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In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle GL(n,\mathbb {R} )} , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix g {\displaystyle g} to an endomorphism of the vector space of all linear transformations of R n {\displaystyle \mathbb {R} ^{n}} defined by: x ↦ g x g − 1 {\displaystyle x\mapsto gxg^{-1}} . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.
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https://en.wikipedia.org/wiki/Adjoint_endomorphism
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In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group G(k((t))) and which describes the representation theory of the Langlands dual group LG through what is known as the geometric Satake correspondence.
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https://en.wikipedia.org/wiki/Infinite_grassmannian
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In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line. Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars the real or complex field, then the affine group is a Lie group.
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https://en.wikipedia.org/wiki/Affine_general_linear_group
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In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff(S) of S is the set of all affine combinations of elements of S, that is, aff ( S ) = { ∑ i = 1 k α i x i | k > 0 , x i ∈ S , α i ∈ R , ∑ i = 1 k α i = 1 } . {\displaystyle \operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}\,{\Bigg |}\,k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.}
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https://en.wikipedia.org/wiki/Affine_span
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In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Affine_q-Krawtchouk_polynomials
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In mathematics, the affinely extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two infinity elements: + ∞ {\displaystyle +\infty } and − ∞ , {\displaystyle -\infty ,} where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R ¯ {\displaystyle {\overline {\mathbb {R} }}} or {\displaystyle } or R ∪ { − ∞ , + ∞ } .
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https://en.wikipedia.org/wiki/Upper-extended_real_line
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{\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}.} It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol + ∞ {\displaystyle +\infty } is often written simply as ∞ . {\displaystyle \infty .} There is also the projectively extended real line where + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } are not distinguished so the infinity is denoted by only ∞ {\displaystyle \infty } .
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https://en.wikipedia.org/wiki/Upper-extended_real_line
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In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being ∅ {\displaystyle \varnothing } and the top being the universe set under consideration.
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https://en.wikipedia.org/wiki/Algebra_of_sets
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In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation x 6 + y 6 = x 2 . {\displaystyle x^{6}+y^{6}=x^{2}.} The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus seven.
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https://en.wikipedia.org/wiki/Butterfly_curve_(algebraic)
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The only plane curves of genus seven are singular, since seven is not a triangular number, and the minimum degree for such a curve is six. The butterfly curve has branching number and multiplicity two, and hence the singularity link has two components, pictured at right. The area of the algebraic butterfly curve is given by (with gamma function Γ {\displaystyle \Gamma } ) 4 ⋅ ∫ 0 1 ( x 2 − x 6 ) 1 6 d x = Γ ( 1 6 ) ⋅ Γ ( 1 3 ) 3 π ≈ 2.804 , {\displaystyle 4\cdot \int _{0}^{1}(x^{2}-x^{6})^{\frac {1}{6}}dx={\frac {\Gamma ({\frac {1}{6}})\cdot \Gamma ({\frac {1}{3}})}{3{\sqrt {\pi }}}}\approx 2.804,} and its arc length s by s ≈ 9.017. {\displaystyle s\approx 9.017.}
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https://en.wikipedia.org/wiki/Butterfly_curve_(algebraic)
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In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G. This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are not images of points in the closure in the algebraic topology. This fundamental distinction is behind the phenomenon of hyperbolic Dehn surgery and plays an important role in the general theory of hyperbolic 3-manifolds.
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https://en.wikipedia.org/wiki/Algebraic_topology_(object)
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In mathematics, the amoeba order is the partial order of open subsets of 2ω of measure less than 1/2, ordered by reverse inclusion. Amoeba forcing is forcing with the amoeba order; it adds a measure 1 set of random reals. There are several variations, where 2ω is replaced by the real numbers or a real vector space or the unit interval, and the number 1/2 is replaced by some positive number ε. The name "amoeba order" come from the fact that a subset in the amoeba order can "engulf" a measure zero set by extending a "pseudopod" to form a larger subset in the order containing this measure zero set, which is analogous to the way an amoeba eats food. The amoeba order satisfies the countable chain condition.
