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Together with an important generalization, Schunck classes, and an important dualization, Fischer classes, formations formed the major research themes of the late 20th century in the theory of finite soluble groups. A dual notion to Carter subgroups was introduced by Bernd Fischer in (Fischer 1966). A Fischer subgroup ... | https://en.wikipedia.org/wiki/Carter_subgroup |
In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of ab... | https://en.wikipedia.org/wiki/Reduced_abelian_group |
In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, (Doerk & Hawkes 1992, I.§6). A subgroup is pronormal if each of its conjugate... | https://en.wikipedia.org/wiki/Pronormal_subgroup |
That is, H is pronormal in G if for every g in G, there is some k in the subgroup generated by H and Hg such that Hk = Hg. (Here Hg denotes the conjugate subgroup gHg-1.) Here are some relations with other subgroup properties: Every normal subgroup is pronormal. | https://en.wikipedia.org/wiki/Pronormal_subgroup |
Every Sylow subgroup is pronormal. Every pronormal subnormal subgroup is normal. Every abnormal subgroup is pronormal. Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property. Every pronormal subgroup is paranormal, and hence polynormal. | https://en.wikipedia.org/wiki/Pronormal_subgroup |
In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the p... | https://en.wikipedia.org/wiki/Central_product |
In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those module... | https://en.wikipedia.org/wiki/Pure_subring |
In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior powers and symmetric powers of a vector space. Schur functors are indexed by... | https://en.wikipedia.org/wiki/Schur_functor |
In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely genera... | https://en.wikipedia.org/wiki/Free_ideal_ring |
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero. A directly irreducible ring is a ring which cannot be written as the direct sum of two non-zero ring... | https://en.wikipedia.org/wiki/Irreducible_ring |
A ring with an irreducible spectrum is a ring whose spectrum is irreducible as a topological space. "Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. | https://en.wikipedia.org/wiki/Irreducible_ring |
This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irredu... | https://en.wikipedia.org/wiki/Irreducible_ring |
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G. The spect... | https://en.wikipedia.org/wiki/Lyndon–Hochschild–Serre_spectral_sequence |
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Li... | https://en.wikipedia.org/wiki/Upper_central_series |
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such functions play a basic role in representation theory. | https://en.wikipedia.org/wiki/Class_function |
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Bra... | https://en.wikipedia.org/wiki/Frobenius_ring |
Jean Dieudonné used this to characterize Frobenius algebras (Dieudonné 1958). Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum f... | https://en.wikipedia.org/wiki/Frobenius_ring |
In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universal algebra since then. | https://en.wikipedia.org/wiki/Graph_algebra |
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient... | https://en.wikipedia.org/wiki/Projective_linear_group |
Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F (where n is the dimension of V and F is the base field). PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, p... | https://en.wikipedia.org/wiki/Projective_linear_group |
If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used. Note that PGL(n, F) and PSL(n, F) are isomorphic if and only if every element of F has an nth root in F. As an example, note that PGL(2, C) = PSL(2, C), but that PGL(2, R) > PSL(2, R); th... | https://en.wikipedia.org/wiki/Projective_linear_group |
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the e... | https://en.wikipedia.org/wiki/Stable_manifold_theorem |
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A sys... | https://en.wikipedia.org/wiki/Limit_set |
In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal locally nilpotent subgroups of the group. It was named by Gruenberg (1961) after Kurt Hirsch and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpoten... | https://en.wikipedia.org/wiki/Hirsch–Plotkin_radical |
In a finite group, the Hirsch–Plotkin radical coincides with the Fitting subgroup but for infinite groups the two subgroups can differ. The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case. == References == | https://en.wikipedia.org/wiki/Hirsch–Plotkin_radical |
In mathematics, especially in topology, a Kuranishi structure is a smooth analogue of scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified with the zero set of a smooth map ( f 1 , … , f k ): R n + k → R k {\displaystyle (f_{1},\ldots ,f_{k})\colon \mathbb {R... | https://en.wikipedia.org/wiki/Kuranishi_theory |
