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  • Probability Theory | Britannica.com
    the 17th century Independently of Descartes Fermat discovered Gauss Carl Friedrich German mathematician generally regarded as one of the greatest mathematicians of all time for his contributions to number theory geometry probability theory geodesy planetary astronomy the theory of functions and potential theory including electromagnetism indifference in the mathematical theory of probability a classical principle stated by the Swiss mathematician Jakob Bernoulli and formulated and named by the English economist John Maynard Keynes in A Treatise on Probability 1921 two cases are equally likely Kolmogorov Andrey Nikolayevich Russian mathematician whose work influenced many branches of modern mathematics especially harmonic analysis probability set theory information theory and number theory A man of broad culture with interests in technology history and education Lévy Paul French mining engineer and mathematician noted for his work in the theory of probability After serving as a professor at the École des Mines de Saint Étienne Paris from 1910 to 1913 Lévy joined the faculty 1914 51 of the École Nationale Supérieure Markov Andrey Andreyevich Russian mathematician who helped to develop the theory of stochastic processes especially those called Markov chains Based on the study of the probability of mutually dependent events his work has been developed and widely applied in the biological Markov process sequence of possibly dependent random variables x 1 x 2 x 3 identified by increasing values of a parameter commonly time with the property that any prediction of the next value of the sequence x n knowing the preceding states x 1 x 2 Mises Richard von Austrian born American mathematician engineer and positivist philosopher who notably advanced statistics and probability theory Von Mises s early work centred on geometry and mechanics especially the theory of turbines In 1913 during his appointment Moivre Abraham de French mathematician who was a pioneer in

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  • Set Theory | Britannica.com
    collection of subsets In symbols a finite set S with n elements contains 2 n subsets so that the cardinality of the Cohen Paul Joseph American mathematician who was awarded the Fields Medal in 1966 for his proof of the independence of the continuum hypothesis from the other axioms of set theory Cohen attended the University of Chicago M S 1954 Ph D 1958 He held appointments continuum hypothesis statement of set theory that the set of real number s the continuum is in a sense as small as it can be In 1873 the German mathematician Georg Cantor proved that the continuum is uncountable that is the real numbers are a larger infinity than the Erdős Paul Hungarian freelance mathematician known for his work in number theory and combinatorics and legendary eccentric who was arguably the most prolific mathematician of the 20th century in terms of both the number of problems he solved and the number Kripke Saul American logician and philosopher who from the 1960s was one of the most powerful thinkers in Anglo American philosophy see analytic philosophy Kripke began his important work on the semantics of modal logic the logic of modal notions such as necessity partition in mathematics and logic division of a set of objects into a family of subsets that are mutually exclusive and jointly exhaustive that is no element of the original set is present in more than one of the subsets and all the subsets together contain set theory branch of mathematics that deals with the properties of well defined collections of objects which may or may not be of a mathematical nature such as numbers or functions The theory is less valuable in direct application to ordinary experience than Venn diagram graphical method of representing categorical propositions and testing the

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  • Statistics | Britannica.com
    Cox Sir David British statistician best known for his proportional hazards model Cox studied at St John s College Cambridge and from 1944 to 1946 he worked at the Royal Aircraft Establishment at Farnborough From 1946 to 1950 he worked at the Wool Industries Research Davenport Charles Benedict American zoologist who contributed substantially to the study of eugenics the improvement of populations through breeding and heredity and who pioneered the use of statistical techniques in biological research After receiving a doctorate in zoology decision theory in statistics a set of quantitative methods for reaching optimal decisions A solvable decision problem must be capable of being tightly formulated in terms of initial conditions and choices or courses of action with their consequences In general distribution function mathematical expression that describes the probability that a system will take on a specific value or set of values The classic examples are associated with games of chance The binomial distribution gives the probabilities that heads will come up a Edgeworth Francis Ysidro Irish economist and statistician who innovatively applied mathematics to the