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https://en.wikipedia.org/wiki/Amoeba_order
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In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually.
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https://en.wikipedia.org/wiki/Amplitwist
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In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik Ivar Fredholm.
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https://en.wikipedia.org/wiki/Analytic_Fredholm_theorem
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In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s. It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences.
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https://en.wikipedia.org/wiki/Analytic_subgroup_theorem
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In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODE's). It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique. The phrase undetermined coefficients can also be used to refer to the step in the annihilator method in which the coefficients are calculated. The annihilator method is used as follows.
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https://en.wikipedia.org/wiki/Annihilator_method
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Given the ODE P ( D ) y = f ( x ) {\displaystyle P(D)y=f(x)} , find another differential operator A ( D ) {\displaystyle A(D)} such that A ( D ) f ( x ) = 0 {\displaystyle A(D)f(x)=0} . This operator is called the annihilator, hence the name of the method.
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https://en.wikipedia.org/wiki/Annihilator_method
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Applying A ( D ) {\displaystyle A(D)} to both sides of the ODE gives a homogeneous ODE ( A ( D ) P ( D ) ) y = 0 {\displaystyle {\big (}A(D)P(D){\big )}y=0} for which we find a solution basis { y 1 , … , y n } {\displaystyle \{y_{1},\ldots ,y_{n}\}} as before. Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combination to satisfy the ODE. This method is not as general as variation of parameters in the sense that an annihilator does not always exist.
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https://en.wikipedia.org/wiki/Annihilator_method
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In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal.
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https://en.wikipedia.org/wiki/Annihilator_(ring_theory)
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In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable.
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https://en.wikipedia.org/wiki/Annulus_theorem
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In mathematics, the antilimit is the equivalent of a limit for a divergent series. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its radius of convergence.
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https://en.wikipedia.org/wiki/Antilimit
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In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized. In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or arguments, at which the function outputs are as large as possible.
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https://en.wikipedia.org/wiki/Arg_max
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In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
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https://en.wikipedia.org/wiki/Arithmetic_genus
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In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
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https://en.wikipedia.org/wiki/Manin–Mumford_conjecture
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In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory.
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https://en.wikipedia.org/wiki/Arithmetic_zeta_function
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In mathematics, the arithmetic–geometric mean of two positive real numbers x and y 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 (an) and (gn) as These two sequences converge to the same number, the arithmetic–geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y) or AGM(x, y). The arithmetic–geometric mean is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing π. The arithmetic–geometric mean can be extended to complex numbers and when the branches of the square root are allowed to be taken inconsistently, it is, in general, a multivalued function.
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https://en.wikipedia.org/wiki/Arithmetic-geometric_mean
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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 the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.
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https://en.wikipedia.org/wiki/Descending_chain_condition
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In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials.
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https://en.wikipedia.org/wiki/Associated_Legendre_function
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In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. In that case the parameters are usually labelled with Greek letters.
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https://en.wikipedia.org/wiki/Associated_Legendre_function
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The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
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https://en.wikipedia.org/wiki/Associated_Legendre_function
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In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: gr I R = ⊕ n = 0 ∞ I n / I n + 1 {\displaystyle \operatorname {gr} _{I}R=\oplus _{n=0}^{\infty }I^{n}/I^{n+1}} .Similarly, if M is a left R-module, then the associated graded module is the graded module over gr I R {\displaystyle \operatorname {gr} _{I}R}: gr I M = ⊕ n = 0 ∞ I n M / I n + 1 M {\displaystyle \operatorname {gr} _{I}M=\oplus _{n=0}^{\infty }I^{n}M/I^{n+1}M} .
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https://en.wikipedia.org/wiki/Associated_graded_module
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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 occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.
