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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 \mathbf {Q} =0,\ \,i>0.} It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.
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https://en.wikipedia.org/wiki/Parshin's_conjecture
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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 ( X ) h o m / Z k ( X ) a l g {\displaystyle \operatorname {Griff} ^{k}(X):=Z^{k}(X)_{\mathrm {hom} }/Z^{k}(X)_{\mathrm {alg} }} where Z k ( X ) {\displaystyle Z^{k}(X)} denotes the group of algebraic cycles of some fixed codimension k and the subscripts indicate the groups that are homologically trivial, respectively algebraically equivalent to zero.This group was introduced by Phillip Griffiths who showed that for a general quintic in P 4 {\displaystyle \mathbf {P} ^{4}} (projective 4-space), the group Griff 2 ( X ) {\displaystyle \operatorname {Griff} ^{2}(X)} is not a torsion group.
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https://en.wikipedia.org/wiki/Griffiths_group
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In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties.
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https://en.wikipedia.org/wiki/Universal_morphism
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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 § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below).
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https://en.wikipedia.org/wiki/Universal_morphism
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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 R by a prime ideal p can be identified with the residue field of the localization of R at p; that is R p / p R p ≅ Frac ( R / p ) {\displaystyle R_{p}/pR_{p}\cong \operatorname {Frac} (R/p)} (all these constructions can be defined by universal properties). Other objects that can be defined by universal properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces.
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https://en.wikipedia.org/wiki/Universal_morphism
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In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal categories consist of a pair of objects in the ambient category—thought of as the 'object of objects' and 'object of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities. Group objects, are common examples of internal categories. There are notions of internal functors and natural transformations that make the collection of internal categories in a fixed category into a 2-category.
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https://en.wikipedia.org/wiki/Internal_category
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In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory K ( R ) {\displaystyle K(R)} has chromatic level one higher than that of a complex-oriented ring spectrum R. It was formulated by John Rognes in a lecture at Schloss Ringberg, Germany, in January 1999, and made more precise by him in a lecture at Mathematische Forschungsinstitut Oberwolfach, Germany, in September 2000. In July 2022, Burklund, Schlank and Yuan announced a solution of a version of the redshift conjecture for arbitrary E ∞ {\displaystyle E_{\infty }} -ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra BP .
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https://en.wikipedia.org/wiki/Redshift_conjecture
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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 group element g ∈ G if g ∈ L, where g is encoded by a word in S and its inverses. Intuitively, an SLP computing some g ∈ G is an efficient way of storing g as a group word over S; observe that if g is constructed in i steps, the word length of g may be exponential in i, but the length of the corresponding SLP is linear in i. This has important applications in computational group theory, by using SLPs to efficiently encode group elements as words over a given generating set. Straight-line programs were introduced by Babai and Szemerédi in 1984 as a tool for studying the computational complexity of certain matrix group properties.
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https://en.wikipedia.org/wiki/Straight-line_program
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Babai and Szemerédi prove that every element of a finite group G has an SLP of length O(log2|G|) in every generating set. An efficient solution to the constructive membership problem is crucial to many group-theoretic algorithms. It can be stated in terms of SLPs as follows.
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https://en.wikipedia.org/wiki/Straight-line_program
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Given a finite group G = ⟨S⟩ and g ∈ G, find a straight-line program computing g over S. The constructive membership problem is often studied in the setting of black box groups. The elements are encoded by bit strings of a fixed length. Three oracles are provided for the group-theoretic functions of multiplication, inversion, and checking for equality with the identity.
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https://en.wikipedia.org/wiki/Straight-line_program
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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.
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https://en.wikipedia.org/wiki/Straight-line_program
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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.
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https://en.wikipedia.org/wiki/Milnor–Wood_inequality
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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
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https://en.wikipedia.org/wiki/Maps_of_manifolds
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In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale.
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https://en.wikipedia.org/wiki/Method_of_averaging
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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.}
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https://en.wikipedia.org/wiki/Method_of_averaging
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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 {\displaystyle 1/\varepsilon } . Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for y {\displaystyle y} . In this way, qualitative methods for autonomous dynamical systems may be employed to analyze the equilibria and more complex structures, such as slow manifold and invariant manifolds, as well as their stability in the phase space of the averaged system. In addition, in a physical application it might be reasonable or natural to replace a mathematical model, which is given in the form of the differential equation for x ˙ {\displaystyle {\dot {x}}} , with the corresponding averaged system y ˙ {\displaystyle {\dot {y}}} , in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ).