fields of economics and statistics Edgeworth was educated at Trinity College in Dublin and Balliol College Oxford graduating in 1869 In 1877 he qualified as a barrister Engel Ernst German statistician remembered for the Engel curve or Engel s law which states that the lower a family s income the greater is the proportion of it spent on food His conclusion was based on a budget study of 153 Belgian families and was later verified estimation in statistics any of numerous procedures used to calculate the value of some property of a population from observations of a sample drawn from the population A point estimate for example is the single number most likely to express the value of the Fisher Sir Ronald Aylmer British statistician and geneticist who pioneered the application of statistical procedures to the design of scientific experiments In 1909 Fisher was awarded a scholarship to study mathematics at the University of Cambridge from which he graduated Fogel Robert William American economist who with Douglass C North was awarded the Nobel Prize for Economics in 1993 The two were cited for having developed cliometrics the application of statistical analysis to the study of economic history Fogel attended Cornell University freedom degree of in mathematics any of the number of independent quantities necessary to express the values of all the variable properties of a system A system composed of a point moving without constraints in space for example has three degrees of freedom because Gini Corrado Italian statistician and demographer Gini was educated at Bologna where he studied law mathematics economics and biology He was a statistics professor at Cagliari in 1909 and at Padua in 1913 After founding the statistical journal Metron 1920 Graunt John English statistician generally considered to be the founder of the science of demography the statistical study of human populations His analysis of the vital statistics of the London populace influenced the pioneer demographic work of

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  • Topology | Britannica.com
    equations developed by the noted French mathematician Évariste Galois 1811 32 Brouwer Luitzen Egbertus Jan Dutch mathematician who founded mathematical intuitionism a doctrine that views the nature of mathematics as mental constructions governed by self evident laws and whose work completely transformed topology the study of the most basic properties of catastrophe theory in mathematics a set of methods used to study and classify the ways in which a system can undergo sudden large changes in behaviour as one or more of the variables that control it are changed continuously Catastrophe theory is generally considered Dehn Max German mathematician and educator whose study of topology in 1910 led to his theorem on topological manifolds known as Dehn s lemma Dehn was educated in Germany and received his doctorate from the University of Göttingen in 1900 He was influenced Dieudonné Jean French mathematician and educator known for his writings on abstract algebra functional analysis topology and his theory of Lie groups Dieudonné was educated in Paris receiving both his bachelor s degree 1927 and his doctorate 1931 from the Donaldson Simon Kirwan British mathematician who was awarded the Fields Medal in 1986 for his work in topology Donaldson attended Pembroke College Cambridge B A 1979 and Worcester College Oxford Ph D 1983 From 1983 to 1985 he was a Junior Research Fellow at All Freedman Michael Hartley American mathematician who was awarded the Fields Medal in 1986 for his solution of the Poincaré conjecture in four dimensions Freedman received his Ph D from Princeton N J University in 1973 Following appointments at the University of California Grothendieck Alexandre German French mathematician who was awarded the Fields Medal in 1966 for his work in algebraic geometry After studies at the University of Montpellier France and a year at the École Normale Supérieure in Paris Grothendieck received his doctorate Jordan curve theorem in topology a theorem first proposed in 1887 by French mathematician Camille Jordan that any simple closed curve that is a continuous closed curve that does not cross itself now known as a Jordan curve divides the plane into exactly two regions Klein bottle topological space named for the German mathematician Felix Klein obtained by identifying two ends of a cylindrical surface in the direction opposite that is necessary to obtain a torus The surface is not constructible in three dimensional Euclidean knot theory in mathematics the study of closed curves in three dimensions and their possible deformations without one part cutting through another Knots may be regarded as formed by interlacing and looping a piece of string in any fashion and then joining the Mac Lane Saunders American mathematician who was a cocreator of category theory an architect of homological algebra and an advocate of categorical foundations for mathematics Mac Lane graduated from Yale University in 1930 and then began graduate work at the University Möbius August Ferdinand German mathematician and theoretical astronomer who is best known for his work in analytic geometry and in topology In the latter field

