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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.
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https://en.wikipedia.org/wiki/Dedekind–Hasse_norm
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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.
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https://en.wikipedia.org/wiki/Rational_mapping
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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 reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.
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https://en.wikipedia.org/wiki/Weyl_chamber
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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 by collapsing I into a single element while the elements of S outside of I retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I. The concept of Rees factor semigroup was introduced by David Rees in 1940.
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https://en.wikipedia.org/wiki/Rees_factor_semigroup
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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 introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".
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https://en.wikipedia.org/wiki/Gödel_constructible_universe
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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 are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
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https://en.wikipedia.org/wiki/Gödel_constructible_universe
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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, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
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https://en.wikipedia.org/wiki/Galois_group
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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 the set of all products of two elements in the group, substituting any element for x and any element for y, and hence would be the group itself. On the other hand, the verbal subgroup for the set of words { x 2 , x y 2 x − 1 } {\displaystyle \{x^{2},xy^{2}x^{-1}\}} is generated by the set of squares and their conjugates.
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https://en.wikipedia.org/wiki/Verbal_subgroup
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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 subgroup.
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https://en.wikipedia.org/wiki/Verbal_subgroup
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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.
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https://en.wikipedia.org/wiki/A-group
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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 Paul Erdős and Wolfgang Heinrich Johannes Fuchs, who published it in 1956.
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https://en.wikipedia.org/wiki/Erdős–Fuchs_theorem
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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 solvable (Suzuki 1957). Further progress was made showing that CN-groups, groups in which the centralizer of a non-identity element is nilpotent, of odd order are solvable (Feit, Thompson & Hall 1960). The complete solution was given in (Feit & Thompson 1963), but further work on CN-groups was done in (Suzuki 1961), giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group G is such that its largest solvable normal subgroup O∞(G) is a 2-group, and the quotient is a group of even order.
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https://en.wikipedia.org/wiki/CN-group
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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 extension and quotient groups, but not under subgroups. If G is a group, N, M are normal subgroups with G/N and G/M imperfect, then G/(N∩M) is imperfect, showing that the class of imperfect groups is a formation.
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https://en.wikipedia.org/wiki/Imperfect_group
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The (restricted or unrestricted) direct product of imperfect groups is imperfect. Every solvable group is imperfect. Finite symmetric groups are also imperfect.
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https://en.wikipedia.org/wiki/Imperfect_group
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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).
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https://en.wikipedia.org/wiki/Imperfect_group
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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 solvable by (Taketa 1930), presented in textbook in (Isaacs 1994, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4).
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https://en.wikipedia.org/wiki/Monomial_group
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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 textbook form in (Bray et al. 1982, Ch 2.4). The symmetric group S 4 {\displaystyle S_{4}} is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group SL 2 ( F 3 ) {\displaystyle \operatorname {SL} _{2}(\mathbb {F} _{3})} is the smallest finite group that is not monomial: since the abelianization of this group has order three, its irreducible characters of degree two are not monomial.
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https://en.wikipedia.org/wiki/Monomial_group
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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.
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https://en.wikipedia.org/wiki/Homotopy_extension_property
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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) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). The following relation holds: Both Dirichlet eta function and Riemann zeta function are special cases of polylogarithm. While the Dirichlet series expansion for the eta function is convergent only for any complex number s with real part > 0, it is Abel summable for any complex number.
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https://en.wikipedia.org/wiki/Dirichlet_eta_function
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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 1 − s {\displaystyle 1-2^{1-s}} , although in fact these hypothetical additional poles do not exist.) Equivalently, we may begin by defining which is also defined in the region of positive real part ( Γ ( s ) {\displaystyle \Gamma (s)} represents the gamma function). This gives the eta function as a Mellin transform. Hardy gave a simple proof of the functional equation for the eta function, which is From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
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https://en.wikipedia.org/wiki/Dirichlet_eta_function
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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 signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.
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https://en.wikipedia.org/wiki/Forgetful_functor
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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.
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https://en.wikipedia.org/wiki/Polar_set_(potential_theory)
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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).
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https://en.wikipedia.org/wiki/Q-derivative
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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} . Hence p {\displaystyle p} is a prime number.
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https://en.wikipedia.org/wiki/Tight_closure
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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.
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https://en.wikipedia.org/wiki/Tight_closure
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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 reduced, then one can instead consider all e > 0 {\displaystyle e>0} .Here I {\displaystyle I^{}} is used to denote the ideal of R {\displaystyle R} generated by the p e {\displaystyle p^{e}} 'th powers of elements of I {\displaystyle I} , called the e {\displaystyle e} th Frobenius power of I {\displaystyle I} . An ideal is called tightly closed if I = I ∗ {\displaystyle I=I^{*}} .
