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Group (mathematics)

Group (mathematics)
Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity. One of the most familiar groups is the set of integers Z which consists of the numbers ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...,[3] together with addition. The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. Closure Associativity Related:  Abstract AlgebraSET THEORY

Algebra "Algebraist" redirects here. For the novel by Iain M. Banks, see The Algebraist. The quadratic formula expresses the solution of the degree two equation in terms of its coefficients , where is not equal to Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.[6] For example, in the letter is unknown, but the law of inverses can be used to discover its value: . , the letters and are variables, and the letter The word algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. How to distinguish between different meanings of "algebra" For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Algebra as a branch of mathematics can be any numbers whatsoever (except that cannot be Etymology History Early history of algebra History of algebra

Dynkin diagram The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups; the and directed diagrams yield the same undirected diagram, correspondingly named In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named. Classification of semisimple Lie algebras[edit] The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields. Dropping the direction on the graph edges corresponds to replacing a root system by the finite reflection group it generates, the so-called Weyl group, and thus undirected Dynkin diagrams classify Weyl groups. Related classifications[edit] The root lattice generated by the root system, as in the E8 lattice. corresponds to Example: A2[edit] for or

Set-builder notation Direct, ellipses, and informally specified sets[edit] A set is an unordered list of elements. The elements are also called set 'members'. Elements can be any mathematical entity. is a set holding the four numbers 3, 7, 15, and 31. is the set containing 'a','b', and 'c'. is the set of natural numbers. is the set of integers between 1 and 100 inclusive. all addresses on Pine Street is the set of all addresses on Pine Street. The ellipses means that the simplest interpretation should be applied for continuing a sequence. In the last example we use simple prose to describe what is in the set. The ellipses and simple prose approaches give the reader rules for building the set rather than directly presenting the elements. Formal set builder notation sets[edit] A set in set builder notation has three parts, a variable, a colon or vertical bar separator, and a logical predicate. or The x is taken to be a free variable. All values of x where the rule is true belong in the set. The is the set ,

Modular arithmetic Time-keeping on this clock uses arithmetic modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. For a positive integer n, two integers a and b are said to be congruent modulo n, and written as History[edit] In the Third Century B.C.E., Euclid formalized, in his book Elements, the fundamentals of arithmetic, as well as showing his lemma, which he used to prove the Fundamental theorem of arithmetic. Congruence relation[edit] For example, because 38 − 14 = 24, which is a multiple of 12. The same rule holds for negative values: Equivalently, can also be thought of as asserting that the remainders of the division of both and by are the same. If then:

Orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere ( ), the frieze groups and wallpaper groups of the Euclidean plane ( ), and their analogues on the hyperbolic plane ( Definition of the notation[edit] The following types of Euclidean transformation can occur in a group described by orbifold notation: All translations which occur are assumed to form a discrete subgroup of the group symmetries being described. Each symbol corresponds to a distinct transformation: Good orbifolds[edit] John H.

Cardinality The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A. Comparing sets[edit] Definition 1: | A | = | B |[edit] For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. Definition 2: | A | ≥ | B |[edit] A has cardinality greater than or equal to the cardinality of B if there exists an injective function from B into A. Definition 3: | A | > | B |[edit] A has cardinality strictly greater than the cardinality of B if there is an injective function, but no bijective function, from B to A. If | A | ≥ | B | and | B | ≥ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). Cardinal numbers[edit] Above, "cardinality" was defined functionally. For each .

Newton's laws of motion First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.[2][3]Second law: F = ma. The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object.Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.[4] Newton used them to explain and investigate the motion of many physical objects and systems.[5] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. Overview Newton's first law Impulse

Orthogonality The line segments AB and CD are orthogonal to each other. The concept of orthogonality has been broadly generalized in mathematics, science, and engineering, especially since the beginning of the 16th century. Much of the generalizing has taken place in the areas of mathematical functions, calculus and linear algebra. Etymology[edit] The word comes from the Greek ὀρθός (orthos), meaning "upright", and γωνία (gonia), meaning "angle". Mathematics[edit] Definitions[edit] A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. A vector space with a bilinear form generalizes the case of an inner product. Euclidean vector spaces[edit] Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin (by the definition of a vector subspace). Orthogonal functions[edit] where

Giuseppe Peano Giuseppe Peano (Italian: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. Biography[edit] Giuseppe Peano and his wife Carola Crosio in 1887 In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891.[3] In 1891 Peano started the Formulario Project. In 1898 he presented a note to the Academy about binary numeration and its ability to be used to represent the sounds of languages. Paris was the venue for the Second International Congress of Mathematicians in 1900. Peano's students Mario Pieri and Alessandro Padoa had papers presented at the philosophy congress also. The year 1908 was important for Peano. Milestones and honors received[edit] See also[edit] Bibliography[edit] 1889.