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  • Trigonometry | Britannica.com
    served as the basic guide for Islamic and European astronomers until about the beginning of the 17th century Its original name was Mathematike Syntaxis The Euler Leonhard Swiss mathematician and physicist one of the founders of pure mathematics He not only made decisive and formative contributions to the subjects of geometry calculus mechanics and number theory but also developed methods for solving problems in observational Gregory James Scottish mathematician and astronomer who discovered infinite series representations for a number of trigonometry functions although he is mostly remembered for his description of the first practical reflecting telescope now known as the Gregorian Hipparchus Greek astronomer and mathematician who made fundamental contributions to the advancement of astronomy as a mathematical science and to the foundations of trigonometry Although he is commonly ranked among the greatest scientists of antiquity very little Menelaus of Alexandria Greek mathematician and astronomer who first conceived and defined a spherical triangle a triangle formed by three arcs of great circles on the surface of a sphere Menelaus s most important work is Sphaerica on the geometry of the sphere extant Moivre Abraham de French mathematician who was a pioneer in the development of analytic trigonometry and in the theory of probability A French Huguenot de Moivre was jailed as a Protestant upon the revocation of the Edict of Nantes in 1685 When he was released shortly Regiomontanus the foremost mathematician and astronomer of 15th century Europe a sought after astrologer and one of the first printers Königsberg means King s Mountain which is what the Latinized version of his name Joannes de Regio monte or Regiomontanus trigonometry the branch of mathematics concerned with specific functions of angles and their application to calculations There are six functions of an angle commonly used in trigonometry Their names and abbreviations are

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  • Mathematics - 3 | Britannica.com
    variables x and y generally stand for values of real numbers The algebra of complex numbers complex analysis uses the complex variable z to represent computer science the study of computers including their design architecture and their uses for computations data processing and systems control The field of computer science includes engineering activities such as the design of computers and of the hardware and cone in mathematics the surface traced by a moving straight line the generatrix that always passes through a fixed point the vertex The path to be definite is directed by some closed plane curve the directrix along which the line always glides conformal mapping In mathematics a transformation of one graph into another in such a way that the angle of intersection of any two lines or curves remains unchanged The most common example is the Mercator map a two dimensional representation of the surface of the conic section in geometry any curve produced by the intersection of a plane and a right circular cone Depending on the angle of the plane relative to the cone the intersection is a circle an ellipse a hyperbola or a parabola Special degenerate cases of intersection connectedness in mathematics fundamental topological property of sets that corresponds with the usual intuitive idea of having no breaks It is of fundamental importance because it is one of the few properties of geometric figures that remains unchanged after a homeomorphism connective in logic a word or group of words that joins two or more propositions together to form a connective proposition Commonly used connectives include but and or if then and if and only if The various types of logical connectives include Connes Alain French mathematician who won the Fields Medal in 1982 for his work in operator theory Connes received a bachelor s degree 1970 and a doctorate 1973 from the École Normale Supérieure now part of the University of Paris He held appointments at Conon of Samos mathematician and astronomer whose work on conic sections curves of the intersections of a right circular cone with a plane served as the basis for the fourth book of the Conics of Apollonius of Perga c 262 190 bce From his observations in Italy constant a number value or object that has a fixed magnitude physically or abstractly as a part of a specific operation or discussion In mathematics the term refers to a quantity often represented by a symbol e g π the ratio of a circle s circumference continuity in mathematics rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps A function is a relationship in which every value of an independent variable say x is associated with a value of a dependent variable say continuum hypothesis statement of set theory that the set of real number s the continuum is in a sense as small as it can be In 1873 the German mathematician Georg Cantor proved that the continuum is uncountable that is the real numbers are a larger infinity than the control theory field of applied mathematics that is