826", "http://doi.org/10.1006%2Fjtbi.2001.2423", "http://doi.org/10.1016%2Fj.compag.2011.11.009", "http://doi.org/10.1038%2Fsrep41621", "http://doi.org/10.1073%2Fpnas.1019192108", "http://doi.org/10.1080%2F1747423X.2011.637136", "http://doi.org/10.1090%2FS0273-0979-09-01249-X", "http://doi.org/10.1090%2Fs0273-0979-1983-15080-2", "http://doi.org/10.2307%2F27642974", "http://www.jstor.org/stable/27642974", "http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014", "http://www.worldcat.org/oclc/8897542", "http://zbmath.org/?format=complete&q=an:0208.48701", "http://www.pdmi.ras.ru/~olegviro/topoman/index.html", "http://pages.bangor.ac.uk/~mas010/topgpds.html", "https://www.futilitycloset.com/2012/06/23/the-figure-8-puzzle/", "https://books.google.com/books?id=SHBj2oaSALoC&pg=PA204", "https://math.stackexchange.com/questions/644676/how-to-reason-about-disentanglement-tavern-puzzles", "https://d-nb.info/gnd/4060425-1", "https://id.ndl.go.jp/auth/ndlna/00573284", "https://web.archive.org/web/20090713073050/http://www.ornl.gov/sci/ortep/topology/defs.txt", "https://curlie.org/Science/Math/Topology", "https://www.encyclopediaofmath.org/index.php?title=p/t093200", "https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html", "https://www.wikidata.org/wiki/Q42989"]}, "Trigonometry": {"categories": ["All articles needing additional references", "All articles with unsourced statements", "Articles needing additional references from March 2016", "Articles with unsourced statements from November 2011", "Trigonometry", "Wikipedia articles with NDL identifiers", "Wikipedia indefinitely move-protected pages"], "title": "Trigonometry", "method": "Trigonometry", "url": "https://en.wikipedia.org/wiki/Trigonometry", "summary": "Trigonometry (from Greek trig\u014dnon, \"triangle\" and metron, \"measure\") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic  world during the 3rd century BC from applications of geometry to astronomical studies.The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying.\nTrigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry (a fundamental part of astronomy and navigation). Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.\nTrigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5d/Csc_drawing_process.gif", "https://upload.wikimedia.org/wikipedia/commons/2/2d/Frieberger_drum_marine_sextant.jpg", "https://upload.wikimedia.org/wikipedia/commons/5/50/Hipparchos_1.jpeg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/b/b5/Sin-cos-defn-I.png", "https://upload.wikimedia.org/wikipedia/commons/0/08/Sine_curve_drawing_animation.gif", "https://upload.wikimedia.org/wikipedia/commons/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg", "https://upload.wikimedia.org/wikipedia/commons/e/ee/Tan_drawing_process.gif", "https://upload.wikimedia.org/wikipedia/commons/2/24/Triangle_ABC_with_Sides_a_b_c_2.png", "https://upload.wikimedia.org/wikipedia/commons/4/4f/TrigonometryTriangle.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/2/24/Wikinews-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/1/1b/Wikiversity-logo-en.svg", "https://upload.wikimedia.org/wikipedia/commons/1/1b/Wikiversity-logo-en.svg", "https://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", "https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg", "https://upload.wikimedia.org/wikipedia/en/0/06/Wiktionary-logo-v2.svg"], "links": ["Abstract algebra", "Acoustics", "Adjacent side (right triangle)", "Algebra", "Algebraic geometry", "Almagest", "Analytic geometry", "Ancient Greek", "Angle", "Applied mathematics", "Archimedes", "Architecture", "Areas of mathematics", "Arithmetic", "Aryabhata", "Aryabhata's sine table", "Astronomers", "Astronomy", "Astronomy in medieval Islam", "Audio synthesis", "Babylonians", "Bartholomaeus Pitiscus", "Basilios Bessarion", "Biology", "Brook Taylor", "Byzantine", "Byzantine scholars in Renaissance", "CT scan", "Calculus", "Carl Benjamin Boyer", "Cartography", "Category theory", "Chemistry", "Chinese mathematics", "Chord (geometry)", "Circumcircle", "Circumscribed circle", "Civil engineering", "Clark University", "Colin Maclaurin", "Complementary angles", "Complex number", "Computational mathematics", "Computer graphics", "Computer science", "Control theory", "Cosine", "Cryptology", "Crystallography", "Curvature", "De Moivre's formula", "De revolutionibus orbium coelestium", "Degree (angle)", "Differential equation", "Differential geometry", "Differentiation of trigonometric functions", "Discrete geometry", "Discrete mathematics", "Dynamical systems theory", "E (mathematics)", "Economics", "Electrical engineering", "Electronics", "Elementary algebra", "Elliptic geometry", "Encyclopedia of Mathematics", "Eric W. 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Curlie links", "Articles with specifically marked weasel-worded phrases from July 2012", "Articles with unsourced statements from April 2011", "Articles with unsourced statements from February 2007", "Articles with unsourced statements from May 2010", "Philosophy of mathematics", "Webarchive template wayback links", "Wikipedia articles needing clarification from August 2010", "Wikipedia further reading cleanup", "Wikipedia spam cleanup from August 2010"], "title": "Philosophy of mathematics", "method": "Philosophy of mathematics", "url": "https://en.wikipedia.org/wiki/Philosophy_of_mathematics", "summary": "The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/79/Hilbert.jpg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", "https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg"], "links": ["A Mathematician's Apology", "A priori and a posteriori", "Abductive reasoning", "Absolute idealism", "Abstract algebra", "Abstract object", "Abstract object theory", "Achintya Bheda Abheda", "Action theory (philosophy)", "Advaita Vedanta", "Aesthetic emotions", "Aesthetics", "African philosophy", "Agriculturalism", "Aj\u00f1ana", "Alan Turing", "Alexandre George", "Alfred North Whitehead", "Alfred Tarski", "Algebra", "Algebraic geometry", "Algorithm", "Allegory of the cave", "American philosophy", "Analytic-synthetic distinction", "Analytic geometry", "Analytic philosophy", "Analytic proposition", "Analytical Marxism", "Analytical feminism", "Analytic\u2013synthetic distinction", "Anarchism", "Ancient Egyptian philosophy", "Ancient Greece", "Ancient Greek philosophy", "Ancient philosophy", "Anekantavada", "Anti-psychologism", "Anti-realism", "Anti-realist", "Antihumanism", "Antinomy", "Applied ethics", "Applied mathematics", "ArXiv", "Areas of mathematics", "Arend Heyting", "Argumentation theory", "Aristotelian logic", "Aristotelianism", "Aristotle", "Aristotle's theory of universals", "Arithmetic", "Arthur W. 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It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.  The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.\nMathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.  These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.\nSince its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt G\u00f6del, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/7/7c/Logic_portal.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["(\u03b5, \u03b4)-definition of limit", "A. M. Turing", "A. S. 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L. Austin", "JSTOR", "Jahrbuch \u00fcber die Fortschritte der Mathematik", "Jan Lukasiewicz", "Jean Van Heijenoort", "Jean van Heijenoort", "Johann Heinrich Lambert", "John von Neumann", "Jon Barwise", "Joseph Henry Woodger", "Joseph R. 