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wiki_2500_chunk_0 | Complex Numbers | The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2501_chunk_0 | Complex reflection group | In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2502_chunk_0 | Complex representation | In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2503_chunk_0 | Normal matrix | In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2504_chunk_0 | Normal matrix | The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A = AA* is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces. The left and right singular vec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2505_chunk_0 | Linear complex structure | In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space. Every complex vector space can be equipped with a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2506_chunk_0 | Complex torus | In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N must be the even number 2n, where n is the complex dimension of M. All such complex structures can be obtained as follow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2507_chunk_0 | Complex torus | For n > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when n >... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2508_chunk_0 | Conjugate bundle | In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the complexification E ⊗ C ; {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2509_chunk_0 | Conjugate bundle | The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2510_chunk_0 | Multiplicative quadratic form | In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2511_chunk_0 | Composition (combinatorics) | In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2512_chunk_0 | Composition (combinatorics) | Each positive integer n has 2n−1 distinct compositions. A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak comp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2513_chunk_0 | Concave function | In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2514_chunk_0 | 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 additional structure, and of its morphisms as struct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2515_chunk_0 | 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 definition, which requires there to be a single 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2516_chunk_0 | 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 negating any row or column whose first entry is n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2517_chunk_0 | 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 S is antisymmetric (as is the normalized C i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2518_chunk_0 | Confluent hypergeometric series | 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 singularity. The term confluent refers to the merging of singular points of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2519_chunk_0 | Confluent hypergeometric series | 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 |
wiki_2520_chunk_0 | Confluent hypergeometric series | 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 functions, Whittaker functions, and Coulomb wave functions are essentially... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2521_chunk_0 | Conformal transformation | 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 V} is called conformal (or angle-preserving) at a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2522_chunk_0 | Conformal transformation | 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 authors define conformality to include orientation-r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2523_chunk_0 | 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 entries are even. More generally, the notion of congruence sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2524_chunk_0 | Mathematical 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 much of mathematical history as new areas of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2525_chunk_0 | 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 |
wiki_2526_chunk_0 | 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 speaking, they are those dynamical systems that hav... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2527_chunk_0 | Constant map | 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 |
wiki_2528_chunk_0 | 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.After like terms are combined, an alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2529_chunk_0 | 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 (any non-zero number exponentiated to 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2530_chunk_0 | Non-binding constraint | 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 satisfy all constraints is called the feasible set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2531_chunk_0 | 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 in étale cohomology constructible sheaves are de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2532_chunk_0 | Non-constructive proof | 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 existence theorem), which proves the existence of a part... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2533_chunk_0 | Non-constructive proof | 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 induces a different meaning for some terminology (f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2534_chunk_0 | 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 reciprocal, and so on. In a finite continued fraction (or te... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2535_chunk_0 | 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 coefficients ai can be determined by applying the Euclidea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2536_chunk_0 | 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 continued fraction, whose coefficients can be found using the non-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2537_chunk_0 | Continuous (topology) | 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 discontinuities. More precisely, a function is continuous if ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2538_chunk_0 | Continuous (topology) | 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 the definition of continuity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2539_chunk_0 | Continuous (topology) | 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. In contrast, the function M(t) denoting the amount of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2540_chunk_0 | 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. More generally it can be seen to be a special case of a Markov ren... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2541_chunk_0 | 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 in terms of the performance levels of its components. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2542_chunk_0 | 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 d ( x , y ) . {\displaystyle d(f(x),f(y))\leq k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2543_chunk_0 | 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\leq k<1} such that d ′ ( f ( x ) , f ( y ) ) ≤ k d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2544_chunk_0 | 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 concept is very useful for iterated function sys... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2545_chunk_0 | 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 |
wiki_2546_chunk_0 | 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 |