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https://en.wikipedia.org/wiki/Tight_closure
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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 that all ideals of the ring are still tightly closed in localizations of the ring. Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly F {\displaystyle F} -regular ring is F {\displaystyle F} -regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?
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https://en.wikipedia.org/wiki/Tight_closure
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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–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions.
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https://en.wikipedia.org/wiki/Carlson's_theorem
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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 based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.
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https://en.wikipedia.org/wiki/Nachbin_resummation
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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.
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https://en.wikipedia.org/wiki/Difference_polynomials
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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 domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT).
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https://en.wikipedia.org/wiki/Fractional_Fourier_transform
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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 groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain.
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https://en.wikipedia.org/wiki/Fractional_Fourier_transform
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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 chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.
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https://en.wikipedia.org/wiki/Fractional_Fourier_transform
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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 variables in a term algebra or in a lambda expression. Director strings were introduced by Kennaway and Sleep in 1982 and further developed by Sinot, Fernández and Mackie as a mechanism for understanding and controlling the computational complexity cost of beta reduction.
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https://en.wikipedia.org/wiki/Director_string
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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 related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
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https://en.wikipedia.org/wiki/Gaussian_period
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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 basis functions. Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used: Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions. Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear form to be symmetric and substitutes the energy minimization with orthogonality constraints determined by the same basis functions that are used to approximate the solution.
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https://en.wikipedia.org/wiki/Galerkin's_method
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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 functions used to approximate the solution. Petrov–Galerkin method can be viewed as an extension of Bubnov–Galerkin method, applying a projection that is not necessarily orthogonal in the operator formulation of the differential equation.Examples of Galerkin methods are: the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method, the boundary element method for solving integral equations, Krylov subspace methods.
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https://en.wikipedia.org/wiki/Galerkin's_method
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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.
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https://en.wikipedia.org/wiki/Free_lattice_word_problem
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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 (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned.
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https://en.wikipedia.org/wiki/Antichain
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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 lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is #P-complete.
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https://en.wikipedia.org/wiki/Antichain
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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 directed set is necessarily centered, and any centered set is a linked set. A subset B of a partial order is said to be σ-centered if it is a countable union of centered sets.
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https://en.wikipedia.org/wiki/Centered_set
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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 a solution, particularly when the boundary has a sufficiently sharp edge protruding into the interior of the region.
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https://en.wikipedia.org/wiki/Lebesgue_spine
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In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions.
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https://en.wikipedia.org/wiki/Pluripolar_set
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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 identical to the Fubini–Study metric when restricted to the pure states alone.
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https://en.wikipedia.org/wiki/Helstrom_metric
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In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
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https://en.wikipedia.org/wiki/Bispectrum
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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 pseudoholomorphic curves.
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https://en.wikipedia.org/wiki/Relative_contact_homology
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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 \sum _{k=0}^{N-1}h_{k}\cdot \varphi (2\cdot x-k)} This condition is called refinement equation, dilation equation or two-scale equation. Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator D {\displaystyle D} one can write more concisely: φ = 2 ⋅ D 1 / 2 ( h ∗ φ ) {\displaystyle \varphi =2\cdot D_{1/2}(h*\varphi )} It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems and de Rham curves.
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https://en.wikipedia.org/wiki/Refinable_function
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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 function c ⋅ φ {\displaystyle c\cdot \varphi } is refinable, too. These functions play a fundamental role in wavelet theory as scaling functions.
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https://en.wikipedia.org/wiki/Refinable_function
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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 Shirshov introduced Lyndon words in 1953 calling them regular words. Lyndon words are a special case of Hall words; almost all properties of Lyndon words are shared by Hall words.
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https://en.wikipedia.org/wiki/Lyndon_word
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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 almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with Hall trees.
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https://en.wikipedia.org/wiki/Hall_word
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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 theorem used in the construction of a universal enveloping algebra.
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https://en.wikipedia.org/wiki/Hall_word
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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.
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https://en.wikipedia.org/wiki/Hall_word
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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.
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https://en.wikipedia.org/wiki/Hall_word
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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 due to Philip Hall and Ernst Witt.
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https://en.wikipedia.org/wiki/Hall_word
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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 set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable.
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https://en.wikipedia.org/wiki/Chain_decomposition
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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 decomposition is its number of chains. The width of the partial order is defined as the common size of the antichain and chain decomposition. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of the smallest chain decomposition are again equal.
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https://en.wikipedia.org/wiki/Chain_decomposition
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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 orders, to the perfection of comparability graphs, to the Gallai–Hasse–Roy–Vitaver theorem relating longest paths and colorings in graphs, and to the Erdős–Szekeres theorem on monotonic subsequences.
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https://en.wikipedia.org/wiki/Mirsky's_theorem
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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.
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https://en.wikipedia.org/wiki/Kadec_norm
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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 < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1.The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value.Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.
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https://en.wikipedia.org/wiki/Graded_poset
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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.
