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wiki_6700_chunk_0 | Digit sum | In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 {\displaystyle 9045} would be 9 + 0 + 4 + 5 = 18. {\displaystyle 9+0+4+5=18.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6701_chunk_0 | Infinite-dimensional vector space | In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6702_chunk_0 | Infinite-dimensional vector space | We say V {\displaystyle V} is finite-dimensional if the dimension of V {\displaystyle V} is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space V {\displaystyle V} over the field F {\displaystyle F} can be written as dim F ( V ) {\displaystyle \dim _{F}(V)} or as , {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6703_chunk_0 | Multidimensional geometry | In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. In other words, the dimension is the number of independent parameters or coordinates that are needed for defining the position of a point that is constrained to be on the object. For e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6704_chunk_0 | Multidimensional geometry | This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" One answer is that to cov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6705_chunk_0 | Multidimensional geometry | This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6706_chunk_0 | Multidimensional geometry | A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4" or: 4D. Although the notion of higher dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6707_chunk_0 | Dimension theorem for vector spaces | In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6708_chunk_0 | Dimension theorem for vector spaces | While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker (the proof given below, however, assumes trichotomy, i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6709_chunk_0 | Direct image | In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6710_chunk_0 | Direct method in calculus of variations | In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6711_chunk_0 | Discrete Fourier Transform | In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6712_chunk_0 | Discrete Fourier Transform | The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6713_chunk_0 | Discrete Fourier Transform | The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6714_chunk_0 | Discrete Fourier Transform | The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6715_chunk_0 | Number-theoretic transform | In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6716_chunk_0 | Discrete Laplace operator | In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6717_chunk_0 | Discrete Poisson equation | In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6718_chunk_0 | Discrete exterior calculus | In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6719_chunk_0 | Discrete sine transform | In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6720_chunk_0 | Discrete-time Fourier transform | In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6721_chunk_0 | Discrete-time Fourier transform | Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6722_chunk_0 | Discrete-time Fourier transform | The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6723_chunk_0 | Discriminant of a polynomial | In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6724_chunk_0 | Brill's theorem | In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6725_chunk_0 | Radon–Hurwitz number | In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6726_chunk_0 | Disintegration theorem | In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disinteg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6727_chunk_0 | Distortion (mathematics) | In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6728_chunk_0 | Distortion (mathematics) | Here Df is the weak derivative of ƒ, and |Df| is the Hilbert–Schmidt norm. For functions on a higher-dimensional Euclidean space Rn, there are more measures of distortion because there are more than two principal axes of a symmetric tensor. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6729_chunk_0 | Distortion (mathematics) | The pointwise information is contained in the distortion tensor G ( x , f ) = { | J ( x , f ) | − 2 / n D T f ( x ) D f ( x ) if J ( x , f ) ≠ 0 I if J ( x , f ) = 0. {\displaystyle G(x,f)={\begin{cases}|J(x,f)|^{-2/n}D^{T}f(x)Df(x)&{\text{if }}J(x,f)\not =0\\I&{\text{if }}J(x,f)=0.\end{cases}}} The outer distortion KO... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6730_chunk_0 | Antidistributive | In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality is always true in elementary algebra. For example, in elementary arithmetic, one has Therefore, one would say that multiplication distributes over addition. This basic property of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6731_chunk_0 | Domain (function) | In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname {dom} (f)} or dom f {\displaystyle \operatorname {dom} f} , where f is the function. In layman's terms, the domain of a function can generally be thought of as "w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6732_chunk_0 | Domain (function) | In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis. For a function f: X → Y {\displaystyle f\colon X\to Y} , the set Y is called the codomain, and the set of values attained by the function (which is a subset of Y) is called its range or im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6733_chunk_0 | Vector dot product | In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6734_chunk_0 | Vector dot product | Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6735_chunk_0 | Double Fourier sphere method | In mathematics, the double Fourier sphere (DFS) method is a simple technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude directions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6736_chunk_0 | Dual bundle | In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6737_chunk_0 | Dual module | In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically denoted M∗ or HomR(M, R). If the base ring R is a field, then a dual module is a dual vector sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6738_chunk_0 | Dual module | A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ( A ) {\displaystyle G=\operatorname {Spec} (A)} is a finite commutative group scheme represented by a Hopf algebra A over a commutative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6739_chunk_0 | Dual quaternion | In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6740_chunk_0 | Dual quaternion | In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebrai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6741_chunk_0 | Dyadic cubes | In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of disc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6742_chunk_0 | Eccentricity (geometry) | In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. {\displaystyle 0.