relevant to the control of certain physical processes and systems Although control theory has deep connections with classical areas of mathematics such as the calculus of variations and the theory of differential convergence in mathematics property exhibited by certain infinite series and functions of approaching a limit more and more closely as an argument variable of the function increases or decreases or as the number of terms of the series increases For example Coolidge Julian Lowell U S mathematician and educator who published numerous works on theoretical mathematics along the lines of the Study Segre school Coolidge was born to a family of well established Bostonians his paternal grandmother was Thomas Jefferson s granddaughter coordinate system Arrangement of reference lines or curves used to identify the location of points in space In two dimensions the most common system is the Cartesian after René Descartes system Points are designated by their distance along a horizontal x and vertical Copson Edward Thomas mathematician known for his contributions to analysis and partial differential equations especially as they apply to mathematical physics Copson studied at St John s College Oxford and then was a lecturer of mathematics first at the University of Coriolis Gustave Gaspard French engineer and mathematician who first described the Coriolis force an effect of motion on a rotating body of paramount importance to meteorology ballistics and oceanography An assistant professor of analysis and mechanics at the École Polytechnique correlation In statistics the degree of association between two random variables The correlation between the graphs of two data sets is the degree to which they resemble each other However correlation is not the same as causation and even a very close correlation Courant Richard German born American mathematician and educator who made significant advances in the calculus of variations Courant received his secondary education in Germany and Switzerland and his doctorate from the University of Göttingen in 1910 under David Hilbert Cournot Antoine Augustin French economist and mathematician Cournot was the first economist who with competent knowledge of both subjects endeavoured to apply mathematics to the treatment of economics His main work in economics is Recherches sur les principes mathématiques Crelle August Leopold German mathematician and engineer who advanced the work and careers of many young mathematicians of his day and founded the Journal für die reine und angewandte Mathematik Journal for Pure and Applied Mathematics now known as Crelle s Journal A Cremona Luigi Italian mathematician who was an originator of graphical statics the use of graphical methods to study forces in equilibrium Following his appointment as professor of higher geometry at the University of Bologna in 1860 he published Introduzione cross ratio in projective geometry ratio that is of fundamental importance in characterizing projections In a projection of one line onto another from a central point see the double ratio of lengths on the first line AC AD BC BD

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  • Mathematics - 4 | Britannica.com
    Wilhelm University of Münster Ph D 1978 Following a visiting research fellowship at Harvard University Cambridge Fefferman Charles Louis American mathematician who was awarded the Fields Medal in 1978 for his work in classical analysis Fefferman attended the University of Maryland B S 1966 and Princeton N J University After receiving his Ph D in 1969 he remained at Princeton Fermat Pierre de French mathematician who is often called the founder of the modern theory of numbers Together with René Descartes Fermat was one of the two leading mathematicians of the first half of the 17th century Independently of Descartes Fermat discovered Fermat s last theorem the statement that there are no natural numbers 1 2 3 x y and z such that x n y n z n in which n is a natural number greater than 2 For example if n 3 Fermat s theorem states that no natural numbers x y and z exist such that Fermat s theorem in number theory the statement first given in 1640 by French mathematician Pierre de Fermat that for any prime number p and any integer a such that p does not divide a the pair are relatively prime p divides exactly into a p a Although a number Ferrari Lodovico Italian mathematician who was the first to find an algebraic solution to the biquadratic or quartic equation an algebraic equation that contains the fourth power of the unknown quantity but no higher power From a poor family Ferrari was taken into Ferro Scipione Italian mathematician who is believed to have found a solution to the cubic equation x 3 px q where p and q are positive numbers Ferro attended the University of Bologna and in 1496 accepted a position at the university as a lecturer in arithmetic Fibonacci number the elements of the sequence of numbers 1 1 2 3 5 8 13 21 each of which after the second is the sum of the two previous numbers These numbers were first noted by the medieval Italian mathematician Leonardo Pisano Fibonacci in his Liber fixed point theorem any of various theorems in mathematics