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https://en.wikipedia.org/wiki/Method_of_averaging
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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 converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.
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https://en.wikipedia.org/wiki/Banach_Spaces
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In mathematics, more specifically in functional analysis, a K-space is an F-space V {\displaystyle V} such that every extension of F-spaces (or twisted sum) of the form is equivalent to the trivial one where R {\displaystyle \mathbb {R} } is the real line.
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https://en.wikipedia.org/wiki/K-space_(functional_analysis)
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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,} it holds that In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f ( v ) {\displaystyle f(v)} is real.
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https://en.wikipedia.org/wiki/Positive_linear_functional
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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 V {\displaystyle V} are the positive elements of W , {\displaystyle W,} by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x ∈ V {\displaystyle x\in V} equal to s ∗ s {\displaystyle s^{\ast }s} for some s ∈ V {\displaystyle s\in V} to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such x . {\displaystyle x.} This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
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https://en.wikipedia.org/wiki/Positive_linear_functional
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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 for all positive elements x {\displaystyle x} of X , {\displaystyle X,} that is x ≥ 0 , {\displaystyle x\geq 0,} it holds that f ( x ) ≥ 0. {\displaystyle f(x)\geq 0.} In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain. Every positive linear functional is a type of positive linear operator. The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.
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https://en.wikipedia.org/wiki/Positive_linear_operator
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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 theorems of functional analysis.
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https://en.wikipedia.org/wiki/Total_subset
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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 generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces.
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https://en.wikipedia.org/wiki/Moore–Smith_limit
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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 sequence in X {\displaystyle X} converging to x , {\displaystyle x,} the composition of f {\displaystyle f} with this sequence converges to f ( x ) {\displaystyle f(x)} (continuous in the sequential sense).While condition 1 always guarantees condition 2, the converse is not necessarily true if the topological spaces are not both first-countable. In particular, the two conditions are equivalent for metric spaces. The spaces for which the converse holds are the sequential spaces.
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https://en.wikipedia.org/wiki/Moore–Smith_limit
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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 countable linearly ordered set, a net is defined on an arbitrary directed set. This allows for theorems similar to the assertion that the conditions 1 and 2 above are equivalent to hold in the context of topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point.
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https://en.wikipedia.org/wiki/Moore–Smith_limit
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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 certain concepts that may not be general enough in the context of metric spaces. A related notion, that of the filter, was developed in 1937 by Henri Cartan.
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https://en.wikipedia.org/wiki/Moore–Smith_limit
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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 arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the product of collections of arbitrary compact spaces. This result is now known as Tychonoff's theorem and is considered one of the most important results in general topology.
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https://en.wikipedia.org/wiki/Tychonoff_cube
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In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
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https://en.wikipedia.org/wiki/Kirby–Siebenmann_class
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In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G(1) = G (the commutator subgroup equals the group), or equivalently one such that Gab = {1} (its abelianization is trivial).
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https://en.wikipedia.org/wiki/Grün's_lemma
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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 developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.
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https://en.wikipedia.org/wiki/Orthogonality_relations
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This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.
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https://en.wikipedia.org/wiki/Orthogonality_relations
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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 counterpart of the continuous, analog system of trigonometric functions on the unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions.
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https://en.wikipedia.org/wiki/Walsh_function
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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 physics and engineering when analyzing digital signals. Historically, various numerations of Walsh functions have been used; none of them is particularly superior to another. This articles uses the Walsh–Paley numeration.
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https://en.wikipedia.org/wiki/Walsh_function
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In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf Hom ( − , U ) {\displaystyle \operatorname {Hom} (-,U)} .
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https://en.wikipedia.org/wiki/Homotopy_sheaf
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Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf B G {\displaystyle BG} .
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https://en.wikipedia.org/wiki/Homotopy_sheaf
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For example, one might set B GL = lim → B G L n {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} } . These types of examples appear in K-theory. If f: X → Y {\displaystyle f:X\to Y} is a local weak equivalence of simplicial presheaves, then the induced map Z f: Z X → Z Y {\displaystyle \mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y} is also a local weak equivalence.