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https://en.wikipedia.org/wiki/Associative_property
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That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
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https://en.wikipedia.org/wiki/Associative_property
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Associativity is not the same as commutativity, which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. However, operations such as function composition and matrix multiplication are associative, but not (generally) commutative.
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https://en.wikipedia.org/wiki/Associative_property
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Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.
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https://en.wikipedia.org/wiki/Associative_property
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In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatment because random dynamical systems are necessarily non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense.
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https://en.wikipedia.org/wiki/Pullback_attractor
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In mathematics, the authors are usually listed in alphabetical order (the so-called Hardy-Littlewood Rule).
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https://en.wikipedia.org/wiki/Academic_authorship
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In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group Aut ( X ) {\displaystyle \operatorname {Aut} (X)} is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory.
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https://en.wikipedia.org/wiki/Automorphism_group
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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 construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an indexed set ( x i ) i ∈ I {\displaystyle (x_{i})_{i\in I}} such that x i ∈ S i {\displaystyle x_{i}\in S_{i}} for every i ∈ I {\displaystyle i\in I} . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishing property that happens to hold for exactly one element in each set.
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https://en.wikipedia.org/wiki/Axiom_of_Choice
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An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function.
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https://en.wikipedia.org/wiki/Axiom_of_Choice
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Even if infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements.
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https://en.wikipedia.org/wiki/Axiom_of_Choice
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However, no definite choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly.
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https://en.wikipedia.org/wiki/Axiom_of_Choice
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For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that forms a set out of selecting one sock from each pair, without invoking the axiom of choice.Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
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https://en.wikipedia.org/wiki/Axiom_of_Choice
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In mathematics, the axiom of dependent choice, denoted by D C {\displaystyle {\mathsf {DC}}} , is a weak form of the axiom of choice ( A C {\displaystyle {\mathsf {AC}}} ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.
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https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
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In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis.
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https://en.wikipedia.org/wiki/Axiom_of_determinacy
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Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research. Assuming the existence of some uncountable cardinal numbers analogous to ℵ 0 {\displaystyle \aleph _{0}} , they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
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https://en.wikipedia.org/wiki/Axiom_of_determinacy
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In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if ( S α ) α ∈ A {\displaystyle (S_{\alpha })_{\alpha \in A}} is a family of non-empty finite sets, then ∏ α ∈ A S α ≠ ∅ {\displaystyle \prod _{\alpha \in A}S_{\alpha }\neq \emptyset } (set-theoretic product). : 14 If every set can be linearly ordered, the axiom of finite choice follows. : 17
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https://en.wikipedia.org/wiki/Axiom_of_finite_choice
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In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: ∀ x ∃ y ∀ z {\displaystyle \forall x\,\exists y\,\forall z\,} where y is the power set of x, P ( x ) {\displaystyle {\mathcal {P}}(x)} . In English, this says: Given any set x, there is a set P ( x ) {\displaystyle {\mathcal {P}}(x)} such that, given any set z, this set z is a member of P ( x ) {\displaystyle {\mathcal {P}}(x)} if and only if every element of z is also an element of x.More succinctly: for every set x {\displaystyle x} , there is a set P ( x ) {\displaystyle {\mathcal {P}}(x)} consisting precisely of the subsets of x {\displaystyle x} . Note the subset relation ⊆ {\displaystyle \subseteq } is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, ∈ {\displaystyle \in } .
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https://en.wikipedia.org/wiki/Power_set_axiom
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By the axiom of extensionality, the set P ( x ) {\displaystyle {\mathcal {P}}(x)} is unique. The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
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https://en.wikipedia.org/wiki/Power_set_axiom
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In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following: The axiom of real determinacy is a stronger version of the axiom of determinacy (AD), which makes the same statement about games where both players choose integers; ADR is inconsistent with the axiom of choice. It also implies the existence of inner models with certain large cardinals. ADR is equivalent to AD plus the axiom of uniformization.
