id stringlengths 14 19 | title stringlengths 1 124 | text stringlengths 12 2.83k | source stringclasses 1
value |
|---|---|---|---|
wiki_3800_chunk_0 | Autonomous system (mathematics) | In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3801_chunk_0 | Effaceable functor | In mathematics, an effaceable functor is an additive functor F between abelian categories C and D for which, for each object A in C, there exists a monomorphism u: A → M {\displaystyle u:A\to M} , for some M, such that F ( u ) = 0 {\displaystyle F(u)=0} . Similarly, a coeffaceable functor is one for which, for each A, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3802_chunk_0 | Eigenform | In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, .... Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3803_chunk_0 | Eigenfunction | In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as for some scalar eigenvalue λ . {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3804_chunk_0 | Eigenoperator | In mathematics, an eigenoperator, A, of a matrix H is a linear operator such that = λ A {\displaystyle =\lambda A\,} where λ {\displaystyle \lambda } is a corresponding scalar called an eigenvalue. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3805_chunk_0 | Eigenplane | In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term eigenvector for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term eigenplane can be used to describe a two-dimensional plane (a 2-p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3806_chunk_0 | Eigenvalue perturbation | In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system A x = λ x {\displaystyle Ax=\lambda x} that is perturbed from one with known eigenvectors and eigenvalues A 0 x 0 = λ 0 x 0 {\displaystyle A_{0}x_{0}=\lambda _{0}x_{0}} . This is useful for studying how se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3807_chunk_0 | Membership (set theory) | In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3808_chunk_0 | Normal element | In mathematics, an element x of a *-algebra is normal if it satisfies x x ∗ = x ∗ x . {\displaystyle xx^{*}=x^{*}x.} This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator A from a Hilbert space into itself is called unitary if A A ∗ = A ∗ A , {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3809_chunk_0 | Unitary element | In mathematics, an element x of a *-algebra is unitary if it satisfies x ∗ = x − 1 . {\displaystyle x^{*}=x^{-1}.} In functional analysis, a linear operator A from a Hilbert space into itself is called unitary if it is invertible and its inverse is equal to its own adjoint A∗ and that the domain of A is the same as tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3810_chunk_0 | Engel group | In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator ,y], ..., y] with n copies of y is trivial (where means xyx−1y−1 or the Lie bracket). It is called an Engel element if it satisfies ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3811_chunk_0 | Engel group | Every nilpotent group or Lie algebra is Engel. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. (Cohn 1955) gave examples of non-nilpotent Engel groups and algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3812_chunk_0 | Topological divisor of zero | In mathematics, an element z {\displaystyle z} of a Banach algebra A {\displaystyle A} is called a topological divisor of zero if there exists a sequence x 1 , x 2 , x 3 , . . . {\displaystyle x_{1},x_{2},x_{3},...} of elements of A {\displaystyle A} such that The sequence z x n {\displaystyle zx_{n}} converges to the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3813_chunk_0 | Elementary functions | In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, lo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3814_chunk_0 | Elementary row operation | In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3815_chunk_0 | Elementary proof | In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invok... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3816_chunk_0 | Elementary proof | While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3817_chunk_0 | Gardener's ellipse | Analytically, the equation of a standard ellipse centered at the origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: Assuming a ≥ b {\displaystyle a\geq b} , the foci are ( ± c , 0 ) {\displaystyle (\pm c,0)} for c = a 2 − b 2 {\textstyle c={\sqrt {a^{2}-b^{2}}}} . The standard parametric equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3818_chunk_0 | Gardener's ellipse | A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3819_chunk_0 | Elliptic Gauss sum | In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an ellipt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3820_chunk_0 | Elliptic boundary value problem | In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on. Differe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3821_chunk_0 | Elliptic boundary value problem | For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler points. Boundary value problems can involve space, time and other quantities such as tempe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3822_chunk_0 | Elliptic curves | Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3823_chunk_0 | Elliptic curves | If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. Mor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3824_chunk_0 | Elliptic curves | Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3825_chunk_0 | Elliptic curves | They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3826_chunk_0 | Elliptic divisibility sequence | In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward in the 1940s. They attracted only sporadic attention until aro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3827_chunk_0 | Elliptic hypergeometric series | In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They wer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3828_chunk_0 | Quasi-elliptic surface | In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3829_chunk_0 | Quasi-elliptic surface | The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3830_chunk_0 | Empty product | In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3831_chunk_0 | Empty sum | Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a zero element). For the same reason, the empty product is taken to be the multiplicative iden... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3832_chunk_0 | Endomorphism | In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3833_chunk_0 | Equaliser (mathematics) | In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3834_chunk_0 | Mathematical equations | In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3835_chunk_0 | Mathematical equations | There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing cou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3836_chunk_0 | Equidistant set | In mathematics, an equidistant set (also called a midset, or a bisector) is a set whose elements have the same distance (measured using some appropriate distance function) from two or more sets. The equidistant set of two singleton sets in the Euclidean plane is the perpendicular bisector of the segment joining the two... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3837_chunk_0 | Equidistant set | The concept of equidistant set is used to define frontiers in territorial domain controversies. For instance, the United Nations Convention on the Law of the Sea (Article 15) establishes that, in absence of any previous agreement, the delimitation of the territorial sea between countries occurs exactly on the median li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3838_chunk_0 | Equivalence relation | In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3839_chunk_0 | Essentially finite vector bundle | In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. Nori, as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3840_chunk_0 | Evasive Boolean function | In mathematics, an evasive Boolean function ƒ (of n variables) is a Boolean function for which every decision tree algorithm has running time of exactly n. Consequently, every decision tree algorithm that represents the function has, at worst case, a running time of n. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3841_chunk_0 | Exact differential equation | In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in Physics and engineering. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3842_chunk_0 | Exceptional Lie algebra | In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: g 2 , f 4 , e 6 , e 7 , e 8 {\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}} ; their r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3843_chunk_0 | Accidental isomorphism | In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms. These coincidences are at times conside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3844_chunk_0 | Purely existential proof | In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase "there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all x, y, ... there exist(s) ..."). In the formal terms of symbo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3845_chunk_0 | Purely existential proof | For example, the statement that the sine function is continuous everywhere, or any theorem written in big O notation, can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used. A controversy that goes back to the early twentieth century c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3846_chunk_0 | Existence theorem | In mathematics, an existence theorem is purely theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive, since the whole approach may not lend itself to construction. In terms of algorithms, purely theoretical existence theorems b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3847_chunk_0 | Existence theorem | Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, John Nash's original proof of the existence of a Nash equilibrium in 1951 was such an existence theorem. An approach which is constructive was also later found in 1962. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3848_chunk_0 | Exp ring | In mathematics, an exp algebra is a Hopf algebra Exp(G) constructed from an abelian group G, and is the universal ring R such that there is an exponential map from G to the group of the power series in R] with constant term 1. In other words the functor Exp from abelian groups to commutative rings is adjoint to the fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3849_chunk_0 | Polynomial expansion | In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3850_chunk_0 | Explicit reciprocity law | In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather rou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3851_chunk_0 | Exponential sum | In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function e ( x ) = exp ( 2 π i x ) . {\displaystyle e(x)=\exp(2\pi ix).\,} Therefore, a typical exponential sum may take the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3852_chunk_0 | Solution in closed form | In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, ÷, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3853_chunk_0 | Narrowing of algebraic value sets | In mathematics, an expression must represent a single value. For example consider the equation, x 2 = 4 {\displaystyle x^{2}=4} which implies, x = 2 ∨ x = − 2 {\displaystyle x=2\lor x=-2} But this is a bit long winded, and it does not allow us to work with multiple values at the same time. If further conditions or cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3854_chunk_0 | Mathematical expression | In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3855_chunk_0 | Mathematical expression | For example, 8 x − 5 {\displaystyle 8x-5} is an expression, while 8 x − 5 ≥ 5 x − 8 {\displaystyle 8x-5\geq 5x-8} is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3856_chunk_0 | Extraneous solution | In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem. A missing solution is a solution that is a valid solution to the problem, but disappeared during the process of solvi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3857_chunk_0 | Extremal point | In mathematics, an extreme point of a convex set S {\displaystyle S} in a real or complex vector space is a point in S {\displaystyle S} which does not lie in any open line segment joining two points of S . {\displaystyle S.} In linear programming problems, an extreme point is also called vertex or corner point of S . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3858_chunk_0 | Idempotent relation | In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. This notion generalizes that of an idempotent function to relations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3859_chunk_0 | Idempotent measure | In mathematics, an idempotent measure on a metric group is a probability measure that equals its convolution with itself; in other words, an idempotent measure is an idempotent element in the topological semigroup of probability measures on the given metric group. Explicitly, given a metric group X and two probability ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3860_chunk_0 | Identity operator | In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(X) = X is true for all values of X to which f can be applied. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3861_chunk_0 | Immersed plane curve | In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f: M → N is an immersion if D p f: T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function at every point p of M (where TpX den... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3862_chunk_0 | Immersed plane curve | The function f itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion f: M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3863_chunk_0 | Implicit curve | In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} . In general, every implicit curve is defined by an equation of the form F ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3864_chunk_0 | Implicit curve | If F ( x , y ) {\displaystyle F(x,y)} is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it. Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3865_chunk_0 | Implicit curve | The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve. The implicit function theorem describes conditions under which an equation F ( x , y ) = 0 {\displaystyle F(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3866_chunk_0 | Implicit curve | This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3867_chunk_0 | Implicit differentiation | In mathematics, an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 , {\displaystyle R(x_{1},\dots ,x_{n})=0,} where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0. {\displaystyle x^{2}+y^{2}-1=0.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3868_chunk_0 | Implicit differentiation | An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3869_chunk_0 | Implicit surface | In mathematics, an implicit surface is a surface in Euclidean space defined by an equation F ( x , y , z ) = 0. {\displaystyle F(x,y,z)=0.} An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z. The graph of a function is usually describ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3870_chunk_0 | Implicit surface | The implicit function theorem describes conditions under which an equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3871_chunk_0 | Incidence matrix | In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3872_chunk_0 | Incidence structure | In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which point... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3873_chunk_0 | Incidence structure | Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3874_chunk_0 | Indexing set | In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled by means of the elements of a set J, then J is an index set. The indexing consists of a surjective function from J onto A, and the indexed collection is typical... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3875_chunk_0 | Indicator random variable | In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then 1 A ( x ) = 1 {\displaystyle \mathbf {1} _{A}(x)=1} if x ∈ A , {\displaystyle x\in A,} and 1 A ( x ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3876_chunk_0 | Indicator random variable | The indicator function of A is the Iverson bracket of the property of belonging to A; that is, 1 A ( x ) = . {\displaystyle \mathbf {1} _{A}(x)=.} For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3877_chunk_0 | Induced character | In mathematics, an induced character is the character of the representation V of a finite group G induced from a representation W of a subgroup H ≤ G. More generally, there is also a notion of induction Ind ( f ) {\displaystyle \operatorname {Ind} (f)} of a class function f on H given by the formula Ind ( f ) ( s )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3878_chunk_0 | Strict inequality | In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: The notation a < b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3879_chunk_0 | Strict inequality | The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).The relation not greater than can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and a ≮ b. The notation a ≠ b means that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3880_chunk_0 | Strict inequality | In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. The notation a ≪ b means that a is much less than b. The notation a ≫ b means that a is much greater than b.This implies that the lesser value can be neglected ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3881_chunk_0 | Inequation | In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are: a < b {\displaystyle a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3882_chunk_0 | Divergent geometric series | In mathematics, an infinite geometric series of the form ∑ k = 0 ∞ a r k = a + a r + a r 2 + a r 3 + ⋯ {\displaystyle \sum _{k=0}^{\infty }ar^{k}=a+ar+ar^{2}+ar^{3}+\cdots } is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3883_chunk_0 | Periodic continued fraction | In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a k + 1 + ⋱ ⋱ a k + m − 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3884_chunk_0 | Absolute convergence | In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0 ∞ a n {\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}} is said to converge absolutely if ∑ n = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3885_chunk_0 | Absolute convergence | {\displaystyle \textstyle \int _{0}^{\infty }|f(x)|dx=L.} Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3886_chunk_0 | Absolute convergence | For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + ⋯ {\textstyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots } converges to ln 2 , {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3887_chunk_0 | Infinitesimal | In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence. Infinitesimals do not exist in the sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3888_chunk_0 | Infinitesimal | In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3889_chunk_0 | Infinitesimal | An infinite number of infinitesimals are summed to calculate an integral. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3890_chunk_0 | Infinitesimal | Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3891_chunk_0 | Infinitesimal | The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3892_chunk_0 | Infinitesimal | A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed nonstandard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle impleme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3893_chunk_0 | Infinitesimal operator | In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3894_chunk_0 | Information source (mathematics) | In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution. The uncertainty, or entropy rate, of an information source is defined as H { X } = lim n → ∞ H ( X n | X 0 , X 1 , … , X n − 1 ) {\displaystyle H\{\mathbf {X} \}=\lim _{n\to \infty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3895_chunk_0 | Infrastructure (number theory) | In mathematics, an infrastructure is a group-like structure appearing in global fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3896_chunk_0 | Initial algebra | In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3897_chunk_0 | One-to-one mapping | In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) ≠ f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3898_chunk_0 | One-to-one mapping | The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3899_chunk_0 | One-to-one mapping | However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details. A function f {\displaystyle f} that is not injective is so... | 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.