dealing with a transformation of the points of a set into points of the same set where it can be proved that at least one point remains fixed For example if each real number is squared the numbers zero and one fluxion in mathematics the original term for derivative introduced by Isaac Newton in 1665 Newton referred to a varying flowing quantity as a fluent and to its instantaneous rate of change as a fluxion Newton stated that the fundamental problems of the formalism in mathematics school of thought introduced by the 20th century German mathematician David Hilbert which holds that all mathematics can be reduced to rules for manipulating formulas without any reference to the meanings of the formulas Formalists Forsyth Andrew Russell British mathematician best known for his mathematical textbooks In 1877 Forsyth entered Trinity College Cambridge where he studied mathematics under Arthur Cayley Forsyth graduated in 1881 as first wrangler first place in the annual Mathematical four colour map problem problem in topology originally posed in the early 1850s and not solved until 1976 that required finding the minimum number of different colours required to colour a map such that no two adjacent regions i e with a common boundary segment are of Fourier Joseph Baron French mathematician known also as an Egyptologist and administrator who exerted strong influence on mathematical physics through his Théorie analytique de la chaleur 1822 The Analytical Theory of Heat He showed how the conduction of heat in solid Fourier series In mathematics an infinite series used to solve special types of differential equations It consists of an infinite sum of sines and cosines and because it is periodic i e its values repeat over fixed intervals it is a useful tool in analyzing fractal in mathematics any of a class of complex geometric shapes that commonly have fractional dimension a concept first introduced by the mathematician Felix Hausdorff in 1918 Fractals are distinct from the simple figures of classical or Euclidean geometry the fraction In arithmetic a number expressed as a quotient in which a numerator is divided by a denominator In a simple fraction both are integers A complex fraction has a fraction in the numerator or denominator In a proper fraction the numerator is less Fréchet Maurice French mathematician known chiefly for his contributions to real analysis He is credited with being the founder of the theory of abstract spaces Fréchet was professor of mechanics at the University of Poitiers 1910 19 before moving to the University Fredholm Ivar Swedish mathematician who founded modern integral equation theory Fredholm entered the University of Uppsala in 1886 There and later at the University of Stockholm 1888 93 he was mainly interested in mathematical physics After receiving his Ph D Freedman Michael Hartley American mathematician who was awarded the Fields Medal in 1986 for his solution of the Poincaré conjecture in four dimensions Freedman received his Ph D from Princeton N J University in 1973 Following appointments at the University of California Frege Gottlob German mathematician and logician who founded modern mathematical logic Working on the borderline between philosophy and mathematics viz in the philosophy of mathematics and mathematical logic in which no intellectual precedents existed Frege discovered Friedmann Aleksandr Aleksandrovich Russian mathematician and physical scientist After graduating from the University of St Petersburg in 1910 Friedmann joined the Pavlovsk Aerological Observatory and during World War I did aerological work for the Russian army After the war he was Frisi Paolo Italian mathematician astronomer and physicist who is best known for his work in hydraulics His most significant contributions to science however were in the compilation interpretation and dissemination of the work of other scientists Frisi was Frobenius Georg German mathematician who made major contributions to group theory Frobenius studied for one year at the University of Göttingen before returning home in 1868 to study at the

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  • Mathematics - 5 | Britannica.com
    served for some 30 years as burgomaster of Amsterdam In his De reductione aequationum Huygens Christiaan Dutch mathematician astronomer and physicist who founded the wave theory of light discovered the true shape of the rings of Saturn and made original contributions to the science of dynamics the study of the action of forces on bodies Huygens was Huygens Constantijn the most versatile and the last of the true Dutch Renaissance virtuosos who made notable contributions in the fields of diplomacy scholarship music poetry and science His diplomatic service took him several times to England where he met and was Hypatia Egyptian mathematician astronomer and philosopher who lived in a very turbulent era in Alexandria s history She is the earliest female mathematician of whose life and work reasonably detailed knowledge exists Hypatia was the daughter of Theon of hyperbolic functions the hyperbolic sine of z written sinh z the hyperbolic cosine of z cosh z the hyperbolic tangent of z tanh z and the hyperbolic