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In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset U ⊆ R n {\di... | https://en.wikipedia.org/wiki/Spaces_of_test_functions_and_distributions |
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If U = R n {\displaystyle U=\mathbb {R} ^{n}} then the use of Schwartz functions as test functions gives rise to a certain subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} whose elem... | https://en.wikipedia.org/wiki/Spaces_of_test_functions_and_distributions |
The set of tempered distributions forms a vector subspace of the space of distributions D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of ... | https://en.wikipedia.org/wiki/Spaces_of_test_functions_and_distributions |
In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. It consists of a sequence of "staircase" polygonal chains in a unit square, formed from horizontal and vertical line segments of decreasing length, so that these staircases c... | https://en.wikipedia.org/wiki/Staircase_paradox |
It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.For any smooth curve, polygonal chains with segment lengths decreasing to zero, connecting consecutive vertices along the curve, always converge to the arc length. The failure of the staircase curves to ... | https://en.wikipedia.org/wiki/Staircase_paradox |
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f {\displaystyle f} defined on a set S {\displaystyle S} the non-negative number ‖ f ‖ ∞ = ‖ f ‖ ∞ , S = sup { | f ( s ) |: s ∈ S } . {\displaystyle \|f\|_{\infty }=\|f\|_{\infty ,S}=\sup \left\{\,|f(s)|:s\in S... | https://en.wikipedia.org/wiki/Chebyshev_norm |
The name "uniform norm" derives from the fact that a sequence of functions { f n } {\displaystyle \left\{f_{n}\right\}} converges to f {\displaystyle f} under the metric derived from the uniform norm if and only if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly.If f {\displaystyle f} is a continuo... | https://en.wikipedia.org/wiki/Chebyshev_norm |
In mathematical analysis, the universal chord theorem states that if a function f is continuous on and satisfies f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} , then for every natural number n {\displaystyle n} , there exists some x ∈ {\displaystyle x\in } such that f ( x ) = f ( x + b − a n ) {\displaystyle f(x)=f\lef... | https://en.wikipedia.org/wiki/Universal_chord_theorem |
In mathematical and scientific research, multimedia is mainly used for modeling and simulation. For example, a scientist can look at a molecular model of a particular substance and manipulate it to arrive at a new substance. Representative research can be found in journals such as the Journal of Multimedia. | https://en.wikipedia.org/wiki/Multimedia |
One well known example of this being applied would be in the movie Interstellar where Executive Director Kip Thorne helped create one of the most realistic depictions of a blackhole in film. The visual effects team under Paul Franklin took Kip Thorne's mathematical data and applied it into their own visual effects engi... | https://en.wikipedia.org/wiki/Multimedia |
In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q-analog of the Pochhammer symbol ( x ) n = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)} , in the sense that The q-Poc... | https://en.wikipedia.org/wiki/Q-Pochhammer_symbol |
In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point. For example, the Lotka–Volterra predator–prey model is d x d t = x ( α − β y ) d y d t = − y ( γ ... | https://en.wikipedia.org/wiki/Community_matrix |
By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form = A , {\displaystyle {\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},} where u = x −... | https://en.wikipedia.org/wiki/Community_matrix |
In mathematical biology, van den Driessche's contributions include important work on delay differential equations and on Hopf bifurcations, and the effects of changing population size and immigration on epidemics.She has also done more fundamental research in linear algebra, motivated by applications in mathematical bi... | https://en.wikipedia.org/wiki/Pauline_van_den_Driessche |
In mathematical combinatorics, the Transylvania lottery is a lottery where players selected three numbers from 1-14 for each ticket, and then three numbers are chosen randomly. A ticket wins if two of the numbers match the random ones. The problem asks how many tickets the player must buy in order to be certain of winn... | https://en.wikipedia.org/wiki/Transylvania_lottery |
Each set of seven uses every line of a Fano plane, labelled with the numbers 1 to 7, and 8 to 14. At least two of the three randomly chosen numbers must be in one Fano plane set, and any two points on a Fano plane are on a line, so there will be a ticket in the collection containing those two numbers. There is a (6/13)... | https://en.wikipedia.org/wiki/Transylvania_lottery |
In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface... | https://en.wikipedia.org/wiki/Radó's_theorem_(Riemann_surfaces) |
In mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0). Schottky's original theore... | https://en.wikipedia.org/wiki/Schottky_theorem |
In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | https://en.wikipedia.org/wiki/Quasiconformal_map |