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https://en.wikipedia.org/wiki/Axiom_of_real_determinacy
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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: ∀ x ( x ≠ ∅ → ∃ y ( y ∈ x ∧ y ∩ x = ∅ ) ) . {\displaystyle \forall x\,(x\neq \varnothing \rightarrow \exists y(y\in x\ \land y\cap x=\varnothing )).} The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
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https://en.wikipedia.org/wiki/Axiom_of_Regularity
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The axiom was introduced by von Neumann (1925); it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on { ( n , α ) ∣ n ∈ ω ∧ α is an ordinal } .
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https://en.wikipedia.org/wiki/Axiom_of_Regularity
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{\displaystyle \{(n,\alpha )\mid n\in \omega \land \alpha {\text{ is an ordinal }}\}\,.} Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent. In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.
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https://en.wikipedia.org/wiki/Axiom_of_Regularity
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In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction (geometry) of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.
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https://en.wikipedia.org/wiki/Axis_angle
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By Rodrigues' rotation formula, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the right-hand rule. The rotation axis is sometimes called the Euler axis. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis. It is one of many rotation formalisms in three dimensions.
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https://en.wikipedia.org/wiki/Axis_angle
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In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy-plane. A special case of an azimuth angle is the angle in polar coordinates of the component of the vector in the xy-plane, although this angle is normally measured in radians rather than degrees and denoted by θ rather than φ.
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https://en.wikipedia.org/wiki/Azimuth_angle
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In mathematics, the ba space b a ( Σ ) {\displaystyle ba(\Sigma )} of an algebra of sets Σ {\displaystyle \Sigma } is the Banach space consisting of all bounded and finitely additive signed measures on Σ {\displaystyle \Sigma } . The norm is defined as the variation, that is ‖ ν ‖ = | ν | ( X ) . {\displaystyle \|\nu \|=|\nu |(X).}
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https://en.wikipedia.org/wiki/Ca_space
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If Σ is a sigma-algebra, then the space c a ( Σ ) {\displaystyle ca(\Sigma )} is defined as the subset of b a ( Σ ) {\displaystyle ba(\Sigma )} consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then r c a ( X ) {\displaystyle rca(X)} is the subspace of c a ( Σ ) {\displaystyle ca(\Sigma )} consisting of all regular Borel measures on X.
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https://en.wikipedia.org/wiki/Ca_space
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In mathematics, the bagpipe theorem of Peter Nyikos (1984) describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes".
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https://en.wikipedia.org/wiki/Bagpipe_theorem
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In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool in algebraic topology.
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https://en.wikipedia.org/wiki/Barycentric_subdivision
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In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: g ∗ ( R r f ∗ F ) → R r f ∗ ′ ( g ′ ∗ F ) {\displaystyle g^{*}(R^{r}f_{*}{\mathcal {F}})\to R^{r}f'_{*}(g'^{*}{\mathcal {F}})} where X ′ → g ′ X f ′ ↓ ↓ f S ′ → g S {\displaystyle {\begin{array}{rcl}X'&{\stackrel {g'}{\to }}&X\\f'\downarrow &&\downarrow f\\S'&{\stackrel {g}{\to }}&S\end{array}}} is a Cartesian square of topological spaces and F {\displaystyle {\mathcal {F}}} is a sheaf on X. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth.
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https://en.wikipedia.org/wiki/Base_change_theorems
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In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system.
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https://en.wikipedia.org/wiki/Base_flow_(random_dynamical_systems)
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In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t {\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt} for complex number inputs z 1 , z 2 {\displaystyle z_{1},z_{2}} such that ℜ ( z 1 ) , ℜ ( z 2 ) > 0 {\displaystyle \Re (z_{1}),\Re (z_{2})>0} . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.
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https://en.wikipedia.org/wiki/Euler_beta_function
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In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic monoid describing the Dyck language of balanced pairs of parentheses. Thus, it finds common applications in combinatorics, such as describing binary trees and associative algebras.
