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wiki_7500_chunk_0 | Srinivasa Ramanujan | Examples of the most intriguing of these formulae include infinite series for π, one of which is given below: 1 π = 2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7501_chunk_0 | Srinivasa Ramanujan | Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places; truncating it to the first two terms gives a value ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7502_chunk_0 | Srinivasa Ramanujan | His intuition also led him to derive some previously unknown identities, such as ( 1 + 2 ∑ n = 1 ∞ cos ( n θ ) cosh ( n π ) ) − 2 + ( 1 + 2 ∑ n = 1 ∞ cosh ( n θ ) cosh ( n π ) ) − 2 = 2 Γ 4 ( 3 4 ) π = 8 π 3 Γ 4 ( 1 4 ) {\displaystyle {\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7503_chunk_0 | Srinivasa Ramanujan | They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asympt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7504_chunk_0 | Stone's duality | In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7505_chunk_0 | Sobolev inequality | In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7506_chunk_0 | Theta series | In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.The most common form of theta function is that occurring in the theory of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7507_chunk_0 | Time-scale calculus | In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7508_chunk_0 | Finite difference coefficients | In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7509_chunk_0 | Solution (equation) | In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7510_chunk_0 | Solution (equation) | A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7511_chunk_0 | Solution (equation) | Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7512_chunk_0 | Solution (equation) | It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solutio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7513_chunk_0 | Solution (equation) | The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7514_chunk_0 | Solution (equation) | Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given interval. When the task is to find the solution that is the best under some criterion, this is an opti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7515_chunk_0 | Topological Galois theory | In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory. It connects many ideas from algebra to ideas in to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7516_chunk_0 | Stably isomorphic | In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirze... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7517_chunk_0 | Topological complexity | In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 2003. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7518_chunk_0 | Topological degree theory | In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7519_chunk_0 | Graph topology | In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7520_chunk_0 | Graph topology | Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem. Other applications can be found in printing electronic circuits where the aim is to p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7521_chunk_0 | Closed subgroup | In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.Topological ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7522_chunk_0 | Topological modular forms | In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space tmf n {\displaystyle \operatorname {tmf} ^{n}} , and these spaces are equipped with certain maps between them, so that for any topol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7523_chunk_0 | Topological modular forms | The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7524_chunk_0 | Topological recursion | In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7525_chunk_0 | Topology | The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7526_chunk_0 | Trace diagram | In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7527_chunk_0 | Trailing zero | The number of trailing zeros in a non-zero base-b integer n equals the exponent of the highest power of b that divides n. For example, 14000 has three trailing zeros and is therefore divisible by 1000 = 103, but not by 104. This property is useful when looking for small factors in integer factorization. Some computer a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7528_chunk_0 | Transform theory | In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7529_chunk_0 | Transformation geometry | In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geome... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7530_chunk_0 | Transformation geometry | For nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. These efforts culminated... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7531_chunk_0 | Triality | In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7532_chunk_0 | Triality | This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8). The vector representation—the natural action of SO(8) (hence Spin(8)) on F8—consists o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7533_chunk_0 | Triality | No other connected Dynkin diagram has an automorphism group of order greater than 2; for other Dn (corresponding to other even Spin groups, Spin(2n)), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation. Roughly speaking... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7534_chunk_0 | Triangulable space | In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7535_chunk_0 | Trigonometric integral | In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7536_chunk_0 | Trigonometric interpolation | In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given per... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7537_chunk_0 | Trigonometric substitution | In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other met... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7538_chunk_0 | Tropical geometry | In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: x ⊕ y = min { x , y } , {\displaystyle x\oplus y=\min\{x,y\},} x ⊗ y = x + y . {\displaystyle x\otimes y=x+y.