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Hugh Woodin", "Well-formed formula", "Well-ordered", "Wilfrid Hodges", "Wolfgang Rautenberg", "Word problem for groups", "Word processor", "World Scientific Publishing", "World Wide Web", "Yuri Matiyasevich", "Zentralblatt MATH", "Zermelo set theory", "Zermelo\u2013Fraenkel set theory"], "references": ["http://euclid.trentu.ca/math/sb/pcml/", "http://www.fecundity.com/logic/", "http://www.springerlink.com/content/978-1-4419-1220-6/", "http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002273500&L=1", "http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266121&L=1", "http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002260018&L=1", "http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002261952&L=1", "http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1", "http://www.mathematik.uni-muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf", "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.5803", "http://plato.stanford.edu/entries/logic-classical/", "http://plato.stanford.edu/entries/modeltheory-fo/", "http://www.people.cs.uchicago.edu/~soare/Turing/frontice.pdf", "http://www.ltn.lv/~podnieks/mlog/ml.htm", "http://www.ams.org/mathscinet-getitem?mr=0237246", "http://www.ams.org/notices/200106/fea-woodin.pdf", "http://arxiv.org/abs/math/0509616", "http://doi.org/10.1007%2F978-1-4419-1221-3", "http://doi.org/10.1007%2FBF01445300", "http://doi.org/10.1007%2FBF01449999", "http://doi.org/10.1007%2FBF01450054", "http://doi.org/10.1007%2FBF01458217", "http://doi.org/10.1007%2FBF01565428", "http://doi.org/10.1007%2FBF01696781", "http://doi.org/10.1007%2FBF01700692", "http://doi.org/10.1007%2FBF01782335", "http://doi.org/10.1007%2FBF02757006", "http://doi.org/10.1090%2Fs0002-9947-07-04297-3", "http://doi.org/10.1111%2Fj.1746-8361.1958.tb01464.x", "http://doi.org/10.1112%2Fplms%2Fs2-45.1.161", "http://doi.org/10.2307%2F1990131", "http://doi.org/10.2307%2F1994188", "http://doi.org/10.2307%2F2318447", "http://doi.org/10.2307%2F2687794", "http://doi.org/10.2307%2F2695743", "http://doi.org/10.2307%2F420992", "http://www.jstor.org/stable/1990131", "http://www.jstor.org/stable/1994188", "http://www.jstor.org/stable/2318447", "http://www.jstor.org/stable/2687794", "http://www.jstor.org/stable/2695743", "http://www.jstor.org/stable/420992", "http://pl.wikipedia.org/wiki/Jan_Salamucha", "http://pl.wikipedia.org/wiki/Stanislaw_Schayer", "http://worldcat.org/oclc/523472", "http://www.worldcat.org/issn/0025-5831", "http://zbmath.org/?format=complete&q=an:0251.02001", "http://zbmath.org/?format=complete&q=an:60.0017.02", "http://www.quantrelog.se/pvlmatrix/index_main.htm", "http://www.ucl.ac.uk/philosophy/LPSG/", "http://www.ucl.ac.uk/philosophy/LPSG/MathLogic.htm", "http://www.ucl.ac.uk/philosophy/LPSG/PhilMath.htm", "http://www.ucl.ac.uk/philosophy/LPSG/SetTheory.htm", "http://www.collegepublications.co.uk/logic/mlf/?00029", "https://books.google.com/books?id=HZAjPwAACAAJ&source=gbs_ViewAPI", "https://link.springer.com/book/10.1007%2F978-1-4471-4558-5", "https://www.springer.com/mathematics/book/978-0-387-94258-2", "https://idus.us.es/xmlui/bitstream/11441/38373/1/The%20road%20to%20modern%20logic.pdf", "https://d-nb.info/gnd/4037951-6", "https://id.ndl.go.jp/auth/ndlna/00565709", "https://www.encyclopediaofmath.org/index.php?title=p/m062660", "https://www.wikidata.org/wiki/Q1166618"]}, "Set theory": {"categories": ["Formal methods", "Georg Cantor", "Mathematical logic", "Set theory", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Set theory", "method": "Set theory", "url": "https://en.wikipedia.org/wiki/Set_theory", "summary": "Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.\nThe modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo\u2013Fraenkel axioms, with or without the axiom of choice, are the best-known.\nSet theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo\u2013Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/7/73/Georg_Cantor_1894.jpg", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/commons/9/99/Venn0001.svg", "https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg", "https://upload.wikimedia.org/wikipedia/commons/5/50/Von_Neumann_Hierarchy.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", 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A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.\nSeveral terms used in category theory, including the term \"morphism\", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.\nSamuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations in 1942\u201345 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.\nCategory theory has practical applications in programming language theory, for example the usage of monads in functional programming.   It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/e/ef/Commutative_diagram_for_morphism.svg", "https://upload.wikimedia.org/wikipedia/commons/f/f2/Natural_transformation.svg", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", 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physics": {"categories": ["CS1 maint: Archived copy as title", "Mathematical physics", "Wikipedia articles needing clarification from January 2018", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Mathematical physics", "method": "Mathematical physics", "url": "https://en.wikipedia.org/wiki/Mathematical_physics", "summary": "Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as \"the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories\". It is a branch of applied mathematics, but deals with physical problems.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/d/d8/Mathematical_Physics_and_other_sciences.png", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/e/e0/StationaryStatesAnimation.gif", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abdus Salam", "Absolute space", "Absolute space and time", "Absolute time", "Abstract algebra", "Ab\u016b Ray\u1e25\u0101n al-B\u012br\u016bn\u012b", "Accelerator physics", "Acoustics", "Action at a distance", "Aerodynamics", "Aether (classical element", "Aether drift", "Agrophysics", "Albert Einstein", "Alexander Markovich Polyakov", "Algebra", "Algebraic geometry", "Algorithm", "Algorithm design", "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism", "Analysis of algorithms", "Analytic geometry", "Analytical mechanics", "Andrey Kolmogorov", "Applied mathematics", "Applied physics", "Approximation theory", "ArXiv", "Archimedes", "Areas of mathematics", "Aristotelian physics", "Arithmetic", "Arnold Sommerfeld", "Arthur Jaffe", "Arthur Strong Wightman", "Ashoke Sen", "Astronomy", "Astroparticle physics", "Astrophysics", "Atmospheric physics", "Atomic, molecular, and optical physics", "Atomic physics", "Atomic spectra", "Ballistics", "Barry Simon", "Bernhard Riemann", "Bertha Swirles", "Bibcode", "Biophysics", "Branches of physics", "Calculus", "Calculus of variations", "Canonical transformations", "Carl Friedrich Gauss", "Carl Gustav Jacobi", "Cartesian coordinates", "Cartesian physics", "Category theory", "Celestial mechanics", "Chemical physics", "Chen-Ning Yang", "Christiaan Huygens", "Classical electromagnetism", "Classical elements", "Classical field theory", "Classical mechanics", "Classical physics", "Cloud physics", "Coding theory", "Combinatorics", "Combinatorics and physics", "Computational mathematics", "Computational number theory", "Computational physics", "Computer science", "Condensed matter physics", "Constantin Carath\u00e9odory", "Continuum mechanics", "Control theory", "Cosmology", "Cryptography", "Daniel Bernoulli", "David Hilbert", "David Ruelle", "Decision theory", "Differential equation", "Differential equations", "Differential geometry", "Discrete geometry", "Discrete mathematics", "Dynamical system", "Dynamical systems", "Dynamical systems theory", "E. 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Marsden", "Joel Lebowitz", "Johannes Kepler", "John Baez", "John Cardy", "John Herapath", "John von Neumann", "Joseph-Louis Lagrange", "Joseph D. Sneed", "Joseph Fourier", "Josiah Willard Gibbs", "Journal of Mathematical Physics", "Juan Mart\u00edn Maldacena", "Julian Schwinger", "J\u00fcrgen Moser", "Lagrangian mechanics", "Lars Onsager", "Law of inertia", "Law of universal gravitation", "Laws of planetary motion", "Leonard Susskind", "Leonhard Euler", "Linear algebra", "List of mathematical physics journals", "List of mathematics topics", "List of publications in physics", "Lists of mathematics topics", "Logic in computer science", "Lord Rayleigh", "Lorentz contraction", "Lorentz transformation", "Lorentzian manifold", "Ludvig Faddeev", "Ludwig Boltzmann", "Luminiferous aether", "M-theory", "Magnetic moment", "Magnetism", "Magnetostatics", "Malliavin calculus", "Mark Kac", "Martin Gutzwiller", "Mary L. 