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https://en.wikipedia.org/wiki/Bicyclic_semigroup
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In mathematics, the big q-Jacobi polynomials Pn(x;a,b,c;q), introduced by Andrews & Askey (1985), are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Big_q-Jacobi_polynomials
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In mathematics, the big q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Big_q-Laguerre_polynomials
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In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q ; q , q ) {\displaystyle \displaystyle P_{n}(x;c;q)={}_{3}\phi _{2}(q^{-n},q^{n+1},x;q,cq;q,q)} .They obey the orthogonality relation ∫ c q q P m ( x ; c ; q ) P n ( x ; c ; q ) d x = q ( 1 − c ) 1 − q 1 − q 2 n + 1 ( c − 1 q ; q ) n ( c q ; q ) n ( − c q 2 ) n q ( n 2 ) δ m n {\displaystyle \int _{cq}^{q}P_{m}(x;c;q)P_{n}(x;c;q)\,dx=q(1-c){\frac {1-q}{1-q^{2n+1}}}{\frac {(c^{-1}q;q)_{n}}{(cq;q)_{n}}}(-cq^{2})^{n}q^{n \choose 2}\delta _{mn}} and have the limiting behavior lim q → 1 P n ( x ; 0 ; q ) = P n ( 2 x − 1 ) {\displaystyle \displaystyle \lim _{q\to 1}P_{n}(x;0;q)=P_{n}(2x-1)} where P n {\displaystyle P_{n}} is the n {\displaystyle n} th Legendre polynomial. == References ==
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https://en.wikipedia.org/wiki/Big_q-Legendre_polynomials
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In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.
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https://en.wikipedia.org/wiki/Biharmonic_equation
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In mathematics, the bimonster is a group that is the wreath product of the monster group M with Z2: B i = M ≀ Z 2 . {\displaystyle Bi=M\wr \mathbb {Z} _{2}.\,} The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16 nodes: John H. Conway conjectured that a presentation of the bimonster could be given by adding a certain extra relation to the presentation defined by the Y555 diagram; this was proved in 1990 by A. A. Ivanov a mathematician not the famous painter and Simon P. Norton.
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https://en.wikipedia.org/wiki/Bimonster_group
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In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, C 2 n {\displaystyle C_{2n}} , thought of as an extension of the cyclic group C n {\displaystyle C_{n}} by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or +. It is the binary polyhedral group corresponding to the cyclic group.In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations ( C n < SO ( 3 ) {\displaystyle C_{n}<\operatorname {SO} (3)} ) under the 2:1 covering homomorphism Spin ( 3 ) → SO ( 3 ) {\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)\,} of the special orthogonal group by the spin group. As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ( 3 ) ≅ Sp ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)} where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
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https://en.wikipedia.org/wiki/Binary_cyclic_group
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In mathematics, the binary game is a topological game introduced by Stanislaw Ulam in 1935 in an addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game. In the binary game, one is given a fixed subset X of the set {0,1}N of all sequences of 0s and 1s. The players take it in turn to choose a digit 0 or 1, and the first player wins if the sequence they form lies in the set X. Another way to represent this game is to pick a subset X {\displaystyle X} of the interval {\displaystyle } on the real line, then the players alternatively choose binary digits x 0 , x 1 , x 2 , . .
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https://en.wikipedia.org/wiki/Binary_game
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. {\displaystyle x_{0},x_{1},x_{2},...} . Player I wins the game if and only if the binary number ( x 0 .
