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c_xwcq325jlu45 | In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains. | Dedekind–Hasse norm |
c_zfult9l4lrya | In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. | Rational mapping |
c_hm6qfr1u8lxl | In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite refl... | Weyl chamber |
c_gslpfwd9q2ic | In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup. Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup... | Rees factor semigroup |
c_llgk098sa2hg | In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L {\displaystyle L} , is a particular class of sets that can be described entirely in terms of simpler sets. L {\displaystyle L} is the union of the constructible hierarchy L α {\displaystyle L_{\alpha }} . It was ... | Gödel constructible universe |
c_bchj3sqjtivx | In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions ar... | Gödel constructible universe |
c_fr0xv19gyudr | In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory,... | Galois group |
c_3epwx8pski1s | In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words. For example, given the word xy, the corresponding verbal subgroup is generated by t... | Verbal subgroup |
c_eyq44rq4kqnl | Verbal subgroups are the only fully characteristic subgroups of a free group and therefore represent the generic example of fully characteristic subgroups, (Magnus, Karrass & Solitar 2004, p. 75). Another example is the verbal subgroup for { x − 1 y − 1 x y } {\displaystyle \{x^{-1}y^{-1}xy\}} , which is the derived su... | Verbal subgroup |
c_abjs3t4fkjqj | In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure. | A-group |
c_0ls35x9ig7x9 | In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this number cannot be too close to being a linear function. The theorem is named after ... | Erdős–Fuchs theorem |
c_3qxhlng6tjvk | In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of (Burnside 1911): are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvab... | CN-group |
c_4m3jsh4570yd | In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients. Some of their basic properties were established in (Berrick & Robinson 1993). The study of imperfect groups apparently began in (Robinson 1972).The class of imperfect groups is closed under ... | Imperfect group |
c_0yw4zndkyvw4 | The (restricted or unrestricted) direct product of imperfect groups is imperfect. Every solvable group is imperfect. Finite symmetric groups are also imperfect. | Imperfect group |
c_9jj2sj0f61ng | The general linear groups PGL(2,q) are imperfect for q an odd prime power. For any group H, the wreath product H wr Sym2 of H with the symmetric group on two points is imperfect. In particular, every group can be embedded as a two-step subnormal subgroup of an imperfect group of roughly the same cardinality (2|H|2). | Imperfect group |
c_p1ch369ronnk | In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 (Isaacs 1994). In this section only finite groups are considered. A monomial group is ... | Monomial group |
c_h6ma4ya8nmff | Every supersolvable group (Bray et al. 1982, Cor 2.3.5) and every solvable A-group (Bray et al. 1982, Thm 2.3.10) is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group, as shown by (Dade & ????) and in textboo... | Monomial group |
c_tk61ihpa78ig | In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. | Homotopy extension property |
c_fsgxwthw6kyx | In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — a... | Dirichlet eta function |
c_ax12a5sn82zi | This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is entire and η ( 1 ) ≠ 0 {\displaystyle \eta (1)\neq 0} together show the zeta function is meromorphic with a simple pole at s = 1, and possibly additional poles at the other zeros of the factor 1 − 2 ... | Dirichlet eta function |
c_srbfq99v6xn7 | In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signat... | Forgetful functor |
c_tz6n5lfcmnf1 | In mathematics, in the area of classical potential theory, polar sets are the "negligible sets", similar to the way in which sets of measure zero are the negligible sets in measure theory. | Polar set (potential theory) |
c_9n37vyi1poqi | In mathematics, in the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see Chung et al. (1994). | Q-derivative |
c_ze3pyeeocfrh | In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke (1988, 1990). Let R {\displaystyle R} be a commutative noetherian ring containing a field of characteristic p > 0 {\displaystyle p>0} . He... | Tight closure |
c_028qav50a3k2 | Let I {\displaystyle I} be an ideal of R {\displaystyle R} . The tight closure of I {\displaystyle I} , denoted by I ∗ {\displaystyle I^{*}} , is another ideal of R {\displaystyle R} containing I {\displaystyle I} . The ideal I ∗ {\displaystyle I^{*}} is defined as follows. | Tight closure |
c_4wkd3wdfgpcv | z ∈ I ∗ {\displaystyle z\in I^{*}} if and only if there exists a c ∈ R {\displaystyle c\in R} , where c {\displaystyle c} is not contained in any minimal prime ideal of R {\displaystyle R} , such that c z p e ∈ I {\displaystyle cz^{p^{e}}\in I^{}} for all e ≫ 0 {\displaystyle e\gg 0} . If R {\displaystyle R} is reduce... | Tight closure |
