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Concrete category Summary Concrete_category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible to think of the objects of the category as sets with addition... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Condensation point Summary Condensation_point In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonymous with " ℵ 1 {\displaystyle \aleph _{1}} -accumulation point". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conference matrix Summary Conference_matrix In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I. Some authors use a more general defini... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conference matrix Summary Conference_matrix Other applications are in statistics, and another is in elliptic geometry.For n > 1, there are two kinds of conference matrix. Let us normalize C by, first (if the more general definition is used), rearranging the rows so that all the zeros are on the diagonal, and then negat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conference matrix Summary Conference_matrix Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal. Let S be the matrix that remains when the first row and column of C are removed. Then either n is evenly even (a multiple of 4), and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Configuration space (mathematics) Summary Configuration_space_(mathematics) In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Confluent hypergeometric series Summary Confluent_hypergeometric_function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Confluent hypergeometric series Summary Confluent_hypergeometric_function There is a different and unrelated Kummer's function bearing the same name. Tricomi's (confluent hypergeometric) function U(a, b, z) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(a; b; z), is another solution to Kummer's equation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Confluent hypergeometric series Summary Confluent_hypergeometric_function This is also known as the confluent hypergeometric function of the second kind. Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation. Coulomb wave functions are solutions to the Coulomb wave equation.The Kummer ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal transformation Summary Conformal_mapping In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U {\displaystyle U} and V {\displaystyle V} be open subsets of R n {\displaystyle \mathbb {R} ^{n}} . A function f: U → V {\displaystyle f:U\to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal transformation Summary Conformal_mapping The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Table of congruences Summary Table_of_congruences In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Congruence subgroup Summary Principal_congruence_subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conic helix Summary Conical_helix In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor projection is a plane spiral. If the floor projection is a logarithmic spiral, it is called conchospiral (from conch). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical conjecture Summary Conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connector (mathematics) Summary Connector_(mathematics) In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conservative system Summary Conservative_system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely spe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant map Summary Constant_function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant term Summary Constant_term In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial x 2 + 2 x + 3 , {\displaystyle x^{2}+2x+3,\ } the 3 is a constant term.A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant term Summary Constant_term If the constant term is 0, then it will conventionally be omitted when the quadratic is written out. Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x 0 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant term Summary Constant_term {\displaystyle x^{0}.} In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant term Summary Constant_term For example, the polynomial x 2 + 2 x y + y 2 − 2 x + 2 y − 4 {\displaystyle x^{2}+2xy+y^{2}-2x+2y-4\ } has a constant term of −4, which can be considered to be the coefficient of x 0 y 0 , {\displaystyle x^{0}y^{0},} where the variables are eliminated by being exponentiated to 0 (an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-binding constraint Summary Mathematical_constraints In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constructible polygon Summary Constructible_polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constructible sheaf Summary Constructible_sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-constructive proof Summary Weak_counterexample In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure exist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-constructive proof Summary Weak_counterexample Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continued fraction Summary Convergent_(continued_fraction) In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continued fraction Summary Convergent_(continued_fraction) The integers a i {\displaystyle a_{i}} are called the coefficients or terms of the continued fraction.It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerator... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continued fraction Summary Convergent_(continued_fraction) Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number p {\displaystyle p} / q {\displaystyle q} has two closely related expressions as a finite continued fraction, whose... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continued fraction Summary Convergent_(continued_fraction) Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α {\displaystyle \alpha } is the value of a unique infinite regular continu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continued fraction Summary Convergent_(continued_fraction) This way of expressing real numbers (rational and irrational) is called their continued fraction representation. The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous (topology) Summary Topological_continuity In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous (topology) Summary Topological_continuity A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous (topology) Summary Topological_continuity Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous (topology) Summary Topological_continuity A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity. As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous module Summary Continuous_module In mathematics, a continuous module is a module M such that every submodule of M is essential in a direct summand and every submodule of M isomorphic to a direct summand is itself a direct summand. The endomorphism ring of a continuous module is a clean ring. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous-time random walk Summary Continuous-time_random_walk In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. Mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuum structure function Summary Continuum_structure_function In mathematics, a continuum structure function (CSF) is defined by Laurence Baxter as a nondecreasing mapping from the unit hypercube to the unit interval. It is used by Baxter to help in the Mathematical modelling of the level of performance of a system... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Contraction mapping Summary Contraction_mapping In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number 0 ≤ k < 1 {\displaystyle 0\leq k<1} such that for all x and y in M, d ( f ( x ) , f ( y ) ) ≤ k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Contraction mapping Summary Contraction_mapping More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then f: M → N {\displaystyle f:M\rightarrow N} is a contractive mapping if there is a constant 0 ≤ k < 1 {\displaystyle 0\le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Contraction mapping Summary Contraction_mapping Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Contraharmonic mean Summary Contraharmonic_mean In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L p {\displaystyle L_{p}} , where p = 2. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convergence group Summary Convergence_group In mathematics, a convergence group or a discrete convergence group is a group Γ {\displaystyle \Gamma } acting by homeomorphisms on a compact metrizable space M {\displaystyle M} in a way that generalizes the properties of the action of Kleinian group by Möbius transformatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convergence space Summary Convergence_space In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergenc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convergence space Summary Convergence_space Many topological properties have generalizations to convergence spaces. Besides its ability to describe notions of convergence that topologies are unable to, the category of convergence spaces has an important categorical property that the category of topological spaces lacks... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convex body Summary Convex_body In mathematics, a convex body in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a compact convex set with non-empty interior. A convex body K {\displaystyle K} is called symmetric if it is centrally symmetric with respect to the origin; that is t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convex body Summary Convex_body Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on R n . {\displaystyle \mathbb {R} ^{n}.} Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convex space Summary Convex_space In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coordinate-induced basis Summary Coordinate-induced_basis In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system x a {\displaystyle x^{a}} , the coordinate-induced basis e a {\displaystyle e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corestriction Summary Corestriction In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corestriction Summary Corestriction Then for any function f: A → B {\displaystyle f:A\to B} , the restriction f | S: S → B {\displaystyle f|_{S}:S\to B} of a function f {\displaystyle f} onto S {\displaystyle S} can be defined as the composition f | S = f ∘ i S {\displaystyle f|_{S}=f\circ i_{S}} . Analogously, for an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corestriction Summary Corestriction In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of f {\displaystyle f} . More generally, one can consider corestriction of a morphism in general categories with images. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corestriction Summary Corestriction The term is well known in category theory, while rarely used in print.Andreotti introduces the above notion under the name coastriction, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if p U: B → U {\displaystyle p^{U}:B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corollary Overview Corollary > Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double covering group Summary Universal_covering_group In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p: G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covering number Summary Covering_number In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric entropy, the number o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covering set Summary Covering_set In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set. The term "covering set" is used only in conjunction with sequences possessing exponential growth. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covering system Summary Covering_system In mathematics, a covering system (also called a complete residue system) is a collection { a 1 ( mod n 1 ) , … , a k ( mod n k ) } {\displaystyle \{a_{1}{\pmod {n_{1}}},\ \ldots ,\ a_{k}{\pmod {n_{k}}}\}} of finitely many residue classes a i ( mod n i ) = { a i + n i x: x ∈ Z } ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Credal set Summary Credal_set In mathematics, a credal set is a set of probability distributions or, more generally, a set of (possibly only finitely additive) probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probabili... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Credal set Summary Credal_set If X {\displaystyle X} is a categorical variable, then the credal set K ( X ) {\displaystyle K(X)} can be considered as a set of probability mass functions over X {\displaystyle X} . If additionally K ( X ) {\displaystyle K(X)} is also closed and convex, then the lower prevision of a funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crunode Summary Crunode In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ordinary double point.For a plane curve, defined as the locus of points f (x, y) = 0, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic root Summary Cube_roots In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, deno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic root Summary Cube_roots In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign 3 . {\displaystyle {\sqrt{~^{~}}}.