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https://en.wikipedia.org/wiki/Binary_game
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x 1 x 2 x 3 . . . )
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https://en.wikipedia.org/wiki/Binary_game
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2 ∈ X {\displaystyle (x_{0}{}.x_{1}{}x_{2}{}x_{3}{}...)_{2}\in {}X} , that is, Σ n = 0 ∞ x n 2 n ∈ X {\displaystyle \Sigma _{n=0}^{\infty }{\frac {x_{n}}{2^{n}}}\in {}X} . See, page 237. The binary game is sometimes called Ulam's game, but "Ulam's game" usually refers to the Rényi–Ulam game. == References ==
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https://en.wikipedia.org/wiki/Binary_game
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In mathematics, the binary icosahedral group 2I or ⟨2,3,5⟩ is a certain nonabelian group of order 120. It is an extension of the icosahedral group I or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism Spin ( 3 ) → SO ( 3 ) {\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)\,} of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.
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https://en.wikipedia.org/wiki/Binary_icosahedral_group
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It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3). The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ( 3 ) ≅ Sp ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)} where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
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https://en.wikipedia.org/wiki/Binary_icosahedral_group
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In mathematics, the binary logarithm (log2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, x = log 2 n ⟺ 2 x = n . {\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.} For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5. The binary logarithm is the logarithm to the base 2 and is the inverse function of the power of two function.
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https://en.wikipedia.org/wiki/Base-2_logarithm
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As well as log2, an alternative notation for the binary logarithm is lb (the notation preferred by ISO 31-11 and ISO 80000-2). Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory.
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https://en.wikipedia.org/wiki/Base-2_logarithm
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In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography.
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https://en.wikipedia.org/wiki/Base-2_logarithm
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Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value. The fractional part of the logarithm can be calculated efficiently.
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https://en.wikipedia.org/wiki/Base-2_logarithm
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In mathematics, the binary logarithm of a number n is often written as log2 n. However, several other notations for this function have been used or proposed, especially in application areas. Some authors write the binary logarithm as lg n, the notation listed in The Chicago Manual of Style. Donald Knuth credits this notation to a suggestion of Edward Reingold, but its use in both information theory and computer science dates to before Reingold was active.
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https://en.wikipedia.org/wiki/Dyadic_logarithm
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The binary logarithm has also been written as log n with a prior statement that the default base for the logarithm is 2. Another notation that is often used for the same function (especially in the German scientific literature) is ld n, from Latin logarithmus dualis or logarithmus dyadis. The DIN 1302, ISO 31-11 and ISO 80000-2 standards recommend yet another notation, lb n. According to these standards, lg n should not be used for the binary logarithm, as it is instead reserved for the common logarithm log10 n.
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https://en.wikipedia.org/wiki/Dyadic_logarithm
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In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism Spin ( 3 ) → SO ( 3 ) {\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)} of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48. The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ( 3 ) ≅ Sp ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)} where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
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https://en.wikipedia.org/wiki/Binary_octahedral_group
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In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C.
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https://en.wikipedia.org/wiki/Binary_tetrahedral_group
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Shephard or 33 and by Coxeter, is isomorphic to the binary tetrahedral group. The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
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https://en.wikipedia.org/wiki/Binary_tetrahedral_group
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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 n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.} It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula ( n k ) = n × ( n − 1 ) × ⋯ × ( n − k + 1 ) k × ( k − 1 ) × ⋯ × 1 , {\displaystyle {\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},} which using factorial notation can be compactly expressed as ( n k ) = n !
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https://en.wikipedia.org/wiki/Binomial_coefficient
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k ! ( n − k ) ! .
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https://en.wikipedia.org/wiki/Binomial_coefficient
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{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.} For example, the fourth power of 1 + x is ( 1 + x ) 4 = ( 4 0 ) x 0 + ( 4 1 ) x 1 + ( 4 2 ) x 2 + ( 4 3 ) x 3 + ( 4 4 ) x 4 = 1 + 4 x + 6 x 2 + 4 x 3 + x 4 , {\displaystyle {\begin{aligned}(1+x)^{4}&={\tbinom {4}{0}}x^{0}+{\tbinom {4}{1}}x^{1}+{\tbinom {4}{2}}x^{2}+{\tbinom {4}{3}}x^{3}+{\tbinom {4}{4}}x^{4}\\&=1+4x+6x^{2}+4x^{3}+x^{4},\end{aligned}}} and the binomial coefficient ( 4 2 ) = 4 × 3 2 × 1 = 4 ! 2 !