c_i1muf45qjh4z | A ring in which all ideals are tightly closed is called weakly F {\displaystyle F} -regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of F {\displaystyle F} -regular, which says tha... | Tight closure |
c_i4us4vf8jsqv | In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lind... | Carlson's theorem |
c_pwhm1ft01r73 | In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates... | Nachbin resummation |
c_m0wpcnbv1qzz | In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases. | Difference polynomials |
c_m6gfyml8yfiw | In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate doma... | Fractional Fourier transform |
c_v4a6j76dxiwi | An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several group... | Fractional Fourier transform |
c_qq7yixyaiqpq | A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear... | Fractional Fourier transform |
c_uphykg0xd4g7 | In mathematics, in the area of lambda calculus and computation, directors or director strings are a mechanism for keeping track of the free variables in a term. Loosely speaking, they can be understood as a kind of memoization for free variables; that is, as an optimization technique for rapidly locating the free varia... | Director string |
c_wl1xinp5bm4y | In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely... | Gaussian period |
c_5tkqc3pi2oar | In mathematics, in the area of numerical analysis, Galerkin methods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basi... | Galerkin's method |
c_vwu2191yevbx | In an operator formulation of the differential equation, Bubnov–Galerkin method can be viewed as applying an orthogonal projection to the operator. Petrov–Galerkin method (after Georgii I. Petrov) allows using basis functions for orthogonality constraints (called test basis functions) that are different from the basis ... | Galerkin's method |
c_dfzhnbjmikbm | In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property. | Free lattice word problem |
c_4qwiyp6a0c5o | In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (t... | Antichain |
c_amf5ihsm5mog | The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive la... | Antichain |
c_j4z40uihfu20 | In mathematics, in the area of order theory, an upwards centered set S is a subset of a partially ordered set, P, such that any finite subset of S has an upper bound in P. Similarly, any finite subset of a downwards centered set has a lower bound. An upwards centered set can also be called a consistent set. Any directe... | Centered set |
c_9py2m316khqy | In mathematics, in the area of potential theory, a Lebesgue spine or Lebesgue thorn is a type of set used for discussing solutions to the Dirichlet problem and related problems of potential theory. The Lebesgue spine was introduced in 1912 by Henri Lebesgue to demonstrate that the Dirichlet problem does not always have... | Lebesgue spine |
c_ntcf6z1j9xs0 | In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions. | Pluripolar set |
c_v7nxg3mdbzmr | In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is i... | Helstrom metric |
c_tuofwxtz4c5k | In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. | Bispectrum |
c_v1ltdektylmu | In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds. It is a part of a more general invariant known as symplectic field theory, and is defined using ... | Relative contact homology |
c_2l0irk0ebahc | In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function φ {\displaystyle \varphi } is called refinable with respect to the mask h {\displaystyle h} if φ ( x ) = 2 ⋅ ∑ k = 0 N − 1 h k ⋅ φ ( 2 ⋅ x − k ) {\displaystyle \varphi (x)=2\cdot \s... | Refinable function |
c_82djec7ki22v | The operator φ ↦ 2 ⋅ D 1 / 2 ( h ∗ φ ) {\displaystyle \varphi \mapsto 2\cdot D_{1/2}(h*\varphi )} is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not uniquely defined. That is, if φ {\displaystyle \varphi } is a refinable function, then for every c {\displaystyle c} the funct... | Refinable function |
c_q240n2jfh4mp | In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly... | Lyndon word |
c_npvx17qkl3ac | In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and ... | Hall word |
c_5sh7exg1w08f | These are binary trees; taken together, they form the Hall set. This set is a particular totally ordered subset of a free non-associative algebra, that is, a free magma. In this form, the Hall trees provide a basis for free Lie algebras, and can be used to perform the commutations required by the Poincaré–Birkhoff–Witt... | Hall word |
c_w7hjc641tfis | As such, this generalizes the same process when done with the Lyndon words. Hall trees can also be used to give a total order to the elements of a group, via the commutator collecting process, which is a special case of the general construction given below. | Hall word |
c_fi7k6edw231h | It can be shown that Lazard sets coincide with Hall sets. The historical development runs in reverse order from the above description. The commutator collecting process was described first, in 1934, by Philip Hall and explored in 1937 by Wilhelm Magnus. | Hall word |
c_73hmmh5nfgtt | Hall sets were introduced by Marshall Hall based on work of Philip Hall on groups. Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory du... | Hall word |