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https://en.wikipedia.org/wiki/Univalent_function
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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 decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
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https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
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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. They are named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.
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https://en.wikipedia.org/wiki/Uniformity_norm
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In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
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https://en.wikipedia.org/wiki/Period_mapping
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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 p of K which have a factor P with inertial degree one (that is, the residue field of P has the same order as the residue field of p). Bauer's theorem states that if M/K is a finite degree Galois extension, then P(M/K) ⊇ P(L/K) if and only if M ⊆ L. In particular, finite degree Galois extensions N of K are characterised by set of prime ideals which split completely in N. An extension F/K is Bauerian if it obeys Bauer's theorem: that is, for every finite extension L of K, we have P(F/K) ⊇ P(L/K) if and only if L contains a subfield K-isomorphic to F. All field extensions of degree at most 4 over Q are Bauerian. An example of a non-Bauerian extension is the Galois extension of Q by the roots of 2x5 − 32x + 1, which has Galois group S5.
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https://en.wikipedia.org/wiki/Bauerian_extension
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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.
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https://en.wikipedia.org/wiki/Modulus_(algebraic_number_theory)
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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.
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https://en.wikipedia.org/wiki/S-unit_equation
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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 original paper addresses this for singular homology.
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https://en.wikipedia.org/wiki/Eilenberg–Moore_spectral_sequence
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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 the global field. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the Brauer group of X to define it. For abelian varieties the Manin obstruction is just the Tate–Shafarevich group and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate–Shafarevich group is finite). There are however examples, due to Alexei Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.
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https://en.wikipedia.org/wiki/Manin_obstruction
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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 condition on the cardinality of the hom-set | homC(X, Y) | is 1 when X = Y and 0 when X is not equal to Y.Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
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https://en.wikipedia.org/wiki/Discrete_category
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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}{\binom {m}{k-j}}_{\!\!q}{\binom {n}{j}}_{\!\!q}q^{j(m-k+j)}.} The nonzero contributions to this sum come from values of j such that the q-binomial coefficients on the right side are nonzero, that is, max(0, k − m) ≤ j ≤ min(n, k).
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https://en.wikipedia.org/wiki/Q-Vandermonde_identity
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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.
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https://en.wikipedia.org/wiki/Holomorphic_curve
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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.
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https://en.wikipedia.org/wiki/Sylvester_equation
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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 rows and m columns.
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https://en.wikipedia.org/wiki/Sylvester_equation
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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 almost the same: There exists a unique solution X exactly when the spectra of A and −B are disjoint.
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https://en.wikipedia.org/wiki/Sylvester_equation
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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 spectral sequence are not isomorphic. As a complex manifold, such an Iwasawa manifold is an important example of a compact complex manifold which does not admit any Kähler metric.
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https://en.wikipedia.org/wiki/Iwasawa_manifold
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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. Used by Vincent Moncrief and Arthur Fischer to study reduced Hamiltonian for Einstein's equations.
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https://en.wikipedia.org/wiki/Yamabe_invariant
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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 of K {\displaystyle K} is defined to be the function p K: X → , {\displaystyle p_{K}:X\to ,} valued in the extended real numbers, defined by where the infimum of the empty set is defined to be positive infinity ∞ {\displaystyle \,\infty \,} (which is not a real number so that p K ( x ) {\displaystyle p_{K}(x)} would then not be real-valued). The set K {\displaystyle K} is often assumed/picked to have properties, such as being an absorbing disk in X , {\displaystyle X,} that guarantee that p K {\displaystyle p_{K}} will be a real-valued seminorm on X . {\displaystyle X.}
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https://en.wikipedia.org/wiki/Minkowski_functional
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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 \{x\in X:p(x)\leq 1\}} (where all three of these sets are necessarily absorbing in X {\displaystyle X} and the first and last are also disks). Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis.
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https://en.wikipedia.org/wiki/Minkowski_functional
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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} ).
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https://en.wikipedia.org/wiki/Minkowski_functional
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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 p_{K}(x)} is a real number if and only if { r > 0: x ∈ r K } {\displaystyle \{r>0:x\in rK\}} is not empty. Consequently, K {\displaystyle K} is usually assumed to have properties (such as being absorbing in X , {\displaystyle X,} for instance) that will guarantee that p K {\displaystyle p_{K}} is real-valued.
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https://en.wikipedia.org/wiki/Minkowski_functional
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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 ,\,\cdot \rangle \,} and a positive semi-definite inner product ( x , y ) = d e f ⟨ x , J y ⟩ , {\displaystyle (x,\,y)\ {\stackrel {\mathrm {def} }{=}}\ \langle x,\,Jy\rangle ,} where the metric operator J {\displaystyle J} is an endomorphism of K {\displaystyle K} obeying J 3 = J . {\displaystyle J^{3}=J.\,} The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on K {\displaystyle K} implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space. An indefinite inner product space is called a Krein space (or J {\displaystyle J} -space) if ( x , y ) {\displaystyle (x,\,y)} is positive definite and K {\displaystyle K} possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).
