text stringlengths 42 3.65k | source stringclasses 1
value |
|---|---|
Adjoint endomorphism Summary Adjoint_action 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,\mathb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite grassmannian Summary Affine_Grassmannian 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine general linear group Summary Affine_general_linear_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine span Summary Affine_span 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) o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine q-Krawtchouk polynomials Summary Affine_q-Krawtchouk_polynomials 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 det... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Upper-extended real line Summary Negative_infinity 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Upper-extended real line Summary Negative_infinity {\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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebra of sets Summary Algebra_of_sets 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Butterfly curve (algebraic) Summary Butterfly_curve_(algebraic) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Butterfly curve (algebraic) Summary Butterfly_curve_(algebraic) 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, pic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic topology (object) Summary Algebraic_topology_(object) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Amoeba order Summary Amoeba_order 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Amplitwist Summary Amplitwist 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic Fredholm theorem Summary Analytic_Fredholm_theorem 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 Hilber... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic subgroup theorem Summary Analytic_subgroup_theorem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Annihilator method Summary Annihilator_method 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Annihilator method Summary Annihilator_method 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Annihilator method Summary Annihilator_method 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 orig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Annihilator (ring theory) Summary Annihilator_(ring_theory) 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 modu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Annulus theorem Summary Annulus_theorem 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 hom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antilimit Summary Antilimit 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arg max Summary Arg_max 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 ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic genus Summary Arithmetic_genus 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Manin–Mumford conjecture Summary Arithmetic_of_elliptic_curves 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 ve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic zeta function Summary Arithmetic_zeta_function 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 functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic-geometric mean Summary Arithmetic-geometric_mean 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Descending chain condition Summary Ascending_chain_condition 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 d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associated Legendre function Summary Associated_Legendre_polynomial 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 polynomia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associated Legendre function Summary Associated_Legendre_polynomial 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associated Legendre function Summary Associated_Legendre_polynomial 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 Lege... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associated graded module Summary Associated_graded_ring 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associative property Summary Left_associative_operator 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 logic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associative property Summary Left_associative_operator 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, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associative property Summary Left_associative_operator 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 num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associative property Summary Left_associative_operator 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 exa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pullback attractor Summary Pullback_attractor 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 sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Academic authorship Authorship in mathematics Academic_authorship > Definition > Authorship in mathematics In mathematics, the authors are usually listed in alphabetical order (the so-called Hardy-Littlewood Rule). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphism group Summary Automorphism_group_functor 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 transformat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Choice Summary Axiom_of_Choice 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Choice Summary Axiom_of_Choice 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Choice Summary Axiom_of_Choice 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Choice Summary Axiom_of_Choice 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 sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Choice Summary Axiom_of_Choice 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 ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of dependent choice Summary Dependent_choice 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of determinacy Summary Axiom_of_determinacy 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 determin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of determinacy Summary Axiom_of_determinacy 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. Hug... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of finite choice Summary Axiom_of_finite_choice 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_{\al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power set axiom Summary Powerset_axiom 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 ) {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power set axiom Summary Powerset_axiom 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 conce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of real determinacy Summary Axiom_of_real_determinacy 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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Regularity Summary Axiom_of_Regularity 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Regularity Summary Axiom_of_Regularity 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 chapte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of Regularity Summary Axiom_of_Regularity {\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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axis angle Summary Axis-angle_representation 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., c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axis angle Summary Axis-angle_representation 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 pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Azimuth angle Polar coordinate Azimuthal_angle > Related coordinates > Polar coordinate 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ca space Summary Ca_space 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ca space Summary Ca_space 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 topolo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bagpipe theorem Summary Bagpipe_theorem 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". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barycentric subdivision Summary Barycentric_subdivision 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 topol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base change theorems Summary Cohomology_base_change_theorem 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 ) {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base flow (random dynamical systems) Summary Base_flow_(random_dynamical_systems) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler beta function Summary Euler_beta_function 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 {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bicyclic semigroup Summary Bicyclic_monoid 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 la... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Big q-Jacobi polynomials Summary Big_q-Jacobi_polynomials 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 detai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Big q-Laguerre polynomials Summary Big_q-Laguerre_polynomials 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Big q-Legendre polynomials Summary Big_q-Legendre_polynomials 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)={}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Biharmonic equation Summary Biharmonic_operator 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 reac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bimonster group Summary Bimonster_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary cyclic group Summary Binary_cyclic_group 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-br... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary game Summary Binary_game 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 pla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary game Summary Binary_game . {\displaystyle x_{0},x_{1},x_{2},...} . Player I wins the game if and only if the binary number ( x 0 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary game Summary Binary_game x 1 x 2 x 3 . . . ) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary game Summary Binary_game 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 g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary icosahedral group Summary Binary_icosahedral_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary icosahedral group Summary Binary_icosahedral_group 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 quatern... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base-2 logarithm Summary Logarithmus_dyadis 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 logari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base-2 logarithm Summary Logarithmus_dyadis 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 mus... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base-2 logarithm Summary Logarithmus_dyadis 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Base-2 logarithm Summary Logarithmus_dyadis 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 valu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dyadic logarithm Notation Dyadic_logarithm > Notation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dyadic logarithm Notation Dyadic_logarithm > Notation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary octahedral group Summary Binary_octahedral_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary tetrahedral group Summary Binary_tetrahedral_group 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binary tetrahedral group Summary Binary_tetrahedral_group 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 multiplicativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients 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 coef... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients k ! ( n − k ) ! . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients {\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}{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial coefficient Summary Binomial_coefficients 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 {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial differential equation Summary Binomial_differential_equation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial series Summary Binomial_series 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 functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bipolar theorem Summary Bipolar_theorem 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 bi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Method of bisection Summary Method_of_bisection 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Method of bisection Summary Method_of_bisection 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 polynom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Blancmange curve Summary Blancmange_curve 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 Lan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bounded inverse theorem Summary Bounded_inverse_theorem 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boustrophedon transform Summary Boustrophedon_transform 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Box-counting content Summary Box-counting_content In mathematics, the box-counting content is an analog of Minkowski content. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.