} So for example, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7539_chunk_0 | Tropical geometry | Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7540_chunk_0 | Twisted K-theory | In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7541_chunk_0 | Twisted K-theory | In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics). In the broader context of K-theory, in each subject it has numerous isomorphic formulations ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7542_chunk_0 | Topological conjugacy | In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct § Topological equivalence of flows, are important in the study of iterated functions and more generally dynamical systems, sinc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7543_chunk_0 | Isospectral flow | In finite-dimensions, one essentially deals with square matrices. In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7544_chunk_0 | Artisan Mannerism | In mathematics, two major works were published in a single year, 1631. Thomas Harriot's Artis analyticae praxis, published ten years posthumously, and William Oughtred's Clavis mathematicae. Both contributed to the evolution of modern mathematical language; the former introduced the × {\displaystyle \times } sign for m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7545_chunk_0 | Commensurability (mathematics) | In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7546_chunk_0 | Antipodal point | In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the intersections of the sphere with a diameter, a straight line passing through its center.Given any point on a sphere, its antipodal point is the unique point at greatest distance, wheth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7547_chunk_0 | Singular measures | {\displaystyle \mu \perp \nu .} A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7548_chunk_0 | The Golden Ratio | In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a {\displaystyle a} and b {\displaystyle b} with a > b > 0 {\displaystyle a>b>0} , where the Greek letter phi ( φ {\displaystyle \varph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7549_chunk_0 | Silver triangle | In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7550_chunk_0 | Supergolden ratio | In mathematics, two quantities are in the supergolden ratio if their quotient equals the unique real solution to the equation x 3 = x 2 + 1. {\displaystyle x^{3}=x^{2}+1.} This solution is commonly denoted ψ . {\displaystyle \psi .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7551_chunk_0 | Supergolden ratio | The name supergolden ratio results of a analogy with the golden ratio φ {\displaystyle \varphi } , which is the positive root of the equation x 2 = x + 1. {\displaystyle x^{2}=x+1.} Using formulas for the cubic equation, one can show that ψ = 1 3 ( 1 + 29 + 3 93 2 3 + 29 − 3 93 2 3 ) , {\displaystyle \psi ={\frac {1}{3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7552_chunk_0 | Hölder conjugates | In mathematics, two real numbers p , q > 1 {\displaystyle p,q>1} are called conjugate indices (or Hölder conjugates) if 1 p + 1 q = 1. {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.} Formally, we also define q = ∞ {\displaystyle q=\infty } as conjugate to p = 1 {\displaystyle p=1} and vice versa. Conjugate indices are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7553_chunk_0 | Proportionality (mathematics) | In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7554_chunk_0 | Proportionality (mathematics) | This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons. Two functions f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are proportional if their ratio f ( x ) g ( x ) {\textstyle {\f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7555_chunk_0 | Equinumerosity | In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y. Equinumerous sets are said to have the same cardina... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7556_chunk_0 | Equinumerosity | Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof). In his controversial 1878 paper, Can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7557_chunk_0 | Matrix congruence | In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = Bwhere "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7558_chunk_0 | Epistemic probability | In mathematics, uncertainty is often characterized in terms of a probability distribution. From that perspective, epistemic uncertainty means not being certain what the relevant probability distribution is, and aleatoric uncertainty means not being certain what a random sample drawn from a probability distribution will... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7559_chunk_0 | Uniform absolute convergence | In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7560_chunk_0 | Unimodal distribution | In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7561_chunk_0 | Value (mathematics) | In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as π or an integer such as 42. The value of a variable or a constant is any number or o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7562_chunk_0 | Value (mathematics) | The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values. The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values.For example, if the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7563_chunk_0 | Exponent pair | In mathematics, van der Corput's method generates estimates for exponential sums. The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate. The processes apply to exponential sums of the form ∑ n = a b e ( f ( n ) ) {\displaystyle \sum _... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7564_chunk_0 | Vanishing cycle | In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth Ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7565_chunk_0 | Vanishing cycle | The loop in the smooth fibers gives an element of the first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop is traversed, i.e. an invertible map of the first homology of a (real) surface of genus g. A classical result is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7566_chunk_0 | Variation of parameters | In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7567_chunk_0 | Variational perturbation theory | In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say s = ∑ n = 0 ∞ a n g n {\displaystyle s=\sum _{n=0}^{\infty }a_{n}g^{n}} ,into a convergent series in powers s = ∑ n = 0 ∞ b n / ( g ω ) n {\displaystyle s=\sum _{n=0}^{\in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7568_chunk_0 | Variational perturbation theory | The convergence is exponentially fast.After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its anomalous dimensions. Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7569_chunk_0 | Vector algebra | In mathematics, vector algebra may mean: Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space R 3 {\displaystyle \mathb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7570_chunk_0 | Vector bundles on algebraic curves | In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points). Some foundational results on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7571_chunk_0 | Vector bundles on algebraic curves | The Riemann–Roch theorem for vector bundles was proved by Weil (1938), before the 'vector bundle' concept had really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7572_chunk_0 | Vector bundles on algebraic curves | He was seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank. This idea would prove fruitful, in terms of moduli spaces of vector bundles. following on the work in the 1960s on geometric invariant theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7573_chunk_0 | Vector multiplication | In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7574_chunk_0 | Vector multiplication | Thus, A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} ⋅ B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} = | A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} | | B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} | cos θMore generally, a bilinear product in an algebra over a field. Cross prod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7575_chunk_0 | Vector multiplication | So, if n̂ is the unit vector perpendicular to the plane determined by vectors A and B, A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} × B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} = | A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} | | B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7576_chunk_0 | Vector spherical harmonics | In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7577_chunk_0 | Von Neumann's theorem | In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7578_chunk_0 | Weak Hopf algebra | In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7579_chunk_0 | Weak Hopf algebra | The first motivations for studying them came from quantum field theory and operator algebras. Weak Hopf algebras have quite interesting representation theory; in particular modules over a semisimple finite weak Hopf algebra is a fusion category (which is a monoidal category with extra properties). It was also shown by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7580_chunk_0 | Banach-Saks theorem | In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7581_chunk_0 | Weak limit | In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7582_chunk_0 | Localization of a topological space | In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in (Sullivan 2005). The reason to do this was in line with an idea of making topolog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7583_chunk_0 | Canonical projection map | This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S {\displaystyle S} by ∼ , {\displaystyle \,\sim \,,} and is denoted by S / ∼ {\displaystyle S/{\sim }} . When the set S {\displaystyle S} has some structure (such as a group operation or a topology) and the equi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7584_chunk_0 | Normal family | Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F. More formally, let X and Y be topological spaces. The set of continuous functions f: X → Y {\displaystyle f:X\to Y} has a natural topology called the co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7585_chunk_0 | Zero is even | Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all.Among the general public, the parity ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7586_chunk_0 | Zero is even | In reaction time experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some teachers—and some children in mathematics classes—think that zero is odd, or both even and odd, or neither. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7587_chunk_0 | Zero-sum Ramsey theory | In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group A {\displaystyle A} ), one seeks for conditions that guarantee the e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7588_chunk_0 | Zero-sum Ramsey theory | (This bound is tight, as a sequence of m − 1 {\displaystyle m-1} zeroes and m − 1 {\displaystyle m-1} ones cannot have any subset of size m {\displaystyle m} summing to zero.) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.Generalizing... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7589_chunk_0 | Sigma approximation | In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.A σ-approximated summation for a series of period T can be written as follows: in terms of the normalized sinc function The term is the Lanczos σ factor, which is responsibl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7590_chunk_0 | Buchholz's ordinal | In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem Π 1 1 {\displaystyle \Pi _{1}^{1}} -CA0 of second-order arithmetic; this is one of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7591_chunk_0 | L-infinity | In mathematics, ℓ ∞ {\displaystyle \ell ^{\infty }} , the (real or complex) vector space of bounded sequences with the supremum norm, and L ∞ = L ∞ ( X , Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum norm, are t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7592_chunk_0 | Blaschke conjecture | In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged. For example, on an ordinar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7593_chunk_0 | Geodesic map | In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ: M → N is said to be a geodesic map if φ is a diffeomorphism of M onto N... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7594_chunk_0 | Pettis theorem | In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7595_chunk_0 | Division by zero | In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab+, in which b+ represents the pseudoinverse of b. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then b+ = 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7596_chunk_0 | Stahl's theorem | In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two impor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7597_chunk_0 | Jacobi's formula | In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.If A is a differentiable map from the real numbers to n × n matrices, then d d t det A ( t ) = tr ( adj ( A ( t ) ) d A ( t ) d t ) = ( det A ( t ) ) ⋅ tr ( A ( t ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7598_chunk_0 | Jacobi's formula | {\displaystyle {\partial \det(A) \over \partial A_{ij}}=\operatorname {adj} (A)_{ji}.} Equivalently, if dA stands for the differential of A, the general formula is d det ( A ) = tr ( adj ( A ) d A ) . {\displaystyle d\det(A)=\operatorname {tr} (\operatorname {adj} (A)\,dA).} The formula is named after the mathemati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7599_chunk_0 | Generalized Hebbian Algorithm | In matrix form, Oja's rule can be written d w ( t ) d t = w ( t ) Q − d i a g w ( t ) {\displaystyle \,{\frac {{\text{d}}w(t)}{{\text{d}}t}}~=~w(t)Q-\mathrm {diag} w(t)} ,and the Gram-Schmidt algorithm is Δ w ( t ) = − l o w e r w ( t ) {\displaystyle \,\Delta w(t)~=~-\mathrm {lower} w(t)} ,where w(t) is any matrix, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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