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is als... | https://en.wikipedia.org/wiki/Mather_stratified_space |
In mathematics, especially in topology, a topological group G {\displaystyle G} is said to have no small subgroup if there exists a neighborhood U {\displaystyle U} of the identity that contains no nontrivial subgroup of G . {\displaystyle G.} An abbreviation '"NSS"' is sometimes used. | https://en.wikipedia.org/wiki/No_small_subgroup |
A basic example of a topological group with no small subgroup is the general linear group over the complex numbers. A locally compact, separable metric, locally connected group with no small subgroup is a Lie group. (cf. Hilbert's fifth problem.) | https://en.wikipedia.org/wiki/No_small_subgroup |
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere. | https://en.wikipedia.org/wiki/Equidimensionality |
In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies A = ( a i j ) ; a i j ≥ 0 ... | https://en.wikipedia.org/wiki/Quasipositive_matrix |
In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices)... | https://en.wikipedia.org/wiki/M-matrix |
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or... | https://en.wikipedia.org/wiki/Exchange_matrix |
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line R ∪ { ± ∞ } , {\displaystyle \mathbb {R} \cup \{\pm \infty \},} which consists of the real numbers R {\displaystyle \mathbb {... | https://en.wikipedia.org/wiki/Set_function |
In mathematics, especially operator theory, a convexoid operator is a bounded linear operator T on a complex Hilbert space H such that the closure of the numerical range coincides with the convex hull of its spectrum. An example of such an operator is a normal operator (or some of its generalization). A closely related... | https://en.wikipedia.org/wiki/Convexoid_operator |
In mathematics, especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator T on a complex Hilbert space H is said to be p-hyponormal ( 0 < p ≤ 1 {\displaystyle 0 | https://en.wikipedia.org/wiki/Hyponormal_operator |
In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator T on a complex Hilbert space H is said to be paranormal if: ‖ T 2 x ‖ ≥ ‖ T x ‖ 2 {\displaystyle \|T^{2}x\|\geq \|Tx\|^{2}} for every unit vector x in H. The class of par... | https://en.wikipedia.org/wiki/Paranormal_operator |
On the other hand, Halmos gave an example of a hyponormal operator T such that T2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.A compact paranormal operator is normal. == References == | https://en.wikipedia.org/wiki/Paranormal_operator |
In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols. | https://en.wikipedia.org/wiki/Subnormal_operator |
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Par... | https://en.wikipedia.org/wiki/Poset_category |
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical systems as a set of functions from time (a totally-ordered set) to some p... | https://en.wikipedia.org/wiki/Prefix_order |
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ... | https://en.wikipedia.org/wiki/Weak_order |
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. | https://en.wikipedia.org/wiki/Covering_relation |
In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2. More formally, a countable poset P = (... | https://en.wikipedia.org/wiki/Interval_dimension |
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R n {\displaystyle R^{n}... | https://en.wikipedia.org/wiki/Harmonic_measure |
On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps. The term harmonic measure was introduced by Rolf Nevanlinna in 1928 for planar domains, although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, F. Riesz, M. Riesz, Carl... | https://en.wikipedia.org/wiki/Harmonic_measure |
In mathematics, especially real analysis, a flat function is a smooth function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } all of whose derivatives vanish at a given point x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } . The flat functions are, in some sense, the antitheses of the analytic functions. An anal... | https://en.wikipedia.org/wiki/Flat_function |
{\displaystyle f(x)\sim \lim _{n\to \infty }\sum _{k=0}^{n}{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}.} In the case of a flat function, all derivatives vanish at x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } , i.e. f ( k ) ( x 0 ) = 0 {\displaystyle f^{(k)}(x_{0})=0} for all k ∈ N {\displaystyle k\in \mathbb {N} } . Thi... | https://en.wikipedia.org/wiki/Flat_function |
In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder R n ( x ) {\displaystyle R_{n}(x)} for all n ∈ N {\displaystyle n\in \mathbb {N} } . The function need not be flat at just one point. Trivially, constant functions on R {\displaystyle \mathbb {R} } are flat everywher... | https://en.wikipedia.org/wiki/Flat_function |