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https://en.wikipedia.org/wiki/Homotopy_sheaf
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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 independent, s p a r k ( A ) {\displaystyle \mathrm {spark} (A)} is usually defined to be 1 more than the number of rows. The concept of matrix spark finds applications in error-correction codes, compressive sensing, and matroid theory, and provides a simple criterion for maximal sparsity of solutions to a system of linear equations. The spark of a matrix is NP-hard to compute.
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https://en.wikipedia.org/wiki/Spark_(mathematics)
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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.
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https://en.wikipedia.org/wiki/Cesaro's_theorem
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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 smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.
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https://en.wikipedia.org/wiki/Baire_set
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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 Borel sets. Baire sets were introduced by Kunihiko Kodaira (1941, Definition 4), Shizuo Kakutani and Kunihiko Kodaira (1944) and Halmos (1950, page 220), who named them after Baire functions, which are in turn named after René-Louis Baire.
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https://en.wikipedia.org/wiki/Baire_set
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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 over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
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https://en.wikipedia.org/wiki/Alternating_multilinear_map
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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 needs to perform multiplications by the conjugate transpose A*.
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https://en.wikipedia.org/wiki/Biconjugate_gradient_method
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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 in large part to study derived sets on the real line.
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https://en.wikipedia.org/wiki/Bendixson_derivative
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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 as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
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https://en.wikipedia.org/wiki/Euclidean_ring
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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 Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
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https://en.wikipedia.org/wiki/Euclidean_ring
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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. Euclidean domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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https://en.wikipedia.org/wiki/Euclidean_ring
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In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.
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https://en.wikipedia.org/wiki/Cyclic_module
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In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R). It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.
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https://en.wikipedia.org/wiki/Maximal_submodule
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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 branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
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https://en.wikipedia.org/wiki/Krull_intersection_theorem
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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: I 1 ⊆ ⋯ I k − 1 ⊆ I k ⊆ I k + 1 ⊆ ⋯ {\displaystyle I_{1}\subseteq \cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq \cdots } there exists an n such that: I n = I n + 1 = ⋯ {\displaystyle I_{n}=I_{n+1}=\cdots } For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.)
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https://en.wikipedia.org/wiki/Commutative_ring_theory
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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 the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.
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https://en.wikipedia.org/wiki/Commutative_ring_theory
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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 conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken (the second paper used symplectic cut) as well as Tian and Zhang. For the formulation due to Teleman, see C. Woodward's notes.
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https://en.wikipedia.org/wiki/Quantization_commutes_with_reduction
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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 associated to quadratic number fields. Roughly speaking, these are possible zeros very near (in a quantifiable sense) to s = 1 {\displaystyle s=1} .
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https://en.wikipedia.org/wiki/Siegel_zero
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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.
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https://en.wikipedia.org/wiki/Solvable_groups
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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.
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https://en.wikipedia.org/wiki/Invariant_basis_number
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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 can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits.
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https://en.wikipedia.org/wiki/Lienard_equation
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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 theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups.
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https://en.wikipedia.org/wiki/PBW_theorem
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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 the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used.
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https://en.wikipedia.org/wiki/Variance_reduction
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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 specialized techniques; for example, particle transport simulations make extensive use of "weight windows" and "splitting/Russian roulette" techniques, which are a form of importance sampling.
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https://en.wikipedia.org/wiki/Variance_reduction
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In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence 0 → H → ı X → π G → 0 {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0} where H , X {\displaystyle H,X} and G {\displaystyle G} are topological groups and i {\displaystyle i} and π {\displaystyle \pi } are continuous homomorphisms which are also open onto their images. Every extension of topological groups is therefore a group extension.
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https://en.wikipedia.org/wiki/Extension_of_a_topological_group
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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 . {\displaystyle Y.} Likewise, a closed map is a function that maps closed sets to closed sets.
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https://en.wikipedia.org/wiki/Closed_map
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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 remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps.
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https://en.wikipedia.org/wiki/Closed_map
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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 pioneered by Simion Stoilow and Gordon Thomas Whyburn.