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6743_chunk_0 | Eccentricity (geometry) | The eccentricity of an ellipse which is not a circle is between 0 {\displaystyle 0} and 1. {\displaystyle 1.} The eccentricity of a parabola is 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6744_chunk_0 | Eccentricity (geometry) | {\displaystyle 1.} The eccentricity of a hyperbola is greater than 1. {\displaystyle 1.} The eccentricity of a pair of lines is ∞ {\displaystyle \infty } Two conic sections with the same eccentricity are similar. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6745_chunk_0 | Elasticity of a function | In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as E f ( a ) = a f ( a ) f ′ ( a ) {\displaystyle Ef(a)={\frac {a}{f(a)}}f'(a)} = lim x → a f ( x ) − f ( a ) x − a a f ( a ) = lim x → a f ( x ) − f ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6746_chunk_0 | Elasticity of a function | Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalisations to multi-input-multi-output cases also exist in the literature.The elasticity of a function is a constant α {\displaystyle \alpha } if and o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6747_chunk_0 | Elliptic gamma function | In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by Jackson (1905), and can be expressed in terms of the triple gamma function. It is given by Γ ( z ; p , q ) = ∏ m = 0 ∞ ∏ n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6748_chunk_0 | Elliptic gamma function | {\displaystyle \Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}}.} It obeys several identities: Γ ( z ; p , q ) = 1 Γ ( p q / z ; p , q ) {\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,} Γ ( p z ; p , q ) = θ ( z ; q ) Γ ( z ; p , q ) {\displaystyle \G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6749_chunk_0 | Endomorphism algebra | In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6750_chunk_0 | Endomorphism algebra | Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0: x ↦ 0 {\textstyle 0:x\mapsto 0} as additive identity and the identity map 1: x ↦ x {\textstyle 1:x\mapsto x} as multiplicative identity.The functions involved are restricted to what is defined as a homomorph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6751_chunk_0 | Endomorphism algebra | An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field, its modules M are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6752_chunk_0 | Graph energy | In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. More precisely, let G be a graph with n vertices. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6753_chunk_0 | Graph energy | It is assumed that G is simple, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let λ i {\displaystyle \lambda _{i}} , i = 1 , … , n {\displaystyle i=1,\ldots ,n} , be the eigenvalues of A. Then the energy of the graph is defined as: E ( G ) = ∑ i = 1 n | λ i | . {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6754_chunk_0 | Epigraph (mathematics) | In mathematics, the epigraph or supergraph of a function f: X → {\displaystyle f:X\to } valued in the extended real numbers = R ∪ { ± ∞ } {\displaystyle =\mathbb {R} \cup \{\pm \infty \}} is the set, denoted by epi f , {\displaystyle \operatorname {epi} f,} of all points in the Cartesian product X × R {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6755_chunk_0 | Epigraph (mathematics) | If the function takes ± ∞ {\displaystyle \pm \infty } as a value then graph f {\displaystyle \operatorname {graph} f} will not be a subset of its epigraph epi f . {\displaystyle \operatorname {epi} f.} For example, if f ( x 0 ) = ∞ {\displaystyle f\left(x_{0}\right)=\infty } then the point ( x 0 , f ( x 0 ) ) = ( x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6756_chunk_0 | Epigraph (mathematics) | Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in {\displaystyle } instead of continuous functions valued in a vector space (such as R {\displaystyle \mathbb {R} } or R 2 {\displaystyle \mathbb {R} ^{2}} ). This is b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6757_chunk_0 | Epsilon number | In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplicati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6758_chunk_0 | Epsilon number | The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6759_chunk_0 | Epsilon number | A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx. Hessenberg (1906) defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6760_chunk_0 | Not equal sign | In mathematics, the equal sign can be used as a simple statement of fact in a specific case ("x = 2"), or to create definitions ("let x = 2"), conditional statements ("if x = 2, then ..."), or to express a universal equivalence ("(x + 1)2 = x2 + 2x + 1"). The first important computer programming language to use the equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6761_chunk_0 | Not equal sign | A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like if x = 2 with essentially the same meaning of = as the conditional usage in mathematics. The e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6762_chunk_0 | Not equal sign | Both usages have remained common in different programming languages into the early 21st century. As well as Fortran, = is used for assignment in such languages as C, Perl, Python, awk, and their descendants. But = is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6763_chunk_0 | Not equal sign | A few languages, such as BASIC and PL/I, have used the equal sign to mean both assignment and equality, distinguished by context. However, in most languages where = has one of these meanings, a different character or, more often, a sequence of characters is used for the other meaning. Following ALGOL, most languages th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6764_chunk_0 | Schlesinger equations | In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinear... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6765_chunk_0 | Equidistribution theorem | In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1is uniformly distributed on the circle R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure μ = d θ 2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6766_chunk_0 | Equilateral dimension | In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6767_chunk_0 | Equioscillation theorem | In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6768_chunk_0 | Equivariant algebraic K-theory | Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K i G ( X ) {\displaystyle K_{i}^{G}(X)} may be defined as the K i {\displaystyle K_{i}} of the category of coherent sheaves on the quotient stack {\displaystyle } . (Hence, the equivariant K-theory is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6769_chunk_0 | Complementary error function | In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: erf z = 2 π ∫ 0 z e − t 2 d t . {\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.