In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. Thi... | https://en.wikipedia.org/wiki/Frobenius_characteristic_map |
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal i {\displaystyle {\mathfrak {i}}} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that e x − x ∈ i {\displaystyle ex-x\in {\mathfrak {i}}}... | https://en.wikipedia.org/wiki/Regular_ideal |
In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings (QF rings), which are in turn generalized by right pseudo-Frobenius rings (PF rings) and right f... | https://en.wikipedia.org/wiki/Quasi-Frobenius_ring |
In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cn of the form P = { z ∈ D: | f j ( z ) | < 1 , 1 ≤ j ≤ N } {\displaystyle P=\{z\in D:|f_{j}(z)|<1,\;\;1\leq j\leq N\}} where D is a bounded connected open subset of Cn, f j {\displaystyle f_{j}} are holomorphi... | https://en.wikipedia.org/wiki/Analytic_polyhedron |
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold. In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization ... | https://en.wikipedia.org/wiki/Behnke–Stein_theorem_on_Stein_manifolds |
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G k ⊂ C n {\displaystyle G_{k}\subset \mathbb {C} ^{n}} (i.e., G k ⊂ G k + 1 {\displaystyle G_{k}\subset G_{k+1}} ) of domains of holomorphy is again a domain of holomorphy. It was proved by Hein... | https://en.wikipedia.org/wiki/Behnke–Stein_theorem |
In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d = 2 , 3 {\displaystyle d=2,3} case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanis... | https://en.wikipedia.org/wiki/Weyl's_law |
In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements. | https://en.wikipedia.org/wiki/Parker_vector |
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. | https://en.wikipedia.org/wiki/Oka–Weil_theorem |
In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it... | https://en.wikipedia.org/wiki/Einstein_summation_notation |
In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps. | https://en.wikipedia.org/wiki/Perfect_map |
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed for... | https://en.wikipedia.org/wiki/Closed_differential_form |
In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev.Basically, essential dimension measures the complexity of algebraic structures vi... | https://en.wikipedia.org/wiki/Essential_dimension |
In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an ordinal α: Every limit ordinal (including 0) is even. The successor of an eve... | https://en.wikipedia.org/wiki/Even_ordinal |
Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n. Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n. Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n. If α = 2β, then α is even. | https://en.wikipedia.org/wiki/Even_ordinal |
Otherwise α = 2β + 1 and α is odd.Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number (ω + 3)2 = ... | https://en.wikipedia.org/wiki/Even_ordinal |
Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ. == References == | https://en.wikipedia.org/wiki/Even_ordinal |
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the po... | https://en.wikipedia.org/wiki/Even_function |
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations. | https://en.wikipedia.org/wiki/Analytic_function_of_a_matrix |
In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory. | https://en.wikipedia.org/wiki/Exponentially_equivalent_measures |
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. | https://en.wikipedia.org/wiki/Exponential_polynomial |
In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n". When n is a positive integer, exponentiation corresponds to repeated multiplication o... | https://en.wikipedia.org/wiki/Laws_of_exponents |
Starting from the basic fact stated above that, for any positive integer n {\displaystyle n} , b n {\displaystyle b^{n}} is n {\displaystyle n} occurrences of b {\displaystyle b} all multiplied by each other, several other properties of exponentiation directly follow. In particular: In other words, when multiplying a b... | https://en.wikipedia.org/wiki/Laws_of_exponents |
For any n {\displaystyle n} , b 0 × b n = b 0 + n = b n {\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}} . Dividing both sides by b n {\displaystyle b^{n}} gives b 0 = b n / b n = 1 {\displaystyle b^{0}=b^{n}/b^{n}=1} . The fact that b 1 = b {\displaystyle b^{1}=b} can similarly be derived from the same rule. | https://en.wikipedia.org/wiki/Laws_of_exponents |
For example, ( b 1 ) 3 = b 1 × b 1 × b 1 = b 1 + 1 + 1 = b 3 {\displaystyle (b^{1})^{3}=b^{1}\times b^{1}\times b^{1}=b^{1+1+1}=b^{3}} . Taking the cube root of both sides gives b 1 = b {\displaystyle b^{1}=b} . The rule that multiplying makes exponents add can also be used to derive the properties of negative integer ... | https://en.wikipedia.org/wiki/Laws_of_exponents |