In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | https://en.wikipedia.org/wiki/Geometric_function_theory |
In mathematical complex analysis, universal Teichmüller space T(1) is a Teichmüller space containing the Teichmüller space T(G) of every Fuchsian group G. It was introduced by Bers (1965) as the set of boundary values of quasiconformal maps of the upper half-plane that fix 0, 1, and ∞. | https://en.wikipedia.org/wiki/Universal_Teichmüller_space |
In mathematical cryptography, a Kleinian integer is a complex number of the form m + n 1 + − 7 2 {\displaystyle m+n{\frac {1+{\sqrt {-7}}}{2}}} , with m and n rational integers. They are named after Felix Klein. The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary q... | https://en.wikipedia.org/wiki/Kleinian_integer |
In the calculus of relations, the semicolon is used in infix notation for the composition of relations: A ; B = { ( x , z ): ∃ y x A y ∧ y B z } . {\displaystyle A;B\ =\ \{(x,z):\exists y\ \ xAy\ \land \ yBz\}~.} The ; Humphrey point is sometimes used as the "decimal point" in duodecimal numbers: 54;612 equals 64.510. | https://en.wikipedia.org/wiki/Semi_colon |
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. | https://en.wikipedia.org/wiki/Continuous-time_signal |
In mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes. The result states that if f is supermodular in (x,θ), and D is a lattic... | https://en.wikipedia.org/wiki/Topkis's_Theorem |
In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent v... | https://en.wikipedia.org/wiki/Isoelastic_function |
In mathematical economics, applied general equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow-up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu model of general equilibrium theory with empirical data, to provide "“a g... | https://en.wikipedia.org/wiki/Applied_general_equilibrium |
Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment” (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solve... | https://en.wikipedia.org/wiki/Applied_general_equilibrium |
Earlier analytic work with these models has examined the distortionary effects of taxes, tariffs, and other policies, along with functional incidence questions. More recent applied models, including those discussed here, provide numerical estimates of efficiency and distributional effects within the same framework. | https://en.wikipedia.org/wiki/Applied_general_equilibrium |
Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models described by functions wi... | https://en.wikipedia.org/wiki/Applied_general_equilibrium |
In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity i... | https://en.wikipedia.org/wiki/Arrow–Debreu_model |
In general, there may be many equilibria. Arrow (1972) and Debreu (1983) were separately awarded the Nobel Prize in Economics for their development of the model. McKenzie however was not awarded. | https://en.wikipedia.org/wiki/Arrow–Debreu_model |
In mathematical elasticity theory, Mikhlin was concerned by three themes: the plane problem (mainly from 1932 to 1935), the theory of shells (from 1954) and the Cosserat spectrum (from 1967 to 1973). Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains. The f... | https://en.wikipedia.org/wiki/Solomon_Mikhlin |
Mikhlin studied its convergence and gave applications to special applied problems. He proved existence theorems for the fundamental problems of plane elasticity involving inhomogeneous anisotropic media: these results are collected in the book (Mikhlin 1957). Concerning the theory of shells, there are several Mikhlin's... | https://en.wikipedia.org/wiki/Solomon_Mikhlin |
He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress. As a result of his study of this problem, Mikhlin also gave a new (invariant) form of the basic equations of the theory. He also proved a theo... | https://en.wikipedia.org/wiki/Solomon_Mikhlin |
Mikhlin studied also the spectrum of the operator pencil of the classical linear elastostatic operator or Navier–Cauchy operator A ( ω ) u = Δ 2 u + ω ∇ ( ∇ ⋅ u ) {\displaystyle {\boldsymbol {\mathcal {A}}}(\omega ){\boldsymbol {u}}=\Delta _{2}{\boldsymbol {u}}+\omega \nabla \left(\nabla \cdot {\boldsymbol {u}}\right)}... | https://en.wikipedia.org/wiki/Solomon_Mikhlin |
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) | https://en.wikipedia.org/wiki/Corner_angle |
as variables denoting the size of some angle (to avoid confusion with its other meaning, the symbol π is typically not used for this purpose). Lower case Roman letters (a, b, c, . . | https://en.wikipedia.org/wiki/Corner_angle |
. ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. | https://en.wikipedia.org/wiki/Corner_angle |
See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} . | https://en.wikipedia.org/wiki/Corner_angle |
Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). Potentially, an angle denoted as, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C ... | https://en.wikipedia.org/wiki/Corner_angle |
In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subse... | https://en.wikipedia.org/wiki/Littlewood–Offord_problem |
This bound is sharp; equality is attained when all vectors in S are equal. In 1966, Kleitman showed that the same bound held for complex numbers. In 1970, he extended this to the setting when V is a normed space.Suppose S = {v1, …, vn}. | https://en.wikipedia.org/wiki/Littlewood–Offord_problem |
