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https://en.wikipedia.org/wiki/Alain%20Connes | Alain Connes (; born 1 April 1947 in Draguignan) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982.
Career
Alain Connes attended high... |
https://en.wikipedia.org/wiki/Arithmetic%20mean | In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. T... |
https://en.wikipedia.org/wiki/Argument%20%28disambiguation%29 | In logic and philosophy, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.
Argument may also refer to:
Mathematics and computer science
Argument (complex analysis), a function which returns the polar angle of a complex number
Command-line ar... |
https://en.wikipedia.org/wiki/Algorithms%20%28journal%29 | Algorithms is a monthly peer-reviewed open-access scientific journal of mathematics, covering design, analysis, and experiments on algorithms. The journal is published by MDPI and was established in 2008. The founding editor-in-chief was Kazuo Iwama (Kyoto University). From May 2014 to September 2019, the editor-in-chi... |
https://en.wikipedia.org/wiki/Algorithm | In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to di... |
https://en.wikipedia.org/wiki/Axiom%20of%20choice | In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to c... |
https://en.wikipedia.org/wiki/Arable%20land | Arable land (from the , "able to be ploughed") is any land capable of being ploughed and used to grow crops. Alternatively, for the purposes of agricultural statistics, the term often has a more precise definition:
A more concise definition appearing in the Eurostat glossary similarly refers to actual rather than pot... |
https://en.wikipedia.org/wiki/Absolute%20value | In mathematics, the absolute value or modulus of a real number , is the non-negative value without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a ... |
https://en.wikipedia.org/wiki/Algebraically%20closed%20field | In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because the polynomial equation has no solution in real numbers, even though all i... |
https://en.wikipedia.org/wiki/Algorithms%20for%20calculating%20variance | Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.
... |
https://en.wikipedia.org/wiki/Algebraic%20number | An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As ... |
https://en.wikipedia.org/wiki/Automorphism | In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely ... |
https://en.wikipedia.org/wiki/Antisymmetric%20relation | In mathematics, a binary relation on a set is antisymmetric if there is no pair of distinct elements of each of which is related by to the other. More formally, is antisymmetric precisely if for all
or equivalently,
The definition of antisymmetry says nothing about whether actually holds or not for any . An an... |
https://en.wikipedia.org/wiki/Angle | In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; ... |
https://en.wikipedia.org/wiki/Almost%20all | In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if is a set, "almost all elements of " means "all elements of but those in a negligible subset of ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.
In c... |
https://en.wikipedia.org/wiki/Associative%20property | In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
Within an expression containing two or more... |
https://en.wikipedia.org/wiki/Kolmogorov%20complexity | In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources n... |
https://en.wikipedia.org/wiki/Augustin-Louis%20Cauchy | Baron Augustin-Louis Cauchy ( , , ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, ... |
https://en.wikipedia.org/wiki/Archimedean%20solid | In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. The convex polyhedra with regular faces and symmetric vertices (the convex uniform polyhedra) include also the five Platonic soli... |
https://en.wikipedia.org/wiki/Antiprism | In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.
Antiprisms ... |
https://en.wikipedia.org/wiki/Algebraic%20geometry | Algebraic geometry is a branch of mathematics which classically studies 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.
The fundamental objects of study in alg... |
https://en.wikipedia.org/wiki/Andr%C3%A9%20Weil | André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is due
both to his original contributions to a remarkably broad
spectrum of mathemat... |
https://en.wikipedia.org/wiki/Atle%20Selberg | Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950 and an honorary Abel Prize in 2002.
Early years
Se... |
https://en.wikipedia.org/wiki/Andrew%20Wiles | Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society. He w... |
https://en.wikipedia.org/wiki/Alexander%20Grothendieck | Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its fou... |
https://en.wikipedia.org/wiki/Associative%20algebra | In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image by the ring homomorphism of ... |
https://en.wikipedia.org/wiki/Axiom%20of%20regularity | In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:
The axiom of regularity together with the axiom of pairing implies that ... |
https://en.wikipedia.org/wiki/Algebraic%20extension | In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, every element of is a root of a non-zero polynomial with coefficients in . A field extension that is not algebraic, is said to be transcendental, and must contain tra... |
https://en.wikipedia.org/wiki/Analytic%20geometry | In 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.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundat... |
https://en.wikipedia.org/wiki/Annals%20of%20Mathematics | The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as The Analyst in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation ... |
https://en.wikipedia.org/wiki/Antiderivative | In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiatio... |
https://en.wikipedia.org/wiki/Convex%20uniform%20honeycomb | In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs are known:
the familiar cubic honeycomb and 7 truncations thereof;
the alternated cubic honeycomb and 4 truncations thereo... |
https://en.wikipedia.org/wiki/Abelian%20group | In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers fo... |
https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric%20mean | In mathematics, the arithmetic–geometric mean of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means:
Begin the sequences with x and y:
Then define the two interdependent sequences and as
These two sequences converge to the same number, the arithm... |
https://en.wikipedia.org/wiki/Asymptote | In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity... |
https://en.wikipedia.org/wiki/Arithmetic | Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Pean... |
https://en.wikipedia.org/wiki/Algebraic%20closure | In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebrai... |
https://en.wikipedia.org/wiki/Alternative%20algebra | In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
for all x and y in the algebra.
Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions.
The asso... |
https://en.wikipedia.org/wiki/Arithmetic%20function | In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetica... |
https://en.wikipedia.org/wiki/Ascending%20chain%20condition | In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important role in the development of the structure theory of commutative rings in th... |
https://en.wikipedia.org/wiki/Baseball%20statistics | Baseball statistics play an important role in evaluating the progress of a player or team.
Since the flow of a baseball game has natural breaks to it, and normally players act individually rather than performing in clusters, the sport lends itself to easy record-keeping and statistics. Statistics have been recorded si... |
https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20total%20bases%20leaders | In baseball statistics, total bases (TB) is the number of bases a player has gained with hits. It is a weighted sum for which the weight value is 1 for a single, 2 for a double, 3 for a triple and 4 for a home run. Only bases attained from hits count toward this total. Reaching base by other means (such as a base on ba... |
https://en.wikipedia.org/wiki/Hit%20%28baseball%29 | In baseball statistics, a hit (denoted by H), also called a base hit, is credited to a batter when the batter safely reaches or passes first base after hitting the ball into fair territory with neither the benefit of an error nor a fielder's choice.
Scoring a hit
To achieve a hit, the batter must reach first base befo... |
https://en.wikipedia.org/wiki/On-base%20percentage | In baseball statistics, on-base percentage (OBP) measures how frequently a batter reaches base. An official Major League Baseball (MLB) statistic since 1984, it is sometimes referred to as on-base average (OBA), as it is rarely presented as a true percentage.
Generally defined as "how frequently a batter reaches base ... |
https://en.wikipedia.org/wiki/Binary | Binary may refer to:
Science and technology
Mathematics
Binary number, a representation of numbers using only two digits (0 and 1)
Binary function, a function that takes two arguments
Binary operation, a mathematical operation that takes two arguments
Binary relation, a relation involving two elements
Binary-co... |
https://en.wikipedia.org/wiki/Binomial%20distribution | In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (wi... |
https://en.wikipedia.org/wiki/Biostatistics | Biostatistics (also known as biometry) is a branch of statistics that applies statistical methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experiments and the interpretation of the results.
History
Biostatistics and genet... |
https://en.wikipedia.org/wiki/Binary%20relation | In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets and is a new set of ordered pairs consisting of elements from and from . It is a generalization of the more widely understood idea of a unary functio... |
https://en.wikipedia.org/wiki/Binary%20function | In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.
Precisely stated, a function is binary if there exists sets such that
where is the Cartesian product of and
Alternative definitions
Set-theoretically, a binary function can be rep... |
https://en.wikipedia.org/wiki/Binary%20operation | In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary operation on a set is a binary operation whose two domains and the codomain are... |
https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can b... |
https://en.wikipedia.org/wiki/Banach%20space | In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always c... |
https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam%20theorem | In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Formally: if is continuous ... |
https://en.wikipedia.org/wiki/BQP | In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. It is the quantum analogue to the complexity class BPP.
A decision problem is a member of BQP... |
https://en.wikipedia.org/wiki/Brouwer%20fixed-point%20theorem | Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself, there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interv... |
https://en.wikipedia.org/wiki/Boltzmann%20distribution | In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is express... |
https://en.wikipedia.org/wiki/Bill%20Schelter | William Frederick Schelter (1947 – July 30, 2001) was a professor of mathematics at The University of Texas at Austin and a Lisp developer and programmer. Schelter is credited with the development of the GNU Common Lisp (GCL) implementation of Common Lisp and the GPL'd version of the computer algebra system Macsyma ca... |
https://en.wikipedia.org/wiki/Borel%20measure | In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
Formal definition
Let be a locally compact Hausdorff space, and let be t... |
https://en.wikipedia.org/wiki/Bilinear%20map | In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let and be three vector spaces over the same base field . A bilinear map is a funct... |
https://en.wikipedia.org/wiki/Bra%E2%80%93ket%20notation | Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mecha... |
https://en.wikipedia.org/wiki/Banach%20algebra | In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the n... |
https://en.wikipedia.org/wiki/Binomial%20coefficient | In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be comput... |
https://en.wikipedia.org/wiki/Binomial%20theorem | In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of eac... |
https://en.wikipedia.org/wiki/Bernoulli%27s%20inequality | In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants:
Integer exponent
Case 1: for every integer and real number . The inequality is strict if and .
Case 2: for every int... |
https://en.wikipedia.org/wiki/Bayesian%20probability | Bayesian probability ( or ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.
The Bayesian interpretation of probability... |
https://en.wikipedia.org/wiki/Naive%20set%20theory | Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (... |
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