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Beyer", "Rodney Baxter", "Roger Penrose", "Roman Jackiw", "Rudolf Haag", "Sagnac effect", "Satyendra Nath Bose", "Schr\u00f6dinger", "Schr\u00f6dinger's equation", "Self-adjoint operator", "Set theory", "Sheldon Lee Glashow", "Sim\u00e9on Denis Poisson", "Sin-Itiro Tomonaga", "Social physics", "Soft matter", "Solar physics", "Solid-state physics", "Solid mechanics", "Sound", "Space physics", "Special relativity", "Special theory of relativity", "Spectral theory", "Statistical mechanics", "Statistics", "Stephen Hawking", "Steven Weinberg", "Stochastic analysis", "Stochastic process", "String theory", "Sublunary sphere", "Subrahmanyan Chandrasekhar", "Symbolic computation", "Symplectic geometry", "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", "The large scale structure of space-time", "Theoretical physics", "Theories", "Theory of computation", "Theory of everything", "Theory of relativity", "Thermodynamics", "Thomas Young (scientist)", "Thought experiment", "Timeline of fundamental physics discoveries", "Topic outline of mathematics", "Topology", "Tosio Kato", "Trigonometry", "Tycho Brahe", "Variational calculus", "Vector analysis", "Vector bundles", "Vibrating string", "Vladimir Arnold", "Walter Thirring", "Werner Heisenberg", "Whittaker and Watson", "William Rowan Hamilton", "William Thomson, 1st Baron Kelvin", "Wolfgang Pauli", "Yakov Sinai"], "references": ["http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/", "http://adsabs.harvard.edu/abs/2005physics...6153Z", "http://plato.stanford.edu/archives/spr2010/entries/galileo", "http://jmp.aip.org/jmp/staff.jsp", "http://arxiv.org/abs/physics/0506153", "https://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA129", "https://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA89&dq=mathematical+reasoning", "https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA213", "https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA220", "https://books.google.com/books?id=Z9K8llQufcMC&pg=PA125&dq=superfluous+learnedness+Einstein+Minkowski+general+relativity", "https://d-nb.info/gnd/4037952-8", "https://id.ndl.go.jp/auth/ndlna/00571738", "https://web.archive.org/web/20061003233339/http://jmp.aip.org/jmp/staff.jsp", "https://catalog.hathitrust.org/Record/011557061?type%5B%5D=author&lookfor%5B%5D=John%20Herapath&ft=ft", "https://www.wikidata.org/wiki/Q156495"]}, "Probability theory": {"categories": ["All articles lacking in-text citations", "All articles lacking reliable references", "All articles with unsourced statements", "Articles lacking in-text citations from September 2009", "Articles lacking reliable references from February 2016", "Articles with unsourced statements from December 2015", "Probability theory", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Probability theory", "method": "Probability theory", "url": "https://en.wikipedia.org/wiki/Probability_theory", "summary": "Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.\nCentral subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.\nAlthough it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.\nAs a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/4/40/Fisher_iris_versicolor_sepalwidth.svg", "https://upload.wikimedia.org/wikipedia/commons/d/d2/Gaussian_distribution_2.jpg", "https://upload.wikimedia.org/wikipedia/commons/2/24/NYW-DK-Poisson%285%29.svg", "https://upload.wikimedia.org/wikipedia/commons/7/77/Nuvola_apps_atlantik.png", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", 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(discrete)", "Variance", "Venn diagram", "YouTube"], "references": ["http://www.leithner.com.au/circulars/circular17.htm", "http://www.galtonboard.com", "http://www.probabilityandfinance.com/articles/04.pdf", "http://plato.stanford.edu/archives/sum2012/entries/probability-interpret/", "http://home.ubalt.edu/ntsbarsh/stat-data/Topics.htm", "http://doi.org/10.1007%2F978-3-642-49888-6", "https://books.google.com/books?id=Bc1FAQAAIAAJ&pg=PA26", "https://physics.stackexchange.com/questions/69718/why-is-quantum-mechanics-based-on-probability-theory", "https://www.youtube.com/watch?v=9eaOxgT5ys0", "https://d-nb.info/gnd/4079013-7", "https://id.ndl.go.jp/auth/ndlna/00564753", "https://web.archive.org/web/20140126113323/http://www.leithner.com.au/circulars/circular17.htm", "https://www.wikidata.org/wiki/Q5862903"]}, "Mathematical statistics": {"categories": ["Actuarial science", "All articles with unsourced statements", "Articles with unsourced statements from March 2018", "CS1 maint: Extra text: authors list", "Statistical theory"], "title": "Mathematical statistics", "method": "Mathematical statistics", "url": "https://en.wikipedia.org/wiki/Mathematical_statistics", "summary": "Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.", "images": ["https://upload.wikimedia.org/wikipedia/commons/3/3a/Linear_regression.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abraham Wald", "Abstract algebra", "Accelerated failure time model", "Actuarial science", "Akaike information criterion", "Algebra", "Algebraic geometry", "Algebraic statistics", "Analysis of covariance", "Analysis of variance", "Analytic geometry", "Anderson\u2013Darling test", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Arithmetic mean", "Asymptotic theory (statistics)", "Autocorrelation", "Autoregressive conditional heteroskedasticity", "Autoregressive\u2013moving-average model", "Average value", "Bar chart", "Bayes estimator", "Bayes factor", "Bayesian inference", "Bayesian information criterion", "Bayesian linear regression", "Bayesian probability", "Bernoulli distribution", "Beta distribution", "Bias of an estimator", "Binomial distribution", "Binomial regression", "Bioinformatics", "Biostatistics", "Biplot", "Blocking (statistics)", "Bootstrapping (statistics)", "Box plot", "Box\u2013Jenkins method", "Breusch\u2013Godfrey test", "Calculus", "Canonical correlation", "Cartography", "Categorical distribution", "Categorical variable", "Category theory", "Census", "Central limit theorem", "Central tendency", "Charles Sanders Peirce", "Chemometrics", "Chi-squared distribution", "Chi-squared test", "Chi squared distribution", "Clinical study design", "Clinical trial", "Cluster analysis", "Cluster sampling", "Coefficient of determination", "Coefficient of variation", "Cohen's kappa", "Cointegration", "Combinatorial design", "Completeness (statistics)", "Computational mathematics", "Computer science", "Conditional expectation", "Confidence interval", "Confounding", "Contingency table", "Continuous probability distribution", "Continuous time", "Continuous uniform distribution", "Control chart", "Control theory", "Correlation and dependence", "Correlogram", "Count data", "Credible interval", "Crime statistics", "Cross-correlation", "Cross-validation (statistics)", "Data", "Data analysis", "Data collection", "David A. 