c_ud3bi9y11ell | In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P. Dilworth (1950). An antichain in a partially ordered set is a s... | Chain decomposition |
c_sgq42t8na6o8 | A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain d... | Chain decomposition |
c_jeb8qgueczy6 | In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains. It is named for Leon Mirsky (1971) and is closely related to Dilworth's theorem on the widths of partial... | Mirsky's theorem |
c_jgo65wuhp3cm | In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by Mikhail Kadets (1966) and Richard Davis Anderson. | Kadec norm |
c_qdke68kyptgq | In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x ... | Graded poset |
c_t0es8wtc16jz | In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. | Univalent function |
c_b3b915z56tf3 | In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely ... | Structure theorem for finitely generated modules over a principal ideal domain |
c_ruvkyewsfulc | In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness. They are used in the study of arithmetic progressions in the group. Th... | Uniformity norm |
c_55tfyur3h15u | In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. | Period mapping |
c_v51c750d1tds | In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized by the prime ideals with inertial degree one in the extension. For a finite degree extension L/K of an algebraic number field K we define P(L/K) to be the set of primes ... | Bauerian extension |
c_637nz6e4w1te | In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. | Modulus (algebraic number theory) |
c_nj0r7l1ncmfd | In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units. | S-unit equation |
c_93emixlbnica | In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's origina... | Eilenberg–Moore spectral sequence |
c_j090y5troyvw | In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety X over a global field, which measures the failure of the Hasse principle for X. If the value of the obstruction is non-trivial, then X may have points over all local fields but not over... | Manin obstruction |
c_c7mr5fi0wyls | In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: homC(X, X) = {idX} for all objects X homC(X, Y) = ∅ for all objects X ≠ YSince by axioms, there is always the identity morphism between the same object, we can express the above as conditi... | Discrete category |
c_qprn3xbqhunw | In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that ( m + n k ) q = ∑ j ( m k − j ) q ( n j ) q q j ( m − k + j ) . {\displaystyle {\binom {m+n}{k}}_{\!\!q}=\sum _{j}{\bin... | Q-Vandermonde identity |
c_plt58uv6p35t | In mathematics, in the field of complex geometry, a holomorphic curve in a complex manifold M is a non-constant holomorphic map f from the complex plane to M.Nevanlinna theory addresses the question of the distribution of values of a holomorphic curve in the complex projective line. | Holomorphic curve |
c_9h1cba9uc3cn | In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: A X + X B = C . {\displaystyle AX+XB=C.} It is named after English mathematician James Joseph Sylvester. Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. | Sylvester equation |
c_su45j9jdxrnd | All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n ... | Sylvester equation |
c_h3rklqsdcl5o | A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B. More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solution X is... | Sylvester equation |
c_7u425fg566oy | In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a nilmanifold, of real dimension 6. Iwasawa manifolds give examples where the first two terms E1 and E2 of the Frölicher ... | Iwasawa manifold |
c_rez606unnfwr | In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe... | Yamabe invariant |
c_2ofo5sbave2l | In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K {\displaystyle K} is a subset of a real or complex vector space X , {\displaystyle X,} then the Minkowski functional or gauge o... | Minkowski functional |
c_1sui5bcsg0k2 | In fact, every seminorm p {\displaystyle p} on X {\displaystyle X} is equal to the Minkowski functional (that is, p = p K {\displaystyle p=p_{K}} ) of any subset K {\displaystyle K} of X {\displaystyle X} satisfying { x ∈ X: p ( x ) < 1 } ⊆ K ⊆ { x ∈ X: p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)<1\}\subseteq K\subseteq... | Minkowski functional |
c_66x8qtxo6f3g | In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of X {\displaystyle X} into certain algebraic properties of a function on X . {\displaystyle X.} The Minkowski function is always non-negative (meaning p K ≥ 0 {\displaystyle p_{K}\geq 0} ... | Minkowski functional |
c_kfj87apcqou0 | This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values. However, p K {\displaystyle p_{K}} might not be real-valued since for any given x ∈ X , {\displaystyle x\in X,} the value p K ( x ) {\displaystyle ... | Minkowski functional |
c_jx1fx9u60a06 | In mathematics, in the field of functional analysis, an indefinite inner product space ( K , ⟨ ⋅ , ⋅ ⟩ , J ) {\displaystyle (K,\langle \cdot ,\,\cdot \rangle ,J)} is an infinite-dimensional complex vector space K {\displaystyle K} equipped with both an indefinite inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\,... | Indefinite inner product space |