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https://en.wikipedia.org/wiki/Closed_map
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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}(R)\subset GL_{n}(R)} is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and σ {\displaystyle \sigma } a permutation matrix thought of as an element in G L n ( R ) {\displaystyle GL_{n}(R)} and acting (superscript) by conjugation. The space is acyclic and the fundamental group π 1 X {\displaystyle \pi _{1}X} is the Steinberg group St ( R ) {\displaystyle \operatorname {St} (R)} of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction B G L ( R ) / X ≃ B G L + ( R ) {\displaystyle BGL(R)/X\simeq BGL^{+}(R)} in algebraic K-theory.
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https://en.wikipedia.org/wiki/Volodin_space
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In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. The field has become more active recently because of its connection to algebraic K-theory.
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https://en.wikipedia.org/wiki/Equivariant_stable_homotopy_theory
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In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength. Three of the most common notions of convergence are described below.
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https://en.wikipedia.org/wiki/Portmanteau_theorem
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In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R.The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space. This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings.
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https://en.wikipedia.org/wiki/Jacobson_density_theorem
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In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold: Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (X is a regular Hausdorff space.) There is a countable collection of open covers of X, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C.
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https://en.wikipedia.org/wiki/Moore_space_(topology)
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(X is a developable space. )Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century.
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https://en.wikipedia.org/wiki/Moore_space_(topology)
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In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in (Levitzki 1945). The result was originally submitted in 1939 as (Levitzki 1950), and a particularly simple proof was given in (Utumi 1963).
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https://en.wikipedia.org/wiki/Levitzky's_theorem
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In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are different from unique factorization domains in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime element. Important examples of atomic domains include the class of all unique factorization domains and all Noetherian domains. More generally, any integral domain satisfying the ascending chain condition on principal ideals (ACCP) is an atomic domain. Although the converse is claimed to hold in Cohn's paper, this is known to be false.The term "atomic" is due to P. M. Cohn, who called an irreducible element of an integral domain an "atom".
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https://en.wikipedia.org/wiki/Atomic_domain
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In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. By I k, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0. The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem. The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
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https://en.wikipedia.org/wiki/Nilpotent_ideal
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In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle R} that annihilate all simple right R {\displaystyle R} -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J ( R ) {\displaystyle J(R)} or rad ( R ) {\displaystyle \operatorname {rad} (R)} ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in (Jacobson 1945).
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https://en.wikipedia.org/wiki/Jacobson_radical
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The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma.
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https://en.wikipedia.org/wiki/Jacobson_radical
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In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.
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https://en.wikipedia.org/wiki/Exceptional_inverse_image_functor
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In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f: X → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle Y.} Local homeomorphisms are used in the study of sheaves.
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https://en.wikipedia.org/wiki/Local_homeomorphism
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Typical examples of local homeomorphisms are covering maps. A topological space X {\displaystyle X} is locally homeomorphic to Y {\displaystyle Y} if every point of X {\displaystyle X} has a neighborhood that is homeomorphic to an open subset of Y . {\displaystyle Y.}
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https://en.wikipedia.org/wiki/Local_homeomorphism
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For example, a manifold of dimension n {\displaystyle n} is locally homeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.} If there is a local homeomorphism from X {\displaystyle X} to Y , {\displaystyle Y,} then X {\displaystyle X} is locally homeomorphic to Y , {\displaystyle Y,} but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane R 2 , {\displaystyle \mathbb {R} ^{2},} but there is no local homeomorphism S 2 → R 2 . {\displaystyle S^{2}\to \mathbb {R} ^{2}.}
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https://en.wikipedia.org/wiki/Local_homeomorphism
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In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in R n {\displaystyle \mathbb {R} ^{n}} . This number depends on the size and shape of the bodies, and their relative orientation to each other.
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https://en.wikipedia.org/wiki/Mixed_volume
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In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. For infinite places, Jean-Pierre Serre gave a recipe in (Serre 1970) for the so-called Gamma factors in terms of the Hodge realization of the motive. It is conjectured that, like other L-functions, that each motivic L-function can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the L-function L(s, M) of a motive M to L(1 − s, M∨), where M∨ is the dual of the motive M.
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https://en.wikipedia.org/wiki/Motivic_L-function
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In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem.
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https://en.wikipedia.org/wiki/Multipliers_and_centralizers_(Banach_spaces)
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In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions.