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Bickel", "Philosophy of mathematics", "Pie chart", "Pivotal quantity", "Plug-in principle", "Point estimation", "Poisson distribution", "Poisson regression", "Population (statistics)", "Population statistics", "Posterior probability", "Power (statistics)", "Prediction interval", "Preferences", "Principal component analysis", "Prior probability", "Probabilistic design", "Probability", "Probability density function", "Probability distribution", "Probability mass function", "Probability measure", "Probability theory", "Proportional hazards model", "Psychometrics", "Pure mathematics", "Quality control", "Quantile", "Quasi-experiment", "Questionnaire", "Q\u2013Q plot", "Radar chart", "Random assignment", "Random sampling", "Random variable", "Random variables", "Random vector", "Randomization test", "Randomized controlled trial", "Randomized experiment", "Range (statistics)", "Rank correlation", "Rank statistics", "Ranking", "Rao\u2013Blackwell theorem", "Recreational mathematics", "Regression analysis", "Regression model validation", "Reliability engineering", "Replication (statistics)", "Resampling (statistics)", "Robust regression", "Robust statistics", "Run chart", "Sample median", "Sample size determination", "Sample space", "Sample variance", "Sampling (statistics)", "Sampling distribution", "Scale parameter", "Scatter plot", "Scientific computing", "Scientific control", "Score test", "Seasonal adjustment", "Semiparametric regression", "Set theory", "Shape of the distribution", "Shape parameter", "Shapiro\u2013Wilk test", "Sign test", "Simple linear regression", "Simultaneous equations model", "Skewness", "Social statistics", "Spatial analysis", "Spearman's rank correlation coefficient", "Spectral density estimation", "Standard deviation", "Standard error", "Standard normal", "Stationary process", "Statistic", "Statistical classification", "Statistical dispersion", "Statistical distance", "Statistical graphics", "Statistical hypothesis testing", "Statistical inference", "Statistical model", "Statistical parameter", "Statistical power", "Statistical process control", "Statistical survey", "Statistical theory", "Statisticians", "Statistics", "Stem-and-leaf display", "Stochastic analysis", "Stochastic processes", "Stratified sampling", "Structural break", "Structural equation modeling", "Student's t-test", "Student's t distribution", "Sufficient statistic", "Survey methodology", "Survival analysis", "Survival function", "System identification", "Theory of computation", "Time domain", "Time series", "Tolerance interval", "Topology", "Trend estimation", "Trigonometry", "U-statistic", "Uniformly most powerful test", "Univariate distribution", "Utility function", "V-statistic", "Variance", "Vector autoregression", "Wald test", "Wavelet", "Whittle likelihood", "Wilcoxon signed-rank test", "Z-test"], "references": ["http://www.trigonella.ch/statibot/english/", "http://www.math.uah.edu/stat/"]}, "Statistics": {"categories": ["All articles with specifically marked weasel-worded phrases", "All articles with unsourced statements", "Articles with specifically marked weasel-worded phrases from April 2014", "Articles with unsourced statements from April 2015", "Articles with unsourced statements from February 2015", "Articles with unsourced statements from March 2013", "Articles with unsourced statements from September 2018", "CS1 maint: Extra text: authors list", "CS1 maint: Uses editors parameter", "Data", "Formal sciences", "Information", "Mathematical and quantitative methods (economics)", "Research methods", "Statistics", "Wikipedia articles needing clarification from October 2016", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NARA identifiers", "Wikipedia articles with NDL identifiers"], "title": "Statistics", "method": "Statistics", "url": "https://en.wikipedia.org/wiki/Statistics", "summary": "Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as \"all people living in a country\" or \"every atom composing a crystal\". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.\nSee glossary of probability and statistics.\nWhen census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.\nTwo main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.\nA standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a \"false positive\") and Type II errors (null hypothesis fails to be rejected and an actual difference between populations is missed giving a \"false negative\"). Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.\nStatistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/4/40/Fisher_iris_versicolor_sepalwidth.svg", "https://upload.wikimedia.org/wikipedia/commons/b/b9/Gretl_screenshot.png", "https://upload.wikimedia.org/wikipedia/commons/9/97/Jer%C3%B4me_Cardan.jpg", "https://upload.wikimedia.org/wikipedia/commons/1/18/Karl_Pearson%2C_1910.jpg", "https://upload.wikimedia.org/wikipedia/commons/7/75/Linear_least_squares%282%29.svg", "https://upload.wikimedia.org/wikipedia/commons/8/8f/NYW-confidence-interval.svg", "https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg", "https://upload.wikimedia.org/wikipedia/commons/3/3a/P-value_in_statistical_significance_testing.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/d/d8/Scatterplot.jpg", "https://upload.wikimedia.org/wikipedia/commons/b/b8/Simple_Confounding_Case.svg", "https://upload.wikimedia.org/wikipedia/commons/2/25/The_Normal_Distribution.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/2/24/Wikinews-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/1/1b/Wikiversity-logo-en.svg", "https://upload.wikimedia.org/wikipedia/commons/1/1b/Wikiversity-logo-en.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg", "https://upload.wikimedia.org/wikipedia/en/0/06/Wiktionary-logo-v2.svg"], "links": ["A.W.F. 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"Probability theory", "Procedure (term)", "Process art", "Proportional hazards model", "Prosecutor's fallacy", "Protocol (natural sciences)", "Psychological statistics", "Psychometrics", "PubMed Central", "PubMed Identifier", "Pure mathematics", "Quality control", "Quasi-experiment", "Questionnaire", "Q\u2013Q plot", "R (programming language)", "Radar chart", "Random assignment", "Random error", "Random sample", "Random variable", "Random vector", "Randomization test", "Randomized assignment", "Randomized controlled trial", "Randomized experiment", "Range (statistics)", "Rank correlation", "Rank statistics", "Rao\u2013Blackwell theorem", "Real data type", "Recreational mathematics", "Regression analysis", "Regression model validation", "Reliability engineering", "Replication (statistics)", "Resampling (statistics)", "Residual sum of squares", "Robust regression", "Robust statistics", "Ronald Fisher", "Root mean square error", "Run chart", "SAS (software)", "SPSS", "Sabermetrics", "Sample (statistics)", "Sample covariance", "Sample mean", "Sample median", "Sample size determination", "Sample statistic", "Sample variance", "Sampling (statistics)", "Sampling distribution", "Scale parameter", "Scatter plot", "Scientific control", "Score test", "Seasonal adjustment", "Semiparametric regression", "Set theory", "Sex ratio", "Sexual selection", "Shape of the distribution", "Shape parameter", "Shapiro\u2013Wilk test", "Sign test", "Significance level", "Simple linear regression", "Simultaneous equations model", "Skewness", "Social research", "Social science", "Social statistics", "Sociology", "Spatial analysis", "Spatial data analysis", "Spearman's rank correlation coefficient", "Specialized terminology", "Spectral density estimation", "Standard deviation", "Standard error", "Standard error (statistics)", "Standardized testing (statistics)", "Stanford Encyclopedia of Philosophy", "Stanley Smith Stevens", "Stationary process", "Statistic", "Statistical Methods for 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"http://onlinestatbook.com/index.html", "http://www.rossmanchance.com/iscam/preface.pdf", "http://www.statsoft.com/textbook/", "http://www.yourstatsguru.com/epar/rp-reviewed/cohen1994/", "http://adsabs.harvard.edu/abs/1877Natur..15..492.", "http://psychstat3.missouristate.edu/Documents/IntroBook3/sbk.htm", "http://www.santafe.edu/news/item/sfnm-wood-big-data/", "http://plato.stanford.edu/entries/statistics/", "http://www.stat.ufl.edu/~aa/articles/agresti_hitchcock_2005.pdf", "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1182327", "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1936999", "http://www.ncbi.nlm.nih.gov/pubmed/16060722", "http://www.ncbi.nlm.nih.gov/pubmed/17608932", "http://www.ncbi.nlm.nih.gov/pubmed/18811377", "http://www.economicsdiscussion.net/essays/essay-on-statistics-meaning-and-definition-of-statistics/2315", "http://doi.org/10.1007%2Fs10260-005-0121-y", "http://doi.org/10.1038%2F015492a0", "http://doi.org/10.1080%2F01621459.1938.10502344", 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2014", "Articles with Russian-language external links", "Articles with dead external links from September 2016", "Articles with permanently dead external links", "Articles with short description", "Articles with specifically marked weasel-worded phrases from July 2012", "Articles with specifically marked weasel-worded phrases from March 2011", "Articles with unsourced statements from April 2015", "Articles with unsourced statements from March 2011", "Articles with unsourced statements from March 2016", "Articles with unsourced statements from October 2017", "Artificial intelligence", "Commons category link from Wikidata", "Formal sciences", "Game theory", "John von Neumann", "Mathematical economics", "Use American English from July 2018", "Use dmy dates from July 2017", "Webarchive template wayback links", "Wikipedia articles with BNE identifiers", "Wikipedia articles with BNF identifiers", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NDL identifiers", "Wikipedia articles with SUDOC identifiers"], "title": "Game theory", "method": "Game theory", "url": "https://en.wikipedia.org/wiki/Game_theory", "summary": "Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.