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https://en.wikipedia.org/wiki/Indefinite_inner_product_space
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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 almost orthogonal pieces. The original version of this lemma (for self-adjoint and mutually commuting operators) was proved by Mischa Cotlar in 1955 and allowed him to conclude that the Hilbert transform is a continuous linear operator in L 2 {\displaystyle L^{2}} without using the Fourier transform. A more general version was proved by Elias Stein.
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https://en.wikipedia.org/wiki/Cotlar-Stein_lemma
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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 following facts are true about mesocompactness: Every compact space, and more generally every paracompact space is mesocompact. This follows from the fact that any locally finite cover is automatically compact-finite. Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that points are compact, and hence any compact-finite cover is automatically point finite.
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https://en.wikipedia.org/wiki/Mesocompact_space
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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 intersection of all open sets in the refinement containing that point is also open. If the number of open sets containing the point is finite, then their intersection is definitionally open. That is, every point-finite open cover is interior preserving.
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https://en.wikipedia.org/wiki/Orthocompact_space
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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.
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https://en.wikipedia.org/wiki/Orthocompact_space
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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 orthocompact if and only if X is countably metacompact. (B.M. Scott) Every orthocompact space is countably orthocompact. Every countably orthocompact Lindelöf space is orthocompact.
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https://en.wikipedia.org/wiki/Orthocompact_space
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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 − 1) and K a division ring. (That is, it is a Desarguesian projective geometry.) For each subspace the corresponding d is called its dimension. The intersection of two subspaces is always a subspace. For each subspace A of dimension n − 1 and each point p not in A, there is a unique subspace B of dimension n − 1 containing p and such that A ∩ B is (n − 2)-dimensional. The points in A ∩ B are exactly the points of A that are in a common subspace of dimension 1 with p. There are at least two disjoint subspaces of dimension n − 1.It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (P,L), so that for each point p ∈ P and each line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.
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https://en.wikipedia.org/wiki/Polar_space
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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 groups: Subgroups of solvable HN groups are solvable HN groups. Metanilpotent A-groups are HN groups.
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https://en.wikipedia.org/wiki/HN_group
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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.
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https://en.wikipedia.org/wiki/T-group_(mathematics)
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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.
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https://en.wikipedia.org/wiki/T-group_(mathematics)
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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 conjugation as a group of power automorphisms by G. A PT-group is a group in which permutability is transitive. A finite T-group is a PT-group.
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https://en.wikipedia.org/wiki/T-group_(mathematics)
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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-subgroup, so the abelian group is the product of its p-components for primes p. These are not components in the sense above, as abelian groups are not quasisimple. A quasisimple subgroup of a finite group is called a standard component if its centralizer has even order, it is normal in the centralizer of every involution centralizing it, and it commutes with none of its conjugates. This concept is used in the classification of finite simple groups, for instance, by showing that under mild restrictions on the standard component one of the following always holds: a standard component is normal (so a component as above), the whole group has a nontrivial solvable normal subgroup, the subgroup generated by the conjugates of the standard component is on a short list, or the standard component is a previously unknown quasisimple group (Aschbacher & Seitz 1976).
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https://en.wikipedia.org/wiki/Component_type
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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 quasinormal subgroup is conjugate-permutable. In fact, it is true that for a finite group: Every maximal conjugate-permutable subgroup is normal.
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https://en.wikipedia.org/wiki/Conjugate_permutable_subgroup
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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-permutable. == References ==
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https://en.wikipedia.org/wiki/Conjugate_permutable_subgroup
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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 general, every subgroup of a group is a contranormal subgroup of a descendant subgroup. Every abnormal subgroup is contranormal.
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https://en.wikipedia.org/wiki/Contranormal_subgroup
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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 {\displaystyle G} itself (the whole group). In the finite case, a group is absolutely simple if and only if it is simple. However, in the infinite case, absolutely simple is a stronger property than simple. The property of being strictly simple is somewhere in between.
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https://en.wikipedia.org/wiki/Absolutely_simple_group
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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 must not have any proper nontrivial normal subgroups, which include characteristic subgroups.
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https://en.wikipedia.org/wiki/Characteristically_simple_group
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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 a characteristically simple group that is not a direct product of simple groups.
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https://en.wikipedia.org/wiki/Characteristically_simple_group
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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 normal.
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https://en.wikipedia.org/wiki/Characteristically_simple_group
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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 {\displaystyle G} itself (the whole group). In the finite case, a group is strictly simple if and only if it is simple. However, in the infinite case, strictly simple is a stronger property than simple.
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https://en.wikipedia.org/wiki/Strictly_simple_group
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