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c_vs030jtjcw3s | Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem). For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For ev... | Undefined term |
c_lpmb4ee1owz6 | In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. | Loop algebra |
c_m3zldalk8foi | In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also random walk for more general treatment of this topic. | Uniform spanning tree |
c_nlh9pqrf3l60 | In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology... | Low-dimensional topology |
c_o6blma9lgnoh | In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling a... | Low-rank approximation |
c_lps7fx4ju9vj | The order of the entries of the vectors x {\displaystyle \mathbf {x} } or y {\displaystyle \mathbf {y} } does not affect the majorization, e.g., the statement ( 1 , 2 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . As a consequence, majoriz... | Majorization |
c_cxmc87kb5fr9 | For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. Various other generalizations of majorization are discussed in chapters 14 and 15 of. | Majorization |
c_m3p81u36xgfk | In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes. | List of logarithmic identities |
c_blxyffedsog5 | In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the Bernoulli numbers and the Bell numbers. In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probabi... | Zero-based numbering |
c_kqb9euuieubx | In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated ex... | Mathematical maturity |
c_it3mf3b0lhfl | In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computi... | Mathematical economics |
c_rwpvqq8yyh6n | More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economic... | Mathematical economics |
c_00rdwv5mpa4x | Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be... | Mathematical economics |
c_9t48php9z2ac | Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.Linear and nonlinear programming have profoundly affected... | Mathematical economics |
c_hcwlrf1omvva | In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R , {\displaystyle f:U\to \mathbb {R} ,} where U is an open subset of R n , {\displaystyle \mathbb {R} ^{n},} that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 ... | Harmonic function |
c_qpt52melv7a2 | In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field the... | Spin current |
c_lfzk2gwvc9qx | In mathematics, mathematical structures can have more than one definition. Therefore, there are several definitions of named sets, each representing a specific construction of named set theory. The informal definition is the most general. | Named set theory |
c_zb4c1jl3mfky | In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} | Matrix addition |
c_jr01n7nvk9yp | , then adding the transformed vectors. A v → + B v → = ( A + B ) v → {\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!} However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. | Matrix addition |
c_rwg52ymvkxlx | In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices ... | Matrix differentiation |
c_xcrvj73oxhgk | Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is ma... | Matrix differentiation |
c_rzfcx1okzrdd | 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. | Matrix differentiation |
c_gcl3hm1toiew | 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... | Matrix differentiation |
c_kuxfcby0lihg | 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... | Mimesis (mathematics) |
c_l08bdnkoda7q | 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 == | Mimesis (mathematics) |
c_jz9asmz6qaly | 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... | Mimetic interpolation |
c_kyvj8dpzzmo6 | 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... | Minimum polynomial extrapolation |
c_1o9zsgwyhvt2 | Given iterates x 1 , x 2 , . . | Minimum polynomial extrapolation |
c_qwnl112xv8nk | . , 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 , . . | Minimum polynomial extrapolation |
c_buz9skidjuua | . , 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 {\... | Minimum polynomial extrapolation |
c_pzew8enn8564 | 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... | Minimum polynomial extrapolation |
c_dzlcktdr1hk1 | 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. | Mirror descent |
c_3f20pqp5oymf | In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as per... | Mirror symmetry conjecture |
c_lv3zwtaf9fyk | 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 ... | Mixed Hodge module |
c_sn5d0h88oto7 | In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, industrial mixing. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dyna... | Strong mixing |
c_06v8so3gbog6 | 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... | Modern triangle geometry |
c_nnsu4nsf38w6 | 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... | Modern triangle geometry |
c_nyyvsj1xdtvu | 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... | Modern triangle geometry |
c_sg66u3a8c8jq | (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... | Modern triangle geometry |
c_1b7ofgyjrq3z | (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. | Modern triangle geometry |
c_u8pk5mttn3f3 | 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. | Modern triangle geometry |
c_h1xf066m1d5a | 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. | Modern triangle geometry |
c_qis3099ghamt | 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 ... | Modern triangle geometry |
c_3kkiw7hg3w8i | 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 ... | Residue class |
c_kzwxs0sk6lsp | If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on ... | Residue class |
c_mgffrvidrx3p | 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. | Modular forms modulo p |
c_vqf71okvcn44 | In mathematics, modular symbols, introduced independently by Bryan John Birch and by Manin (1972), span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms. | Modular symbol |
c_3b7na91axprk | In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by Kubert and Lang (1975). They are functions whose zeroes and poles are confined to the cusps (images of infinity). | Modular unit |
c_fsvf4zu4782m | 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. | Modulus of smoothness |
c_wk4lyvv3nkl6 | 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... | Cutoff function |
c_19ie6vc748ah | 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... | Monodromy theory |