c_p5064lhr9v28 | In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlar and Elias Stein. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another when the operator can be decomposed into almo... | Cotlar-Stein lemma |
c_3igzn83kfrbx | In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many members of the refinement.The followin... | Mesocompact space |
c_tirmd2wcuni7 | In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement. That is, given an open cover of the topological space, there is a refinement that is also an open cover, with the further property that at any point, the inter... | Orthocompact space |
c_d20dgttgvqha | Hence, we have the following: every metacompact space, and in particular, every paracompact space, is orthocompact. Useful theorems: Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms. | Orthocompact space |
c_mu3e6xlbcl68 | Every closed subspace of an orthocompact space is orthocompact. A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has an interior-preserving refinement that is an open cover of X. The product X × of the closed unit interval with an orthocompact space X is orthocompac... | Orthocompact space |
c_5dbguyz2tn5q | In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms: Every subspace is isomorphic to a projective space Pd(K) with −1 ≤ d ≤ (n −... | Polar space |
c_6vpqt5fszgmo | In mathematics, in the field of group theory, a HN group or hypernormalizing group is a group with the property that the hypernormalizer of any subnormal subgroup is the whole group. For finite groups, this is equivalent to the condition that the normalizer of any subnormal subgroup be subnormal. Some facts about HN gr... | HN group |
c_jardzb729zkr | In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups: Every simple group is a T-group. Every quasisimple group is a T-group. Every abelian group is a T-group. | T-group (mathematics) |
c_rpbodtub8pqr | Every Hamiltonian group is a T-group. Every nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal. Every normal subgroup of a T-group is a T-group. | T-group (mathematics) |
c_csk3uaikbiv3 | Every homomorphic image of a T-group is a T-group. Every solvable T-group is metabelian.The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon by con... | T-group (mathematics) |
c_9084j3wsxvht | In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of the group. For finite abelian (or nilpotent) groups, p-component is used in a different sense to mean the Sylow p-subgro... | Component type |
c_2gze679k87s0 | In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997 and arose in the context of the proof that for finite groups, every quasinormal subgroup is a subnormal subgroup. Clearly, every qua... | Conjugate permutable subgroup |
c_ikokqee06dky | Every conjugate-permutable subgroup is a conjugate-permutable subgroup of every intermediate subgroup containing it. Combining the above two facts, every conjugate-permutable subgroup is subnormal.Conversely, every 2-subnormal subgroup (that is, a subgroup that is a normal subgroup of a normal subgroup) is conjugate-pe... | Conjugate permutable subgroup |
c_qrelgmfzg849 | In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group. Some facts: Every subgroup of a finite group is a contranormal subgroup of a subnormal subgroup. In gene... | Contranormal subgroup |
c_xhjlanqrsg11 | In mathematics, in the field of group theory, a group is said to be absolutely simple if it has no proper nontrivial serial subgroups. That is, G {\displaystyle G} is an absolutely simple group if the only serial subgroups of G {\displaystyle G} are { e } {\displaystyle \{e\}} (the trivial subgroup), and G {\displaysty... | Absolutely simple group |
c_sldqj0eg5dty | In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a weaker condition than being a simple group, as simple groups... | Characteristically simple group |
c_lpl5tg5ol52t | A finite group is characteristically simple if and only if it is the direct product of isomorphic simple groups. In particular, a finite solvable group is characteristically simple if and only if it is an elementary abelian group. This does not hold in general for infinite groups; for example, the rational numbers form... | Characteristically simple group |
c_0tir5i2se3nl | A minimal normal subgroup of a group G is a nontrivial normal subgroup N of G such that the only proper subgroup of N that is normal in G is the trivial subgroup. Every minimal normal subgroup of a group is characteristically simple. This follows from the fact that a characteristic subgroup of a normal subgroup is norm... | Characteristically simple group |
c_x8xa1js1zw0r | In mathematics, in the field of group theory, a group is said to be strictly simple if it has no proper nontrivial ascendant subgroups. That is, G {\displaystyle G} is a strictly simple group if the only ascendant subgroups of G {\displaystyle G} are { e } {\displaystyle \{e\}} (the trivial subgroup), and G {\displayst... | Strictly simple group |
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