wiki_2547_chunk_0 | 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 are not categorically dual. Given any ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2548_chunk_0 | 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 inclusion i T: T ↪ B {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2549_chunk_0 | Corollary | 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 from A or is self-evident from its proof. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2550_chunk_0 | 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 probability model that should be used, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2551_chunk_0 | 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 function f {\displaystyle f} of X {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2552_chunk_0 | 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, where f (x, y) is a smoo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2553_chunk_0 | Cubic root | 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 inverse function of the cube func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2554_chunk_0 | Cubic form | 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 cubic rings up to isomorphism. The classi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2555_chunk_0 | 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 function is considered as a real function that m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2556_chunk_0 | 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 points, a local minimum and a local maximum. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2557_chunk_0 | 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 functions. Cubic functions are fundamental... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2558_chunk_0 | Darboux cubic | 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 determined by setting z = 1 in su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2559_chunk_0 | Darboux cubic | 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 intersections are then counted by Bézou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2560_chunk_0 | 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 are generally considered in projectiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2561_chunk_0 | Closed 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 article, these curves are sometim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2562_chunk_0 | Closed 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 indeterminates. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2563_chunk_0 | 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 tangent, changes sign (the direction of the tangen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2564_chunk_0 | 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 property are cusps. The theory of Puiseux series imp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2565_chunk_0 | 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 2 , {\displaystyle y=\pm ax^{\frac {3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2566_chunk_0 | 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 by putting t = y a x . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2567_chunk_0 | 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 every edge belongs to a cycle Dir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2568_chunk_0 | 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 about linear orders. Cycles have more symmet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2569_chunk_0 | Cyclic polytope | 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 Klee, and others. They play an important role in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2570_chunk_0 | Skorokhod space | In mathematics, a càdlàg (French: "continue à droite, limite à gauche"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2571_chunk_0 | Cesàro fractal | In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all special cases of the general de Rham curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2572_chunk_0 | Decomposable measure | In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures. There are several theorems in measure th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2573_chunk_0 | Negative-definite bilinear form | In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2574_chunk_0 | Deformation ring | In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field. In particular for any such lifting problem there is often a universal deformation ring that classifies all such liftings, and whose spectrum is the universal deformation space.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2575_chunk_0 | Constant random variable | In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter definition, it is a deterministic distribution and takes only a single valu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2576_chunk_0 | Delta operator | In mathematics, a delta operator is a shift-equivariant linear operator Q: K ⟶ K {\displaystyle Q\colon \mathbb {K} \longrightarrow \mathbb {K} } on the vector space of polynomials in a variable x {\displaystyle x} over a field K {\displaystyle \mathbb {K} } that reduces degrees by one. To say that Q {\displaystyle Q... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2577_chunk_0 | Delta-matroid | As a special case, an even delta-matroid is a delta-matroid in which either all sets have even number of elements, or all sets have an odd number of elements. If a constraint satisfaction problem has a Boolean variable on each edge of a planar graph, and if the variables of the edges incident to each vertex of the grap... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2578_chunk_0 | Dendroid (topology) | In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of X is unicoherent), arcwise connected, and forms a continuum. The term dendroid was introduced by Bronisław Knaster lecturing at the University of Wrocław, although... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2579_chunk_0 | Homogeneous derivation | In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D ( a b ) = a D ( b ) + D ( a ) b . {\displaystyle D(ab)=aD(b)+D(a)b.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2580_chunk_0 | Homogeneous derivation | The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2581_chunk_0 | Homogeneous derivation | The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. That is, = G + F {\displaystyle =G+F} where {\displaystyle } is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2582_chunk_0 | Locally nilpotent derivation | In mathematics, a derivation ∂ {\displaystyle \partial } of a commutative ring A {\displaystyle A} is called a locally nilpotent derivation (LND) if every element of A {\displaystyle A} is annihilated by some power of ∂ {\displaystyle \partial } . One motivation for the study of locally nilpotent derivations comes from... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2583_chunk_0 | Dessin d'enfant | It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2584_chunk_0 | Determinantal point process | In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, physics, and wireless network modeling. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2585_chunk_0 | Fermat hypersurface | In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is ∑ i = 1 n a i x i m {\displaystyle \sum _{i=1}^{n}a_{i}{x_{i}}^{m}\ } for some given degree m. Such forms F, and the hypersurfaces F = 0 they define in projective space, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2586_chunk_0 | Directed cut | Woodall's conjecture, an unsolved problem in this area, states that in any directed graph the minimum number of edges in a dicut (the unweighted minimum closure) equals the maximum number of disjoint dijoins that can be found in the graph (a packing of dijoins). A fractional weighted version of the conjecture, posed by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2587_chunk_0 | Diffeomorphism group | In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2588_chunk_0 | Continuously differentiable | In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2589_chunk_0 | Continuously differentiable | In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). f is said to be differentiable on U if it is differentiable at every point of U. f is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function f {\displaystyle f} . Gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2590_chunk_0 | Smooth manifold | In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2591_chunk_0 | Smooth manifold | In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2592_chunk_0 | Smooth manifold | A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2593_chunk_0 | Smooth manifold | Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on different... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2594_chunk_0 | Parallelizable manifold | In mathematics, a differentiable manifold M {\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point p {\displaystyle p} of M {\displaystyle M} the tangent vectors provide a basis of the tangent space at p {\displaystyle p} . Equivalently, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2595_chunk_0 | Order (differential equation) | In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2596_chunk_0 | Order (differential equation) | The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are soluble by explicit formulas; however, many properties of solutions of a given differential equati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2597_chunk_0 | Differential invariant | In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. Differential invar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2598_chunk_0 | Ring of differential operators | In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer scien... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2599_chunk_0 | Differential variational inequality | In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving both dynamics and inequality constraints. Examples of such problems include, for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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