\nModern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.\nGame theory was developed extensively in the 1950s by many scholars. It was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of  2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/9/96/Centipede_game.svg", "https://upload.wikimedia.org/wikipedia/commons/f/f3/Emblem-money.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/John_f_nash_20061102_3.jpg", "https://upload.wikimedia.org/wikipedia/commons/2/23/PD_with_outside_option.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/b/be/Ultimatum_Game_Extensive_Form.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg", "https://upload.wikimedia.org/wikipedia/commons/9/99/Wiktionary-logo-en-v2.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5e/JohnvonNeumann-LosAlamos.gif", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["AI planning", "A Beautiful Mind (book)", "A Beautiful Mind (film)", "Aboriginal title", "Abstract algebra", "Accumulation by dispossession", "Acequia", "Adam Smith", "Addison-Wesley", "Agent-based computational economics", "Agent (economics)", "Aggregation problem", "Agricultural economics", "Air rights", "Albert W. 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It was originally proposed by Claude E. Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper entitled \"A Mathematical Theory of Communication\". Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPEGs), and channel coding (e.g. for digital subscriber line (DSL)). Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.\nA key measure in information theory is \"entropy\". Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy.\nThe field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, information engineering, and electrical engineering. The theory has also found applications in other areas, including statistical inference, natural language processing, cryptography, neurobiology, human vision, the evolution and function of molecular codes (bioinformatics), model selection in statistics, thermal physics, quantum computing, linguistics, plagiarism detection, pattern recognition, and anomaly detection. 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"http://www.cs.bu.edu/AboutCS/WhatIsCS.pdf", "http://www.cse.buffalo.edu/~rapaport/computation.html", "http://www.cis.cornell.edu/Dean/Presentations/Slides/bgu.pdf", "http://www.cs.cornell.edu/cis-dean/bgu.pdf", "http://www.cs.mtu.edu/~john/jenning.pdf", "http://www.nap.edu/catalog.php?record_id=11106#toc", "http://wordnetweb.princeton.edu/perl/webwn?s=computer%20scientist", "http://citeseerx.ist.psu.edu/", "http://www.cs.purdue.edu/about/conte.html", "http://www.scottlan.edu/Lriddle/women/ada-love.htm", "http://library.stanford.edu/guides/stanford-university-oral-history", "http://www.stanford.edu/dept/ICME/docs/history/forsythe_knuth.pdf", "http://galton.uchicago.edu/~thisted/Distribute/comparch.pdf", "http://www.soc.ucsb.edu/faculty/mariacharles/documents/WhatGenderisScience.pdf", "http://www.cbi.umn.edu/about/babbage.html", "http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX525961", "http://www.eqanie.eu/media/Como%20Conference/Tanca-Research_Assessment_A_new_Initiative_by_Informatics_Europe.pdf", "http://epublications.uef.fi/pub/urn_isbn_952-458-867-6/urn_isbn_952-458-867-6.pdf", "http://data.bnf.fr/ark:/12148/cb11932109b", "http://cbseacademic.nic.in/web_material/Curriculum19/Main-SeniorSecondary/15_Information_Practices_New.pdf", "http://robroy.dyndns.info/collier/index.html", "http://www.idi.ntnu.no/emner/dif8916/denning.pdf", "http://www.aaai.org/home.html", "http://cacm.acm.org/magazines/2009/8/34492-viewpoint-time-for-computer-science-to-grow-up/fulltext", "http://cacm.acm.org/magazines/2010/2/69363-why-an-informatics-degree", "http://portal.acm.org/citation.cfm?id=1053309&coll=&dl=ACM&CFID=15151515&CFTOKEN=6184618", "http://portal.acm.org/citation.cfm?id=971303&dl=ACM&coll=&CFID=15151515&CFTOKEN=6184618", "http://portal.acm.org/ralston.cfm", "http://runningonempty.acm.org/fullreport2.pdf", "http://www.acm.org/", "http://www.claymath.org/millennium/P_vs_NP/", "http://www.computer.org/", "http://archive.computerhistory.org/resources/text/IBM/IBM.709.1957.102646304.pdf", "http://www.computerhistory.org/babbage/adalovelace/", "http://www.computerhistory.org/revolution/computer-graphics-music-and-art/15/222/633", "http://www.computerhistory.org/timeline/computers/", "http://cra.org/resources/bp-view/evaluating_computer_scientists_and_engineers_for_promotion_and_tenure/", "http://www.csab.org/", "http://www.csab.org/comp_sci_profession.html", "http://doi.org/10.1007%2Fs11023-007-9060-8", "http://doi.org/10.1007%2Fs11023-011-9240-4", "http://doi.org/10.1023%2FA:1018949113292", "http://doi.org/10.1145%2F1498765.1498780", "http://doi.org/10.1145%2F1536616.1536631", "http://doi.org/10.1145%2F365719.366510", "http://doi.org/10.1145%2F368424.368427", "http://doi.org/10.1145%2F368796.368802", "http://doi.org/10.1145%2F63238.63239", "http://doi.org/10.2139%2Fssrn.1686004", "http://www.eden-study.org/articles/2007/three_paradigms_of_computer_science.pdf", "http://www.edweek.org/ew/articles/2014/02/26/22computer_ep.h33.html", "http://www.informatics-europe.org/", "http://blogs.royalsociety.org/history-of-science/2012/06/25/history-of-computing/", "http://tools.wmflabs.org/ftl/cgi-bin/ftl?st=wp&su=Computer+science", "http://tools.wmflabs.org/ftl/cgi-bin/ftl?st=wp&su=Computer+science&library=0CHOOSE0", "http://www.cl.cam.ac.uk/conference/EDSAC99/statistics.html", "http://www.computingatschool.org.uk/data/uploads/internationalcomparisons-v5.pdf", "http://www.sciencemuseum.org.uk/on-line/babbage/index.asp", "https://www.economist.com/news/international/21601250-global-push-more-computer-science-classrooms-starting-bear-fruit", "https://books.google.com/books?id=JTYPKxug49IC&printsec=frontcover", "https://www.nytimes.com/2014/03/24/world/europe/adding-coding-to-the-curriculum.html", "https://www.wired.com/2013/01/kickstarted-scifi-movie/", 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"https://curlie.org/Computers/Computer_Science/", "https://developer.mozilla.org/en-US/docs/multiparadigmlanguage.html", "https://www.wikidata.org/wiki/Q21198", "https://www.telegraph.co.uk/money/main.jhtml?xml=/money/2006/08/27/ccsoft27.xml"]}, "Theory of computation": {"categories": ["Theory of computation"], "title": "Theory of computation", "method": "Theory of computation", "url": "https://en.wikipedia.org/wiki/Theory_of_computation", "summary": "In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm.  The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: \"What are the fundamental capabilities and limitations of computers?\".In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation.  There are several models in use, but the most commonly examined is the Turing machine. Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible \"reasonable\" model of computation (see Church\u2013Turing thesis). It might seem that the potentially infinite memory capacity is an unrealizable attribute, but any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a finite amount of memory.", "images": ["https://upload.wikimedia.org/wikipedia/commons/9/9a/Chomsky-hierarchy.svg", "https://upload.wikimedia.org/wikipedia/commons/6/6e/Complexity_subsets_pspace.svg", "https://upload.wikimedia.org/wikipedia/commons/3/3d/Maquina.png", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Alan Turing", "Alan Turing: The Enigma", "Algebra", "Algebraic geometry", "Algorithm", "Alonzo Church", "Alphabet", "Analytic geometry", "Andrew Hodges", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Asymptotic analysis", "Automata theory", "Beta reduction", "Big O notation", "Calculus", "Carl Herbert Smith", "Category theory", "Chomsky hierarchy", "Church\u2013Turing thesis", "Claude Shannon", "Clay Mathematics Institute", "Combinatory logic", "Computability theory", "Computational complexity theory", "Computational mathematics", "Computational theory of mind", "Computer science", "Context-free grammar", "Context-sensitive grammar", "Control theory", "Decidability (logic)", "Differential equation", "Differential geometry", "Digital object identifier", "Discrete geometry", "Discrete mathematics", "Dynamical systems theory", "Eitan Gurari", "Elementary algebra", "Finite geometry", "Finite state automaton", "Finite state machines", "Formal language", "Foundations of mathematics", "Function composition (computer science)", "Functional analysis", "Game theory", "Geometry", "Grammar", "Graph theory", "G\u00f6del numbering", "Halting problem", "Hartley Rogers, Jr", "Harvard", "History of mathematics", "Information theory", "International Standard Book Number", "Introduction to Automata Theory, Languages, and Computation", "JSTOR", "Jeffrey D. Ullman", "John Hopcroft", "John von Neumann", "Kurt G\u00f6del", "Lambda calculus", "Linear algebra", "Linear bounded automaton", "Linz P", "Lists of mathematics topics", "London Mathematical Society", "Markov algorithm", "Martin Davis", "Massachusetts Institute of Technology", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Michael O. Rabin", "Michael Sipser", "Millennium Prize Problems", "Model of computation", "Multilinear algebra", "NP (complexity)", "Number theory", "Numerical analysis", "Order theory", "Outline of mathematics", "P = NP problem", "P versus NP problem", "Philosophy of mathematics", "Primitive recursion", "Primitive recursive function", "Princeton University Press", "Probability theory", "Program semantics", "Programming language", "Pure mathematics", "Pushdown automaton", "Quantification theory", "Recreational mathematics", "Recursion theory", "Recursively enumerable language", "Register machine", "Regular expressions", "Regular grammar", "Rice's theorem", "Richard L. Epstein", "R\u00f3zsa P\u00e9ter", "S. Barry Cooper", "Set theory", "Statistics", "Stephen Cook", "Stephen Kleene", "String (computer science)", "String rewriting system", "Theoretical computer science", "Theory of Computing", "Topology", "Trigonometry", "Turing Award", "Turing machine", "Walter A. Carnielli", "\u039c-recursive function"], "references": ["http://toc.seas.harvard.edu/", "http://toc.csail.mit.edu/", "http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html", "http://cse.ucdenver.edu/~cscialtman/foundation/Essentials%20of%20Theoretical%20Computer%20Science.pdf", "http://www.cis.upenn.edu/~giorgi/cl.html", "http://videolectures.net/turing100_rabin_turing_church_goedel/", "http://www.claymath.org/sites/default/files/pvsnp.pdf", "http://doi.org/10.1109%2FTIT.1956.1056813", "http://doi.org/10.1112%2Fplms%2Fs2-42.1.230", "http://doi.org/10.2307%2F1990888", "http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1056813&tag=1", "http://www.jstor.org/stable/1990888", "http://www.turingarchive.org/browse.php/B/12", "https://web.archive.org/web/20070107040625/http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html"]}, "Control theory": {"categories": ["All articles with dead external links", "Articles with dead external links from January 2016", "Commons category link is on Wikidata", "Computational mathematics", "Control engineering", "Control theory", "Cybernetics", "Use mdy dates from July 2016", "Wikipedia articles with GND identifiers", "Wikipedia articles with NDL identifiers"], "title": "Control theory", "method": "Control theory", "url": "https://en.wikipedia.org/wiki/Control_theory", "summary": "Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems  in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability.\nTo do this, a controller with the requisite corrective behaviour is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are  controllability and observability. On this is based the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is feedback control, which is usually continuous and involves taking measurements using a sensor and making calculated adjustments to keep the measured variable within a set range by means of a \"final control element\", such as a control valve.Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system.\nControl theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell.  Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky.\nAlthough a major application of control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this.  As the general theory of feedback systems, control theory is useful wherever feedback occurs.", "images": ["https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/c/c7/Boulton_and_Watt_centrifugal_governor-MJ.jpg", "https://upload.wikimedia.org/wikipedia/commons/0/00/Complex-adaptive-system.jpg", "https://upload.wikimedia.org/wikipedia/commons/2/24/Feedback_loop_with_descriptions.svg", "https://upload.wikimedia.org/wikipedia/commons/6/6a/Industrial_control_loop.jpg", "https://upload.wikimedia.org/wikipedia/commons/4/43/PID_en.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/9/90/Simple_feedback_control_loop2.svg", "https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Active optics", "Actuator", "Adaptive control", "Adolf Hurwitz", "Aerospace industry", "Aleksandr Lyapunov", "Alexander Bogdanov", "Alexander Lerner", "Alexander Lyapunov", "Alfred Radcliffe-Brown", "Algebra", "Algebraic equation", "Algebraic geometry", "Ali H. Nayfeh", "Allenna Leonard", "Analytic geometry", "Anatol Rapoport", "Andrey Kolmogorov", "Anthony Stafford Beer", "Anthony Wilden", "Anti-wind up system (control)", "Applied mathematics", "ArXiv", "Areas of mathematics", "Arithmetic", "Artificial intelligence", "Artificial neural networks", "Asymptotic stability", "Audio system", "Automation", "Automation and Remote Control", "Automation and remote control", "Autopilot", "BIBO stability", "Backstepping", "Bandwidth (signal processing)", "Bang-bang control", "Bang\u2013bang control", "Barbara J. Grosz", "Bayesian probability", "Bibcode", "Biocybernetics", "Biological system", "Biomedical cybernetics", "Biorobotics", "Biosemiotics", "Block diagram", "Bode diagram", "Bode plot", "Bond graph", "Boulton & Watt engine", "Bounded function", "Brain\u2013computer interface", "Buckminster Fuller", "B\u00e9la H. B\u00e1n\u00e1thy", "C. West Churchman", "Calculus", "Cambridge University", "Cartesian coordinates", "Catastrophe theory", "Category theory", "Cell (biology)", "Centrifugal governor", "Chaos theory", "Charles A S Hall", "Charles Fran\u00e7ois (systems scientist)", "Circular coordinates", "Classical control theory", "Claude Bernard", "Claude Shannon", "Cliff Joslyn", "Closed-loop controller", "Closed-loop transfer function", "Coefficient diagram method", "Complex adaptive system", "Complex plane", "Complex system", "Complex systems", "Computational mathematics", "Computational neuroscience", "Computer network", "Computer science", "Conceptual system", "Connectionism", "Control (linguistics)", "Control Theory", "Control engineering", "Control loop", "Control reconfiguration", "Control system", "Control theory (sociology)", "Control valve", "Controllability", "Controller (control theory)", "Control\u2013feedback\u2013abort loop", "Coordinate vector", "Coupled human\u2013environment system", "Cruise control", "Cut-insertion theorem", "Cyberneticist", "Cybernetics", "Damping", "Database", "Deadbeat controller", "Decision theory", "Derivative", "Describing function", "Diederich