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https://en.wikipedia.org/wiki/Near_sets
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Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets.
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https://en.wikipedia.org/wiki/Near_sets
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The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near sets.
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https://en.wikipedia.org/wiki/Near_sets
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Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems. Near sets have a variety of applications in areas such as topology, pattern detection and classification, abstract algebra, mathematics in computer science, and solving a variety of problems based on human perception that arise in areas such as image analysis, image processing, face recognition, ethology, as well as engineering and science problems.
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https://en.wikipedia.org/wiki/Near_sets
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From the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition , spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology. As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colour model for varying degrees of nearness between sets of picture elements in pictures (see, e.g., §4.3). The two pairs of ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals in Fig.1 are closer to each other descriptively than the ovals in Fig. 2.
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https://en.wikipedia.org/wiki/Near_sets
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In mathematics, negacyclic convolution is a convolution between two vectors a and b. It is also called skew circular convolution or wrapped convolution. It results from multiplication of a skew circulant matrix, generated by vector a, with vector b.
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https://en.wikipedia.org/wiki/Negacyclic_convolution
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In mathematics, negafibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end.
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https://en.wikipedia.org/wiki/Negafibonacci_coding
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In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: Negative-definite bilinear form Negative-definite quadratic form Negative-definite matrix Negative-definite function
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https://en.wikipedia.org/wiki/Negative_definite
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In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras.
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https://en.wikipedia.org/wiki/Nilpotent_orbit
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In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry. There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property (i.e. with respect to order or magnitude).
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https://en.wikipedia.org/wiki/Non-Archimedean_geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.
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https://en.wikipedia.org/wiki/Non-Euclidean_space
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In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.
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https://en.wikipedia.org/wiki/Non-abelian_class_field_theory
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In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras. Many of the higher-dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theory, and also of Nonabelian Algebraic Topology (NAAT), which generalises to higher dimensions ideas coming from the fundamental group. Such algebraic structures in dimensions greater than 1 develop the nonabelian character of the fundamental group, and they are in a precise sense ‘more nonabelian than the groups'. These noncommutative, or more specifically, nonabelian structures reflect more accurately the geometrical complications of higher dimensions than the known homology and homotopy groups commonly encountered in classical algebraic topology.
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https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology
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An important part of nonabelian algebraic topology is concerned with the properties and applications of homotopy groupoids and filtered spaces. Noncommutative double groupoids and double algebroids are only the first examples of such higher-dimensional structures that are nonabelian. The new methods of Nonabelian Algebraic Topology (NAAT) "can be applied to determine homotopy invariants of spaces, and homotopy classification of maps, in cases which include some classical results, and allow results not available by classical methods".
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https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology
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Cubical omega-groupoids, higher homotopy groupoids, crossed modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy of filtered spaces, higher-dimensional space structures, the construction of the fundamental groupoid of a topos E in the general theory of topoi, and also in their physical applications in nonabelian quantum theories, and recent developments in quantum gravity, as well as categorical and topological dynamics. Further examples of such applications include the generalisations of noncommutative geometry formalizations of the noncommutative standard models via fundamental double groupoids and spacetime structures even more general than topoi or the lower-dimensional noncommutative spacetimes encountered in several topological quantum field theories and noncommutative geometry theories of quantum gravity. A fundamental result in NAAT is the generalised, higher homotopy van Kampen theorem proven by R. Brown, which states that "the homotopy type of a topological space can be computed by a suitable colimit or homotopy colimit over homotopy types of its pieces''.
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https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology
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A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include statements for 2-categories and a topos of topoi . Important results in higher-dimensional algebra are also the extensions of the Galois theory in categories and variable categories, or indexed/'parametrized' categories. The Joyal–Tierney representation theorem for topoi is also a generalisation of the Galois theory. Thus, indexing by bicategories in the sense of Benabou one also includes here the Joyal–Tierney theory.
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https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology
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In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory. The main task is therefore the case of G that is locally compact, not compact and not commutative.
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https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis
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The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations. What to expect is known as the result of basic work of John von Neumann.
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https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis
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He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analogue of the Plancherel theorem is abstractly given by identifying a measure on the unitary dual, the Plancherel measure, with respect to which the direct integral is taken.
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https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis
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