cosecant secant and cotangent of z These functions are most conveniently defined in terms of the exponential function hyperbolic geometry a non Euclidean geometry that rejects the validity of Euclid s fifth the parallel postulate Simply stated this Euclidean postulate is through a point not on a given line there is exactly one line parallel to the given line In hyperbolic geometry hyperboloid the open surface generated by revolving a hyperbola about either of its axes If the tranverse axis of the surface lies along the x axis and its centre lies at the origin and if a b and c are the principal semi axes then the general equation of the hypergeometric distribution in statistics distribution function in which selections are made from two groups without replacing members of the groups The hypergeometric distribution differs from the binomial distribution in the lack of replacements Thus it often is employed hypothesis testing In statistics a method for testing how accurately a mathematical model based on one set of data predicts the nature of other data sets generated by the same process Hypothesis testing grew out of quality control in which whole batches of manufactured Ibn al Haytham mathematician and astronomer who made significant contributions to the principles of optics and the use of scientific experiments Life Conflicting stories are told about the life of Ibn al Haytham particularly concerning his scheme to regulate the ideal in modern algebra a subring of a mathematical ring with certain absorption properties The concept of an ideal was first defined and developed by German mathematician Richard Dedekind in 1871 In particular he used ideals to translate ordinary properties indifference in the mathematical theory of probability a classical principle stated by the Swiss mathematician Jakob Bernoulli and formulated and named by the English economist John Maynard Keynes in A Treatise on Probability 1921 two cases are equally likely inequality In mathematics a statement of an order relationship greater than greater than or equal to less than or less than or equal to between two numbers or algebraic expressions Inequalities can be posed either as questions much like equation s and solved inference in statistics the process of drawing conclusions about a parameter one is seeking to measure or estimate Often scientists have many measurements of an object say the mass of an electron and wish to choose the best measure One principal approach of infinite series the sum of infinitely many numbers related in a given way and listed in a given order Infinite series are useful in mathematics and in such disciplines as physics chemistry biology and engineering For an infinite series a 1 a 2 a 3 infinitesimal in mathematics a quantity less than any finite quantity yet not zero Even though no such quantity can exist in the real number system many early attempts to justify calculus were based on sometimes dubious reasoning about infinitesimals derivatives infinity the concept of something that is unlimited endless without bound The common symbol for infinity was invented by the English mathematician John Wallis in 1657 Three main types of infinity may be distinguished the mathematical the physical and information theory a mathematical representation of the conditions and parameters affecting the transmission and processing of information Most closely associated with the work of the American electrical engineer Claude Shannon in the mid 20th century information theory inner product space In mathematics a vector space or function space in which an operation for combining two vectors or functions whose result is called an inner product is defined and has certain properties Such spaces an essential tool of functional analysis and vector integer Whole valued positive or negative number or 0 The integers are generated from the set of counting numbers 1 2 3 and the operation of subtraction When a counting number is subtracted from itself the result is zero When a larger number is subtracted integral in mathematics either a numerical value equal to the area under the graph of a function for some interval definite integral or a new function the derivative of which is the original function indefinite integral These two meanings are related by integral calculus Branch of calculus concerned with the theory and applications of integral s While differential calculus focuses on rates of change such as slopes of tangent lines and velocities integral calculus deals with total size or value such as lengths areas integral equation in mathematics equation in which the unknown function to be found lies within an integral sign An example of an integral equation is in which f x is known if f x f x for all x one solution is integral transform mathematical operator that produces a new function f y by integrating the product of an existing function F x and a so called kernel function K x y between suitable limits The process which is called transformation is symbolized by the equation integraph mathematical instrument for plotting the integral of a graphically defined function Two such instruments were

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