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wiki_4800_chunk_0 | Matrix differentiation | A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4801_chunk_0 | Matrix differentiation | Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the la... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4802_chunk_0 | Mimesis (mathematics) | In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of numerical analysis is to approximate the continuum, so instead of solving a partial differential equation one aims to solve a discrete version of the continuum problem. Properties of the con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4803_chunk_0 | Mimesis (mathematics) | For example, a mixed finite element method applied to Darcy flows strictly conserves the mass of the flowing fluid. The term geometric integration denotes the same philosophy. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4804_chunk_0 | Mimetic interpolation | In mathematics, mimetic interpolation is a method for interpolating differential forms. In contrast to other interpolation methods, which estimate a field at a location given its values on neighboring points, mimetic interpolation estimates the field's k {\displaystyle k} -form given the field's projection on neighbori... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4805_chunk_0 | Minimum polynomial extrapolation | In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson.While Aitken's method is the most famous, it often fails for vector sequences. An effective method for vector sequences is the minimum polynomial extrapolation. I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4806_chunk_0 | Minimum polynomial extrapolation | . , x k {\displaystyle x_{1},x_{2},...,x_{k}} in R n {\displaystyle \mathbb {R} ^{n}} , one constructs the n × ( k − 1 ) {\displaystyle n\times (k-1)} matrix U = ( x 2 − x 1 , x 3 − x 2 , . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4807_chunk_0 | Minimum polynomial extrapolation | . , x k − x k − 1 ) {\displaystyle U=(x_{2}-x_{1},x_{3}-x_{2},...,x_{k}-x_{k-1})} whose columns are the k − 1 {\displaystyle k-1} differences. Then, one computes the vector c = − U + ( x k + 1 − x k ) {\displaystyle c=-U^{+}(x_{k+1}-x_{k})} where U + {\displaystyle U^{+}} denotes the Moore–Penrose pseudoinverse of U {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4808_chunk_0 | Minimum polynomial extrapolation | The number 1 is then appended to the end of c {\displaystyle c} , and the extrapolated limit is s = X c ∑ i = 1 k c i , {\displaystyle s={Xc \over \sum _{i=1}^{k}c_{i}},} where X = ( x 2 , x 3 , . . . , x k + 1 ) {\displaystyle X=(x_{2},x_{3},...,x_{k+1})} is the matrix whose columns are the k {\displaystyle k} iterate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4809_chunk_0 | Mirror descent | In mathematics, mirror descent is an iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative weights. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4810_chunk_0 | Mixed Hodge module | In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4811_chunk_0 | Modern triangle geometry | In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the ti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4812_chunk_0 | Modern triangle geometry | The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4813_chunk_0 | Modern triangle geometry | Later the theory of correspondences which was an offshoot of the theory of geometric transformations was developed to give coherence to the various isolated results. With its development, the expression "new triangle geometry" indicated not only the many remarkable objects associated with a triangle but also the method... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4814_chunk_0 | Modern triangle geometry | (See the conference paper titled "Teaching new geometrical methods with an ancient figure in the nineteenth and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952)" by Pauline Romera-Lebret presented in 2009.) However, this escalation of interest soon collapsed and triangle geometr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4815_chunk_0 | Modern triangle geometry | (The Development of Mathematics, p. 323) Philip Davis has suggested several reasons for the decline of interest in triangle geometry. These include: The feeling that the subject is elementary and of low professional status. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4816_chunk_0 | Modern triangle geometry | The exhaustion of its methodologic possibilities. The visual complexity of the so-called deeper results of the subject. The downgrading of the visual in favor of the algebraic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4817_chunk_0 | Modern triangle geometry | A dearth of connections to other fields. Competition with other topics with a strong visual content like tessellations, fractals, graph theory, etc.A further revival of interest was witnessed with the advent of the modern electronic computer. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4818_chunk_0 | Modern triangle geometry | The triangle geometry has again become an active area of research pursued by a group of dedicated geometers. As epitomizing this revival, one can point out the formulation of the concept of a "triangle centre" and the compilation by Clark Kimberling of an encyclopedia of triangle centers containing a listing of nearly ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4819_chunk_0 | Residue class | In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4820_chunk_0 | Modular forms modulo p | In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4821_chunk_0 | Modulus of smoothness | In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4822_chunk_0 | Cutoff function | In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a function which is rather irregula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4823_chunk_0 | Monodromy theory | In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4824_chunk_0 | Monstrous Moonshine | In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979.The monstrous mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4825_chunk_0 | Positive current | In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4826_chunk_0 | Positive current | Now, let M be a complex manifold. The Hodge decomposition Λ i ( M ) = ⨁ p + q = i Λ p , q ( M ) {\displaystyle \Lambda ^{i}(M)=\bigoplus _{p+q=i}\Lambda ^{p,q}(M)} is defined on currents, in a natural way, the (p,q)-currents being functionals on Λ c p , q ( M ) {\displaystyle \Lambda _{c}^{p,q}(M)} . A positive current... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4827_chunk_0 | Integral current | In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4828_chunk_0 | Rational singularities | In mathematics, more particularly in the field of algebraic geometry, a scheme X {\displaystyle X} has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map f: Y → X {\displaystyle f\colon Y\rightarrow X} from a regular scheme Y {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4829_chunk_0 | Prosolvable group | In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4830_chunk_0 | Étale group scheme | In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4831_chunk_0 | Solder form | In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4832_chunk_0 | Energetic space | In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energeti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4833_chunk_0 | Arithmetic hyperbolic 3-manifold | In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4834_chunk_0 | Lebesgue's decomposition theorem | In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures μ {\displaystyle \mu } and ν {\displaystyle \nu } on a measurable space ( Ω , Σ ) , {\displaystyle (\Omega ,\Sigma ),} there exist two σ-finite signed measures ν 0 {\displaystyle \nu _{0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4835_chunk_0 | Contact type | In mathematics, more precisely in symplectic geometry, a hypersurface Σ {\displaystyle \Sigma } of a symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} is said to be of contact type if there is 1-form α {\displaystyle \alpha } such that j ∗ ( ω ) = d α {\displaystyle j^{*}(\omega )=d\alpha } and ( Σ , α ) {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4836_chunk_0 | Pseudoconvex domain | In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let G ⊂ C n {\displaystyle G\subset {\mathbb {C} }^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4837_chunk_0 | Hyperbolic 3-manifold | In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4838_chunk_0 | Simplicial group | In fact it can be shown that any simplicial abelian group A {\displaystyle A} is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, ∏ i ≥ 0 K ( π i A , i ) . {\displaystyle \prod _{i\geq 0}K(\pi _{i}A,i).} A commutative monoid in the category of simplicial abelian groups is a simplicial commu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4839_chunk_0 | Dold–Kan correspondence | Example: Let C be a chain complex that has an abelian group A in degree n and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space K ( A , n ) {\displaystyle K(A,n)} . There is also an ∞-category-version of the Dold–Kan correspondence.The book "Nonabelian Algebraic Topology"... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4840_chunk_0 | Nakayama's lemma | In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4841_chunk_0 | Nakayama's lemma | In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah (1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson (1945), and so the noncommutative Nakayama lemma is somet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4842_chunk_0 | Finite ring | In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4843_chunk_0 | Finite ring | For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finite simple ring is isomorphic to the ring M n ( F q ) {\displaystyle \mathrm {M} _{n}(\mathb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4844_chunk_0 | Abstract Algebra | In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4845_chunk_0 | Abstract Algebra | Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4846_chunk_0 | Residue (complex analysis) | In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f: C ∖ { a k } k → C {\displaystyle f\colon \mathbb {C} \setmin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4847_chunk_0 | P-derivation | In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p-derivation is related to that of a derivation in differential algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4848_chunk_0 | Degree of an extension | In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4849_chunk_0 | Unbounded operator | In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should somet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4850_chunk_0 | Frobenius theorem (real division algebras) | In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4851_chunk_0 | Transfinite derived series | In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4852_chunk_0 | Parshin's conjecture | In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion: K i ( X ) ⊗ Q = 0 , i > 0. {\displaystyle K_{i}(X)\otimes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4853_chunk_0 | Griffiths group | In mathematics, more specifically in algebraic geometry, the Griffiths group of a projective complex manifold X measures the difference between homological equivalence and algebraic equivalence, which are two important equivalence relations of algebraic cycles. More precisely, it is defined as Griff k ( X ) := Z k ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4854_chunk_0 | Universal morphism | In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Forma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4855_chunk_0 | Universal morphism | Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a commutative ring R, the field of fractions of the quotient ring of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4856_chunk_0 | Straight-line program | In mathematics, more specifically in computational algebra, a straight-line program (SLP) for a finite group G = ⟨S⟩ is a finite sequence L of elements of G such that every element of L either belongs to S, is the inverse of a preceding element, or the product of two preceding elements. An SLP L is said to compute a gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4857_chunk_0 | Straight-line program | A black box algorithm is one which uses only these oracles. Hence, straight-line programs for black box groups are black box algorithms. Explicit straight-line programs are given for a wealth of finite simple groups in the online ATLAS of Finite Groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4858_chunk_0 | Milnor–Wood inequality | In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4859_chunk_0 | Maps of manifolds | In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4860_chunk_0 | Method of averaging | It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution. More precisely, the system has the following form of a phase space variable x . {\displaystyle x.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4861_chunk_0 | Method of averaging | The fast oscillation is given by f {\displaystyle f} versus a slow drift of x ˙ {\displaystyle {\dot {x}}} . The averaging method yields an autonomous dynamical system which approximates the solution curves of x ˙ {\displaystyle {\dot {x}}} inside a connected and compact region of the phase space and over time of 1 / ε... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4862_chunk_0 | Banach Spaces | In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4863_chunk_0 | Positive linear functional | In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space ( V , ≤ ) {\displaystyle (V,\leq )} is a linear functional f {\displaystyle f} on V {\displaystyle V} so that for all positive elements v ∈ V , {\displaystyle v\in V,} that is v ≥ 0 , {\displaystyle v\geq 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4864_chunk_0 | Positive linear functional | As in the case when V {\displaystyle V} is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W ⊆ V , {\displaystyle W\subseteq V,} and the partial order does not extend to all of V , {\displaystyle V,} in which case the positive elements of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4865_chunk_0 | Positive linear operator | In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} into a preordered vector space ( Y , ≤ ) {\displaystyle (Y,\leq )} is a linear operator f {\displaystyle f} on X {\displaystyle X} into Y {\displaystyle Y} such that f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4866_chunk_0 | Total subset | In mathematics, more specifically in functional analysis, a subset T {\displaystyle T} of a topological vector space X {\displaystyle X} is said to be a total subset of X {\displaystyle X} if the linear span of T {\displaystyle T} is a dense subset of X . {\displaystyle X.} This condition arises frequently in many theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4867_chunk_0 | Moore–Smith limit | In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4868_chunk_0 | Moore–Smith limit | In particular, the following two conditions are, in general, not equivalent for a map f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y}: The map f {\displaystyle f} is continuous in the topological sense; Given any point x {\displaystyle x} in X , {\displaystyle X,} and any sequ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4869_chunk_0 | Moore–Smith limit | The concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922, is to generalize the notion of a sequence so that the above conditions (with "sequence" being replaced by "net" in condition 2) are in fact equivalent for all maps of topological spaces. In particular, rather than being defined on a coun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4870_chunk_0 | Moore–Smith limit | Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behavior. The term "net" was coined by John L. Kelley.Nets are one of the many tools used in topology to generalize c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4871_chunk_0 | Tychonoff cube | In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4872_chunk_0 | Orthogonality relations | In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4873_chunk_0 | Walsh function | In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis. They can thus be viewed as a discrete, digital c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4874_chunk_0 | Walsh function | The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions.Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh. They find various applications in phy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4875_chunk_0 | Spark (mathematics) | In mathematics, more specifically in linear algebra, the spark of a m × n {\displaystyle m\times n} matrix A {\displaystyle A} is the smallest integer k {\displaystyle k} such that there exists a set of k {\displaystyle k} columns in A {\displaystyle A} which are linearly dependent. If all the columns are linearly inde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4876_chunk_0 | Cesaro's theorem | In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4877_chunk_0 | Baire set | In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4878_chunk_0 | Baire set | Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4879_chunk_0 | Alternating multilinear map | In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4880_chunk_0 | Biconjugate gradient method | In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method, this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint, but instead one n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4881_chunk_0 | Bendixson derivative | In mathematics, more specifically in point-set topology, the derived set of a subset S {\displaystyle S} of a topological space is the set of all limit points of S . {\displaystyle S.} It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4882_chunk_0 | Euclidean ring | In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4883_chunk_0 | Euclidean ring | It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4884_chunk_0 | Euclidean ring | So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain. Eu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4885_chunk_0 | Krull intersection theorem | In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4886_chunk_0 | Commutative ring theory | In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4887_chunk_0 | Commutative ring theory | The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4888_chunk_0 | Quantization commutes with reduction | In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of L. This was... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4889_chunk_0 | Siegel zero | In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero (also known as exceptional zero), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions asso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4890_chunk_0 | Solvable groups | In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4891_chunk_0 | Invariant basis number | In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4892_chunk_0 | Lienard equation | In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard. During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4893_chunk_0 | PBW theorem | In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4894_chunk_0 | Variance reduction | In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4895_chunk_0 | Variance reduction | The main ones are common random numbers, antithetic variates, control variates, importance sampling, stratified sampling, moment matching, conditional Monte Carlo and quasi random variables. For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4896_chunk_0 | Closed map | In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f: X → Y {\displaystyle f:X\to Y} is open if for any open set U {\displaystyle U} in X , {\displaystyle X,} the image f ( U ) {\displaystyle f(U)} is open in Y . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4897_chunk_0 | Closed map | A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4898_chunk_0 | Closed map | Recall that, by definition, a function f: X → Y {\displaystyle f:X\to Y} is continuous if the preimage of every open set of Y {\displaystyle Y} is open in X . {\displaystyle X.} (Equivalently, if the preimage of every closed set of Y {\displaystyle Y} is closed in X {\displaystyle X} ). Early study of open maps was pio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4899_chunk_0 | Volodin space | In mathematics, more specifically in topology, the Volodin space X {\displaystyle X} of a ring R is a subspace of the classifying space B G L ( R ) {\displaystyle BGL(R)} given by X = ⋃ n , σ B ( U n ( R ) σ ) {\displaystyle X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })} where U n ( R ) ⊂ G L n ( R ) {\displaystyle U_{n}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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