Hinrichsen", "Differential equation", "Differential geometry", "Digital control", "Digital object identifier", "Digital signal processing", "Dimension (vector space)", "Discrete-time signal", "Discrete geometry", "Discrete mathematics", "Distributed Control System", "Distributed control system", "Distributed parameter systems", "Donella Meadows", "Doubling time", "Duncan McFarlane", "Dynamic programming", "Dynamic system", "Dynamical system", "Dynamical systems theory", "Earth system science", "Economic system", "Ecosystem", "Edsger W. Dijkstra", "Eduardo D. Sontag", "Edward John Routh", "Edward Norton Lorenz", "Edward Routh", "Eigenvalue", "Eigenvalues", "Electric motor", "Electric motors", "Electronics", "Elementary algebra", "Embedded system", "Emergence", "Energy-shaping control", "Energy Conversion and Management", "Energy system", "Engineering cybernetics", "Erich von Holst", "Ernst von Glasersfeld", "Evolutionary computation", "Faina Mihajlovna Kirillova", "Feed forward (control)", "Feedback", "Feedback linearization", "Feedback loop", "Feedback mechanism", "Finite geometry", "Fire-control system", "Formal system", "Foundations of mathematics", "Fourier transform", "Fractional-order control", "Francis Heylighen", "Francisco Varela", "Frederic Vester", "Frequency", "Frequency domain", "Frequency response", "Functional analysis", "Fuzzy control", "Fuzzy logic", "Gain (electronics)", "Game theory", "Genetic algorithms", "Geoffrey Vickers", "Geometry", "George Dantzig", "George Klir", "Gordon Pask", "Gordon S. Brown", "Graph theory", "Gregory Bateson", "Guidance system", "H-infinity loop-shaping", "H infinity", "Hankel singular value", "Harold Stephen Black", "Harry Nyquist", "Heinz von Foerster", "Hendrik Wade Bode", "Hierarchical", "Hierarchical control system", "History of mathematics", "Holon (philosophy)", "Homeostasis", "Howard T. Odum", "Human body", "Humberto Maturana", "Hybrid computer", "Igor Aleksander", "Ilya Prigogine", "Imaginary number", "Impulse response", "Industrial Control Systems", "Information system", "Information theory", "Input-to-state stability", "Instability", "Integral", "Integrated Authority File", "Integrator", "Intelligent control", "International Standard Book Number", "Irmgard Flugge-Lotz", "Irmgard Fl\u00fcgge-Lotz", "JSTOR", "Jacque Fresco", "Jacques Charles Fran\u00e7ois Sturm", "Jakob von Uexk\u00fcll", "James Clerk Maxwell", "James Grier Miller", "James J. Kay", "Jan Camiel Willems", "Jason Jixuan Hu", "Jay Wright Forrester", "Jennifer Wilby", "John N. Warfield", "John R. Ragazzini", "Kalman filter", "Kathleen Carley", "Katia Sycara", "Keck telescopes", "Keith Glover", "Kenneth E. Boulding", "Kevin Warwick", "Krener's theorem", "Laplace transform", "Lead-lag compensator", "Lead\u2013lag compensator", "Least squares", "Lev Pontryagin", "Limit cycle", "Limiting factor", "Linear", "Linear-quadratic-Gaussian control", "Linear algebra", "Linear control theory", "Linear equation", "Linear function", "Linear matrix inequality", "Linear system", "Linear time invariant", "Linearization", "List of national legal systems", "List of systems sciences organizations", "List of systems scientists", "Lists of mathematics topics", "Living systems", "Ludwig von Bertalanffy", "Lyapunov's theory", "Lyapunov function", "Lyapunov stability", "Lydia Kavraki", "MIMO", "MMT Observatory", "Machine learning", "Maleyka Abbaszadeh", "Management cybernetics", "Manfred Clynes", "Manuela M. Veloso", "Margaret Boden", "Margaret Mead", "Marginal stability", "Marian Mazur", "Markov chain approximation method", "Mary Cartwright", "Mathematical analysis", "Mathematical logic", "Mathematical model", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematical system theory", "Mathematics", "Mathematics and art", "Mathematics education", "Mechatronics", "Medical cybernetics", "Metric system", "Mihajlo D. Mesarovic", "Minor loop feedback", "Model Predictive Control", "Model predictive control", "Motion control", "Multi-agent system", "Multilinear algebra", "Multivariable", "Murray Bowen", "N. 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Bellman", "Rise time", "Robert Trappl", "Robotics", "Robust control", "Root Locus", "Root locus", "Routh\u2013Hurwitz theorem", "Rudolf Kalman", "Russell L. Ackoff", "Ruzena Bajcsy", "SCADA", "Sample (statistics)", "Scalar (mathematics)", "Scientific American", "Second-order cybernetics", "Self-organized", "Self-organized criticality control", "Self-oscillation", "Semiotics", "Sensor", "Sensory system", "Sergei P. Kurdyumov", "Servomechanism", "Set theory", "Setpoint (control system)", "Settling time", "Ship stability", "Signal-flow graph", "Simulation", "Simulation language", "Single-input single-output system", "Sliding mode control", "Social system", "Sociocybernetics", "Sociotechnical system", "Space Race", "Stability theory", "Stabilizer (ship)", "Stable polynomial", "Stafford Beer", "Star system", "State-space", "State (controls)", "State observer", "State space (controls)", "State space representation", "State variable", "Statistics", "Steady State", "Stephanie Forrest", "Stochastic control", "Stuart Kauffman", "Stuart Umpleby", "Subjectivity", "Superposition principle", "Synergetics (Haken)", "System", "System Dynamics", "System dynamics", "System identification", "System of measurement", "Systemics", "Systems analysis", "Systems art", "Systems biology", "Systems ecology", "Systems engineering", "Systems neuroscience", "Systems pharmacology", "Systems psychology", "Systems science", "Systems theory", "Systems theory in anthropology", "Systems theory in archaeology", "Systems theory in political science", "Systems thinking", "Talcott Parsons", "Telescope", "Theory of computation", "Throttle", "Time-domain state space representation", "Time-invariant", "Time scale calculus", "Topology", "Transfer function", "Transform (mathematics)", "Tree (data structure)", "Trigonometry", "Twelve leverage points", "Ulla Mitzdorf", "Underactuation", "Unit circle", "Urban metabolism", "Vadim Utkin", "Valentin Turchin", "Valentino Braitenberg", "Variable (mathematics)", "Vector control (motor)", "Viscosity solutions", "Voltage", "W. Ross Ashby", "Walter Bradford Cannon", "Walter Pitts", "Warren Sturgis McCulloch", "Wavefront", "Wiener filter", "William Grey Walter", "William Ross Ashby", "World-systems theory", "World War II", "Wright brothers", "Writing system", "Youla\u2013Kucera parametrization", "Z-transform", "Z Transform", "Z transform", "Zeros and poles"], "references": ["http://www.controlguru.com", "http://www.mathworks.com/help/toolbox/simulink/slref/trim.html", "http://www.nature.com/ncomms/2015/151001/ncomms9414/full/ncomms9414.html", "http://www.nature.com/scientificamerican/journal/v211/n3/pdf/scientificamerican0964-186.pdf", "http://www.PIDlab.com", "http://www.cds.caltech.edu/~murray/books/AM08/pdf/am08-complete_28Sep12.pdf", "http://adsabs.harvard.edu/abs/2015NatCo...6E8414G", "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.302.5633&rep=rep1&type=pdf", "http://www.math.rutgers.edu/~sontag/FTP_DIR/sontag_mathematical_control_theory_springer98.pdf", "http://www.engin.umich.edu/class/ctms/", "http://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202012.pdf", "http://cbasso.pagesperso-orange.fr/Spice.htm", "http://www.dtic.mil/dtic/tr/fulltext/u2/621137.pdf", "http://delivery.acm.org/10.1145/2820000/2814734/p50-li.pdf", "http://arxiv.org/abs/1211.0248", "http://arxiv.org/abs/1406.5197", "http://doi.org/10.1016%2Fj.enconman.2012.11.021", "http://doi.org/10.1016%2Fj.fss.2010.04.009", "http://doi.org/10.1038%2Fncomms9414", "http://doi.org/10.1098%2Frspl.1867.0055", "http://doi.org/10.1111%2Fj.1559-3584.1922.tb04958.x", "http://doi.org/10.1142%2FS0219477502000919", "http://doi.org/10.2307%2F2589614", "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1453566", "http://www.jstor.org/stable/112510", "http://www.maa.org/programs/maa-awards/writing-awards/some-fundamental-control-theory-i-controllability-observability-and-duality-and-some-fundamental", "http://camo.ici.ro/neculai/history.pdf", "https://d-nb.info/gnd/4032317-1", "https://id.ndl.go.jp/auth/ndlna/00570298", "https://www.wikidata.org/wiki/Q6501221", "https://upload.wikimedia.org/wikipedia/commons/b/b1/On_Governors.pdf"]}, "Order theory": {"categories": ["All Wikipedia articles needing clarification", "All articles lacking in-text citations", "Articles lacking in-text citations from December 2015", "Order theory", "Wikipedia articles needing clarification from January 2017"], "title": "Order theory", "method": "Order theory", "url": "https://en.wikipedia.org/wiki/Order_theory", "summary": "Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as \"this is less than that\" or \"this precedes that\". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.", "images": ["https://upload.wikimedia.org/wikipedia/commons/5/51/Lattice_of_the_divisibility_of_60.svg", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg", "https://upload.wikimedia.org/wikipedia/commons/9/99/Wiktionary-logo-en-v2.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["Abstract algebra", "Adjoint functor", "Alexandrov topology", "Algebra", "Algebraic geometry", "Alphabetical order", "Analytic geometry", "Antichain", "Antisymmetric relation", "Applied mathematics", "Areas of mathematics", "Arithmetic", "Bijective", "Binary relation", "Boolean algebra (structure)", "Boolean ring", "Bounded poset", "Calculus", "Cartesian product", "Category theory", "Causal Sets", "Charles Sanders Peirce", "Circles", "Closed set", "Closure operator", "Complement (set theory)", "Complete lattice", "Completeness (order theory)", "Computational mathematics", "Computer science", "Connex relation", "Continuous function (topology)", "Control theory", "Cyclic order", "Differential equation", "Differential geometry", "Directed acyclic graph", "Directed complete partial order", "Directed set", "Discrete geometry", "Discrete mathematics", "Disjoint union", "Distributive lattice", "Distributivity (order theory)", "Division (mathematics)", "Divisor", "Domain theory", "Duality (order theory)", "Dynamical systems theory", "Elementary algebra", "Ellipse", "Ellipsis", "Empty set", "Empty subset", "Equivalence of categories", "Equivalence relation", "Ernst Schr\u00f6der", "Euler characteristic", "Filter (mathematics)", "Finite geometry", "Finite set", "Foundations of mathematics", "Function (mathematics)", "Function space", "Functional analysis", "Galois connection", "Game theory", "Garrett Birkhoff", "Genealogy", "Geometry", "George Boole", "Graph drawing", "Graph theory", "Greatest common divisor", "Greatest element", "Greatest lower bound", "Hasse diagram", "Heyting algebra", "Hierarchy", "Hilary Priestley", "History of mathematics", "Ideal (order theory)", "Idempotent", "Iff", "Incidence algebra", "Infimum", "Information theory", "Integer", "International Standard Book Number", "Interval (mathematics)", "Interval order", "Lattice (order)", "Lawson topology", "Least common multiple", "Least element", "Least upper bound", "Limit-preserving function (order theory)", "Limit (category theory)", "Lineal descent", "Linear algebra", "Lists of mathematics topics", "Mathematical analysis", "Mathematical logic", "Mathematical optimization", "Mathematical physics", "Mathematical statistics", "Mathematics", "Mathematics and art", "Mathematics education", "Maximal element", "Minimal element", "Monotonic function", "Multilinear algebra", "Natural number", "Natural numbers", "Net (mathematics)", "Number", "Number systems", "Number theory", "Numeral systems", "Numerical analysis", "Open set", "Order-embedding", "Order consistent", "Order isomorphism", "Order theory glossary", "Outline of mathematics", "Outline of order theory", "Partial order", "Partial orders with complements", "Partially ordered set", "Pediatricians", "Philosophy of mathematics", "Physicians", "Pointless topology", "Pointwise order", "Poset", "Powerset", "Preorder", "Primary school", "Probability theory", "Product (category theory)", "Product order", "Pure mathematics", "Quantale", "Rational number", "Real number", "Recreational mathematics", "Reflexive relation", "Richard Dedekind", "Scott-continuous", "Scott topology", "Semiorder", "Set (mathematics)", "Set theory", "Specialization order", "Statistics", "Stone duality", "Strict order", "Strict weak ordering", "Subbase", "Subset", "Subtraction", "Supremum", "Surjective", "T0 space", "Theory of computation", "Topology", "Total order", "Transitive relation", "Trigonometry", "Union (set theory)", "Universal algebra", "Upper bound", "Upper topology", "Vertex (graph theory)", "Well-founded relation", "Well partial order", "Zorn's Lemma"], "references": ["http://www.thoralf.uwaterloo.ca/htdocs/ualg.html", "http://www.apronus.com/provenmath/orders.htm", "http://jeff560.tripod.com/p.html", "http://www.felixnagel.org", "http://www.personal.soton.ac.uk/gan/Roller.pdf"]}, "Graph theory": {"categories": ["Graph theory", "Pages using div col with small parameter", "Webarchive template wayback links", "Wikipedia articles with GND identifiers", "Wikipedia articles with LCCN identifiers", "Wikipedia articles with NDL identifiers"], "title": "Graph theory", "method": "Graph theory", "url": "https://en.wikipedia.org/wiki/Graph_theory", "summary": "In mathematics,  graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.\nRefer to the glossary of graph theory for basic definitions in graph theory.", "images": ["https://upload.wikimedia.org/wikipedia/commons/5/5b/6n-graf.svg", "https://upload.wikimedia.org/wikipedia/commons/7/73/Blue_pencil.svg", "https://upload.wikimedia.org/wikipedia/commons/b/b6/Moreno_Sociogram_2nd_Grade.png", "https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg", "https://upload.wikimedia.org/wikipedia/commons/8/89/Symbol_book_class2.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5b/Wikipedia_multilingual_network_graph_July_2013.svg", "https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg", "https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg", "https://upload.wikimedia.org/wikipedia/commons/5/5d/Konigsberg_bridges.png", "https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg"], "links": ["ACM Computing Classification System", "Abstract algebra", "Abstract simplicial complex", "Academic Press", "Adjacency list", "Adjacency matrix", "Alexandre-Th\u00e9ophile Vandermonde", "Alfred Kempe", "Alfr\u00e9d R\u00e9nyi", "Algebra", "Algebraic geometry", "Algebraic graph theory", "Algorithm", "Algorithm design", "Algorithmic efficiency", "Analysis of algorithms", "Analytic geometry", "Anton Kotzig", "Application security", "Applied mathematics", "ArXiv", "Arboricity", "Areas of mathematics", "Arithmetic", "Arthur Cayley", "Artificial intelligence", "Atom", "August Kekul\u00e9", "Augustin-Louis Cauchy", "Augustus De Morgan", "Automata theory", "Automated planning and scheduling", "Automatic theorem prover", "Bellman\u2013Ford algorithm", "Benny Sudakov", "Bibcode", "Biology", "Bor\u016fvka's algorithm", "Breadth-first search", "B\u00e9la Bollob\u00e1s", "Calculus", "Cambridge University Press", "Camille Jordan", "Carsten Thomassen", "Category theory", "Chemical bond", "Chemistry", "Citation graph", "Claude Berge", "Clique problem", "Combinatorics", "Compiler construction", "Complete bipartite graph", "Complete graph", "Computational biology", "Computational chemistry", "Computational complexity theory", "Computational engineering", "Computational geometry", "Computational linguistics", "Computational mathematics", "Computational neuroscience", "Computational physics", "Computational social science", "Computer accessibility", "Computer animation", "Computer architecture", "Computer data storage", "Computer graphics", "Computer hardware", "Computer network", "Computer science", "Computer security", "Computer security compromised by hardware failure", "Computer vision", "Computing platform", "Conceptual graph", "Concurrency (computer science)", "Concurrent computing", "Condensed matter physics", "Control theory", "Covering problem", "Cross-validation (statistics)", "Crossing number (graph theory)", "Cryptography", "Cyberwarfare", "Cycle double cover", "Cytoscape", "Daniel P. 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