} Some authors define erf {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6770_chunk_0 | Complementary error function | In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6771_chunk_0 | Inclusive counting | In mathematics, the essence of counting a set and finding a result n, is that it establishes a one-to-one correspondence (or bijection) of the set with the subset of positive integers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematical induction, is that no bijection can exist between {1, 2, ..., n}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6772_chunk_0 | Inclusive counting | For instance, the set of all integers (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of real numbers, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6773_chunk_0 | Inclusive counting | Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities). Counting, mostly of finite sets, has various appl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6774_chunk_0 | Inclusive counting | One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6775_chunk_0 | Eta invariant | In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by Atiyah, Pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6776_chunk_0 | Riemann's explicit formula | In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6777_chunk_0 | Characterizations of the exponential function | In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these consideration... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6778_chunk_0 | Well function | In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6779_chunk_0 | Exponential response formula | In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6780_chunk_0 | Exponential sequence | In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6781_chunk_0 | Exponential sequence | The exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the logarithm of g in a neighbor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6782_chunk_0 | Exponential sequence | In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing holomorphic function, something that is always locally possible. A further consequence of the sequence is the exactness of ⋯ → H 1 ( O M ) → H 1 ( O M ∗ ) → H 2 ( 2 π i Z ) → ⋯ . {\displaystyle \cdots \to H^{1}(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6783_chunk_0 | Grassmann algebra | In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and thei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6784_chunk_0 | Grassmann algebra | More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = − ( v ∧ u ) {\displaystyle u\wedge v=-(v\wedge u)} for all vectors u {\displaystyle u} and v , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6785_chunk_0 | Grassmann algebra | More generally, the exterior product of any number k {\displaystyle k} of vectors can be defined and is sometimes called a k {\displaystyle k} -blade (or decomposable, or simple, by some authors). It lives in a space known as the k {\displaystyle k} -th exterior power (generalizing exterior square and exterior cubic). ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6786_chunk_0 | Grassmann algebra | If Euclidean product is given for the vectors, the magnitude (that is, a scalar) of the resulting k {\displaystyle k} -blade is the oriented hypervolume of the k {\displaystyle k} -dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6787_chunk_0 | Grassmann algebra | The exterior algebra contains objects that are not only k {\displaystyle k} -blades, but sums of k {\displaystyle k} -blades; such a sum is called a k-vector. Combining k {\displaystyle k} -blades in a linear structure by adding and scalar multiplication is the core of Projective Geometry, (see Plücker coordinates). k ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6788_chunk_0 | Grassmann algebra | For any k {\displaystyle k} -vector more associated objects exist: rank is defined to be the smallest number of simple elements of which it is a sum; support is defined as the minimal subspace the k {\displaystyle k} -vector lives in; the divisor space (some authors might have other names for it, like kernel or factor)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6789_chunk_0 | Grassmann algebra | {\displaystyle \alpha ,\beta ,\gamma .} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6790_chunk_0 | Grassmann algebra | As said, the k {\displaystyle k} -vectors are a lot like homogeneous polynomials of degree k {\displaystyle k} , such that when elements of different degrees are multiplied, the degrees add for the degree of the product. This means that the exterior algebra is a graded algebra. Exterior algebra emerged on two paths: as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6791_chunk_0 | Grassmann algebra | This twofold approach exists in almost all cases, but an exception has to be singled out: when the vector spaces in the construction are over a field of characteristic 2. Hence, whenever antisymmetric/alternating tensors are considered in connection to exterior algebra, the basic field is supposed of 0 or odd character... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6792_chunk_0 | Grassmann algebra | On the first path, the abstract one, both ingredients are clearly given (pretty abstract, though) and this is its main power. On the second path, the vector space is clear and less abstract, but the exterior product can be defined in more (equivalent) ways, and much care is needed to avoid mistakes. The definition of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6793_chunk_0 | Grassmann algebra | Moreover, the field the vector spaces are based on may not be numeric, as real or complex numbers, but other (less usual) field (finite or not) with zero or positive characteristic. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6794_chunk_0 | Grassmann algebra | The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6795_chunk_0 | Factorial function | In mathematics, the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to n {\displaystyle n} . The factorial of n {\displaystyle n} also equals the product of n {\displaystyle n} with the next smaller factorial: For e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6796_chunk_0 | Factorial function | is 1, according to the convention for an empty product.Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, not... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6797_chunk_0 | Factorial function | In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science. Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6798_chunk_0 | Factorial function | Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6799_chunk_0 | Falling factorial power | In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is def... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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