Consider the question of what b − 1 {\displaystyle b^{-1}} should mean. In order to respect the "exponents add" rule, it must be the case that b − 1 × b 1 = b − 1 + 1 = b 0 = 1 {\displaystyle b^{-1}\times b^{1}=b^{-1+1}=b^{0}=1} . Dividing both sides by b 1 {\displaystyle b^{1}} gives b − 1 = 1 / b 1 {\displaystyle b^{... | https://en.wikipedia.org/wiki/Laws_of_exponents |
By a similar argument, b − n = 1 / b n {\displaystyle b^{-n}=1/b^{n}} . The properties of fractional exponents also follow from the same rule. For example, suppose we consider b {\displaystyle {\sqrt {b}}} and ask if there is some suitable exponent, which we may call r {\displaystyle r} , such that b r = b {\displaysty... | https://en.wikipedia.org/wiki/Laws_of_exponents |
From the definition of the square root, we have that b × b = b {\displaystyle {\sqrt {b}}\times {\sqrt {b}}=b} . Therefore, the exponent r {\displaystyle r} must be such that b r × b r = b {\displaystyle b^{r}\times b^{r}=b} . Using the fact that multiplying makes exponents add gives b r + r = b {\displaystyle b^{r+r}=... | https://en.wikipedia.org/wiki/Laws_of_exponents |
The b {\displaystyle b} on the right-hand side can also be written as b 1 {\displaystyle b^{1}} , giving b r + r = b 1 {\displaystyle b^{r+r}=b^{1}} . Equating the exponents on both sides, we have r + r = 1 {\displaystyle r+r=1} . | https://en.wikipedia.org/wiki/Laws_of_exponents |
Therefore, r = 1 2 {\displaystyle r={\frac {1}{2}}} , so b = b 1 / 2 {\displaystyle {\sqrt {b}}=b^{1/2}} . The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices. Expon... | https://en.wikipedia.org/wiki/Laws_of_exponents |
In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a ... | https://en.wikipedia.org/wiki/Extendible_cardinal |
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a ... | https://en.wikipedia.org/wiki/Extrapolation_method |
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a ... | https://en.wikipedia.org/wiki/Factorization |
Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. | https://en.wikipedia.org/wiki/Factorization |
Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has also been studied ... | https://en.wikipedia.org/wiki/Factorization |
In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible... | https://en.wikipedia.org/wiki/Factorization |
In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factoriza... | https://en.wikipedia.org/wiki/Factorization |
A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor ... | https://en.wikipedia.org/wiki/Factorization |
Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, ev... | https://en.wikipedia.org/wiki/Factorization |
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and o... | https://en.wikipedia.org/wiki/Field_Arithmetic |
In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily o... | https://en.wikipedia.org/wiki/Finite_field_arithmetic |
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times). | https://en.wikipedia.org/wiki/Finite-dimensional_distribution |
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely gene... | https://en.wikipedia.org/wiki/Finiteness_properties_of_groups |
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence. More precisely, let E 2 p , q ⇒ H n ( A ) {\displaystyle E_{2}^{p,q}\Rightarrow H^{n}(A)} be a first quadrant spectral sequence, meaning that E 2 p , q {\displaystyle E... | https://en.wikipedia.org/wiki/Five-term_exact_sequence |
In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find... | https://en.wikipedia.org/wiki/Flat_convergence |
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R ( M , x ) , {\displaystyle R(M,\mathbf {x} ),} is defined as:: p. 234 For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x H {\displaystyle \m... | https://en.wikipedia.org/wiki/Hermitian_matrix |
{\displaystyle c.} Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M) when x {\displaystyle \mathbf {x} } is v min {\displaystyle \m... | https://en.wikipedia.org/wiki/Hermitian_matrix |
Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,\mathbf {x} )\leq \lambda _{\max }} and R ( M , v max ) = λ max . {\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.} The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. | https://en.wikipedia.org/wiki/Hermitian_matrix |
It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional anal... | https://en.wikipedia.org/wiki/Hermitian_matrix |
When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed... | https://en.wikipedia.org/wiki/Hermitian_matrix |
In mathematics, for a natural number n ≥ 2 {\displaystyle n\geq 2} , the nth Fibonacci group, denoted F ( 2 , n ) {\displaystyle F(2,n)} or sometimes F ( n ) {\displaystyle F(n)} , is defined by n generators a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\dots ,a_{n}} and n relations: a 1 a 2 = a 3 , {\displaystyle a_{... | https://en.wikipedia.org/wiki/Fibonacci_group |
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