c_8q0oe3oiv71i | 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... | Monstrous Moonshine |
c_fuxhgzne6s5v | In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also de... | Truncated subtraction |
c_0dxfs6u273p9 | 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... | Positive current |
c_v1d0lmzeh433 | 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... | Positive current |
c_blnuo4a1kvpq | 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 ... | Integral current |
c_64j2qvseek3v | 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... | Rational singularities |
c_qgz4jokuizjn | In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorp... | Anosov system |
c_7886d52170vy | 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... | Prosolvable group |
c_xhn8hyi091wg | In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme. | Étale group scheme |
c_n378i4k4ndi2 | 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... | Solder form |
c_v0zq84e3zsof | In mathematics, more precisely in formal language theory, the profinite words are a generalization of the notion of finite words into a complete topological space. This notion allows the use of topology to study languages and finite semigroups. For example, profinite words are used to give an alternative characterizati... | Profinite word |
c_wm6ognc9rkea | 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... | Energetic space |
c_sgfpluuislro | 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 {... | Arithmetic hyperbolic 3-manifold |
c_i9cv3osdcr4y | 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... | Lebesgue's decomposition theorem |
c_o9uhh9v9c65w | In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure)... | Discrete measure |
c_x9o2o4onk71d | In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. | Atom (measure theory) |
c_n5eyiyic4cyz | In mathematics, more precisely in operator theory, a sectorial operator is a linear operator on a Banach space, whose spectrum in an open sector in the complex plane and whose resolvent is uniformly bounded from above outside any larger sector. Such operators might be unbounded. Sectorial operators have applications in... | Sectorial operator |
c_nvjls03pr4d7 | 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... | Contact type |
c_xzkqlcbhwug0 | 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} }^{... | Pseudoconvex domain |
c_ngd1kdo3n15y | 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... | Hyperbolic 3-manifold |
c_kd21i4taaxcu | In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The ... | Simplicial group |
c_w7l901ys5fs5 | 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... | Simplicial group |
c_46y322zp0zm4 | In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the ... | Dold–Kan correspondence |
c_wzezoqjy09ap | 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"... | Dold–Kan correspondence |
c_k56x1wz129mx | 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... | Nakayama's lemma |
c_037fpmcgcw50 | 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... | Nakayama's lemma |
c_amktnhowub3l | 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... | Finite ring |
c_kcu0cgkdpkl7 | 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... | Finite ring |
c_8ph5i4mlgjkq | 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... | Abstract Algebra |
c_mjokp6f9s8h8 | 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... | Abstract Algebra |
c_szvx7u1ky2ro | In mathematics, more specifically algebraic topology, a pair ( X , A ) {\displaystyle (X,A)} is shorthand for an inclusion of topological spaces i: A ↪ X {\displaystyle i\colon A\hookrightarrow X} . Sometimes i {\displaystyle i} is assumed to be a cofibration. A morphism from ( X , A ) {\displaystyle (X,A)} to ( X ′ , ... | Pair of spaces |
c_cgaofwn7f62p | A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A. Pairs of spaces occur centrally in relative homology, homology theory and cohomo... | Pair of spaces |
c_nuojnr6iqome | There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X {\displaystyle X} to the pair ( X , ∅ ) {\displaystyle (X,\varnothing )} . A related concept is that of a triple (X, A, B), with B ⊂ A ⊂ X. Triples are used in homotopy theory. Often, for a pointed space... | Pair of spaces |
c_564421d2iqsi | In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory. Quasi-ca... | Weak Kan complex |
c_uq237w7z2eoe | André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by Jacob Lurie (2009). Quasi-categories are... | Weak Kan complex |
c_9a3edbb3dloh | Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other... | Weak Kan complex |
c_djze62t81f0m | These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc. The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there... | Weak Kan complex |
c_wpyytfng4sy2 | 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... | Residue (complex analysis) |
c_n5gj6tvepoep | 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. | P-derivation |
c_nkfw33wf2ig9 | In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. | Local diffeomorphism |
c_9awprpjuqtxk | 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. | Degree of an extension |
c_bmdrwd65vlhm | 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... | Unbounded operator |
c_q2u2fuzxhu14 | In mathematics, more specifically general topology, the divisor topology is a specific topology on the set X = { 2 , 3 , 4 , . . . } {\displaystyle X=\{2,3,4,...\}} of positive integers greater than or equal to two. The divisor topology is the poset topology for the partial order relation of divisibility of integers on... | Divisor topology |
c_qb9z3fz9la5s | In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set X = R2 ∐ {0*}, where ∐ denotes the disjoint union. | Double origin topology |
c_uui4u9c7r1yg | In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval (0,1), i.e. the set of all real numbers x such that 0 < x < 1. The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1. To give the set ... | Nested interval topology |
c_r60qzwvw13c5 | In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set R of real numbers. | Rational sequence topology |
c_5wlkjotuq6sp | In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity. | Three subgroups lemma |
c_q0rhll722cv9 | 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 ... | Frobenius theorem (real division algebras) |
c_lsctl5bth4zm | 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 ... | Transfinite derived series |
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