} The cube root is the inv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic form Summary Cubic_hypersurface In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve. In (Delone & Faddeev 1964), Boris Delone and Dmitry Faddeev showed th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic form Summary Cubic_hypersurface Their work was generalized in (Gan, Gross & Savin 2002, §4) to include all cubic rings (a cubic ring is a ring that is isomorphic to Z3 as a Z-module), giving a discriminant-preserving bijection between orbits of a GL(2, Z)-action on the space of integral binary cubic forms and cub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic polynomial Summary Cubic_polynomial In mathematics, a cubic function is a function of the form f ( x ) = a x 3 + b x 2 + c x + d , {\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,} that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic polynomial Summary Cubic_polynomial A cubic function with real coefficients has either one or three real roots (which may not be distinct); all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic polynomial Summary Cubic_polynomial Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only three possible graphs for cubic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to homogeneous coordinates ( x: y: z ) {\displaystyle (x:y:z)} for the projective plane; or the inhomogeneous version for the affine space... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the inter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points. The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve Like for conic sections, a line cuts this oval at, at most, two points. A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux cubic Summary Cubic_curve This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when K is the rational number field. The singular points of an irreducible plane cubic curve are quite limited: one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic surface Summary Cubic_surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubic surface Summary Cubic_surface The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface x 3 + y 3 + z 3 + w 3 = 0 {\displaystyle x^{3}+y^{3}+z^{3}+w^{3}=0} in P ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cubical complex Summary Cubical_complex In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homolog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closed curve Summary Space_curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closed curve Summary Space_curve "This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closed curve Summary Space_curve This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closed curve Summary Space_curve This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closed curve Summary Space_curve More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cusp neighborhood Summary Cusp_neighborhood In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cusp (singularity) Summary Cusp_(singularity) In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cusp (singularity) Summary Cusp_(singularity) For a plane curve defined by an analytic, parametric equation x = f ( t ) y = g ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t),\end{aligned}}} a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cusp (singularity) Summary Cusp_(singularity) For a curve defined by an implicit equation F ( x , y ) = 0 , {\displaystyle F(x,y)=0,} which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; however, not all singular points that have this proper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cuspidal cubic Summary Cuspidal_cubic In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form y 2 − a 2 x 3 = 0 {\displaystyle y^{2}-a^{2}x^{3}=0} (with a ≠ 0) in some Cartesian coordinate system. Solving for y leads to the explicit form y = ± a x 3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cuspidal cubic Summary Cuspidal_cubic Solving the implicit equation for x yields a second explicit form x = ( y a ) 2 3 . {\displaystyle x=\left({\frac {y}{a}}\right)^{\frac {2}{3}}.} The parametric equation x = t 2 , y = a t 3 {\displaystyle \quad x=t^{2},\quad y=at^{3}} can also be deduced from the implicit equation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cuspidal cubic Summary Cuspidal_cubic {\textstyle t={\frac {y}{ax}}.} The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closure with a twist Cwatset Closure_with_a_twist > Cwatset In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist. If each string in a cwatset, C, say, is of length n, then C will be a subset of Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} . Thus, two strings in C are adde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closure with a twist Cwatset Closure_with_a_twist > Cwatset Closure with a twist now means that for each element c in C, there exists some permutation p c {\displaystyle p_{c}} such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. Tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic graph Summary Cyclic_graph In mathematics, a cyclic graph may mean a graph that contains a cycle, or a graph that is a cycle, with varying definitions of cycles. See: Cycle (graph theory), a cycle in a graph Forest (graph theory), an undirected graph with no cycles Biconnected graph, an undirected graph in which... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic sequence Summary Cyclic_sequence In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic sequence Summary Cyclic_sequence For example, , but not , cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic sequence Summary Cyclic_sequence Dropping the "connected" requirement results in a partial cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal dire... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic sequence Summary Cyclic_sequence In a finite cycle, each element has a "next element" and a "previous element". There are also cyclic orders with infinitely many elements, such as the oriented unit circle in the plane. Cyclic orders are closely related to the more familiar linear orders, which arrange objects in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic sequence Summary Cyclic_sequence Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with the related constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformed into questions abou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic polytope Summary Gale_evenness_condition In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclically ordered group Summary Cyclically_ordered_group In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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