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https://en.wikipedia.org/wiki/Binomial_coefficient
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2 ! = 6 {\displaystyle {\tbinom {4}{2}}={\tfrac {4\times 3}{2\times 1}}={\tfrac {4!}{2!2! }}=6} is the coefficient of the x2 term.
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https://en.wikipedia.org/wiki/Binomial_coefficient
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Arranging the numbers ( n 0 ) , ( n 1 ) , … , ( n n ) {\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}} in successive rows for n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\ldots } gives a triangular array called Pascal's triangle, satisfying the recurrence relation ( n k ) = ( n − 1 k − 1 ) + ( n − 1 k ) . {\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}}.} The binomial coefficients occur in many areas of mathematics, and especially in combinatorics.
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https://en.wikipedia.org/wiki/Binomial_coefficient
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The symbol ( n k ) {\displaystyle {\tbinom {n}{k}}} is usually read as "n choose k" because there are ( n k ) {\displaystyle {\tbinom {n}{k}}} ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ( 4 2 ) = 6 {\displaystyle {\tbinom {4}{2}}=6} ways to choose 2 elements from { 1 , 2 , 3 , 4 } , {\displaystyle \{1,2,3,4\},} namely { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , {\displaystyle \{1,2\},\,\{1,3\},\,\{1,4\},\,\{2,3\},\,\{2,4\},} and { 3 , 4 } . {\displaystyle \{3,4\}.} The binomial coefficients can be generalized to ( z k ) {\displaystyle {\tbinom {z}{k}}} for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.
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https://en.wikipedia.org/wiki/Binomial_coefficient
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In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions. For example: ( y ′ ) m = f ( x , y ) , {\displaystyle \left(y'\right)^{m}=f(x,y),} when m {\displaystyle m} is a natural number (i.e., a positive integer), and f ( x , y ) {\displaystyle f(x,y)} is a polynomial in two variables (i.e., a bivariate polynomial).
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https://en.wikipedia.org/wiki/Binomial_differential_equation
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In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like ( 1 + x ) n {\displaystyle (1+x)^{n}} for a nonnegative integer n {\displaystyle n} . Specifically, the binomial series is the Taylor series for the function f ( x ) = ( 1 + x ) α {\displaystyle f(x)=(1+x)^{\alpha }} centered at x = 0 {\displaystyle x=0} , where α ∈ C {\displaystyle \alpha \in \mathbb {C} } and | x | < 1 {\displaystyle |x|<1} . Explicitly, where the power series on the right-hand side of (1) is expressed in terms of the (generalized) binomial coefficients ( α k ) := α ( α − 1 ) ( α − 2 ) ⋯ ( α − k + 1 ) k ! . {\displaystyle {\binom {\alpha }{k}}:={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.}
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https://en.wikipedia.org/wiki/Binomial_series
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In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem. : 76–77
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https://en.wikipedia.org/wiki/Bipolar_theorem
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In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow.
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https://en.wikipedia.org/wiki/Method_of_bisection
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Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method.For polynomials, more elaborate methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation.
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https://en.wikipedia.org/wiki/Method_of_bisection
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In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a Blancmange pudding. It is a special case of the more general de Rham curve; see also fractal curve.
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https://en.wikipedia.org/wiki/Blancmange_curve
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In mathematics, the bounded inverse theorem ( also called inverse mapping theorem or Banach isomorphism theorem) is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed graph theorem.
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https://en.wikipedia.org/wiki/Bounded_inverse_theorem
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In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "Raster Scan" sawtooth-like manner.
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https://en.wikipedia.org/wiki/Boustrophedon_transform
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In mathematics, the box-counting content is an analog of Minkowski content.
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https://en.wikipedia.org/wiki/Box-counting_content
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