By subtracting 1 2 ∑ i = 1 n v i {\displaystyle {\frac {1}{2}}\sum _{i=1}^{n}v_{i}} from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form ∑ i = 1 n ε i v i {\displaystyle \su... | https://en.wikipedia.org/wiki/Littlewood–Offord_problem |
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map j: V → V {\displaystyle j\colon V\to V} which satisfies j 2 = − 1. {\displaystyle j^{2}=-1.} Together with the imagi... | https://en.wikipedia.org/wiki/Pseudoreal_representation |
From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the ide... | https://en.wikipedia.org/wiki/Pseudoreal_representation |
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form ω: V × V → F {\displaystyle \omega \colon V\times V\to \mathbb {F}... | https://en.wikipedia.org/wiki/Symplectic_representation |
In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of th... | https://en.wikipedia.org/wiki/Smoluchowski_equation |
In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles. | https://en.wikipedia.org/wiki/Margrabe's_formula |
In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how... | https://en.wikipedia.org/wiki/Monte_Carlo_option_model |
In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has the same cash flows as the reference asset (and no changes need to be ma... | https://en.wikipedia.org/wiki/Replicating_portfolio |
The notion of a replicating portfolio is fundamental to rational pricing, which assumes that market prices are arbitrage-free – concretely, arbitrage opportunities are exploited by constructing a replicating portfolio. In practice, replicating portfolios are seldom, if ever, exact replications. Most significantly, unle... | https://en.wikipedia.org/wiki/Replicating_portfolio |
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives du... | https://en.wikipedia.org/wiki/Equivalent_Martingale_Measure |
In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the... | https://en.wikipedia.org/wiki/Convexity_(finance) |
In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal. Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the ex... | https://en.wikipedia.org/wiki/Kelly_criterion |
If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less t... | https://en.wikipedia.org/wiki/Kelly_criterion |
In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.... | https://en.wikipedia.org/wiki/Black-Scholes_equation |
In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by J... | https://en.wikipedia.org/wiki/Constant_elasticity_of_variance_model |
In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was in... | https://en.wikipedia.org/wiki/Cox–Ingersoll–Ross_model |
In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let X = (X0, X1, . . . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
, XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, . . . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
, FN), and let Q {\displaystyle \mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
. , UN) denote the Snell envelope of X with respect to Q {\displaystyle \mathbb {Q} } . The Snell envelope is the smallest Q {\displaystyle \mathbb {Q} } -supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
Let U = M + A denote the Doob decomposition with respect to Q {\displaystyle \mathbb {Q} } of the Snell envelope U into a martingale M = (M0, M1, . . . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
, MN) and a decreasing predictable process A = (A0, A1, . . . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
, AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way is τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0. {\displaystyle \tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<... | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
. . , N − 1}, hence τmax is indeed a stopping time. It gives the last moment before the discounted value of the American option will drop in expectation; up to time τmax the discounted value process U is a martingale with respect to Q {\displaystyle \mathbb {Q} } . | https://en.wikipedia.org/wiki/Doob_decomposition_theorem |
In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are... | https://en.wikipedia.org/wiki/Option_delta |
In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especial... | https://en.wikipedia.org/wiki/SABR_volatility_model |
In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form d S t = ( r t − d t ) S t d t + σ t S t d W t {\displaystyle dS_{t}=(r_{t}-d_{t})S_{t}\,dt+\sigma _{t}S_{t}\,dW_{t}} ,under the risk neutral measure, where r t {\disp... | https://en.wikipedia.org/wiki/Local_volatility |
And when such volatility is merely a function of the current underlying asset level St and of time t, we have a local volatility model. The local volatility model is a useful simplification of the stochastic volatility model. | https://en.wikipedia.org/wiki/Local_volatility |
"Local volatility" is thus a term used in quantitative finance to denote the set of diffusion coefficients, σ t = σ ( S t , t ) {\displaystyle \sigma _{t}=\sigma (S_{t},t)} , that are consistent with market prices for all options on a given underlying, yielding an asset price model of the type d S t = ( r t − d t ) S t... | https://en.wikipedia.org/wiki/Local_volatility |
In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates. This model fits the observed implied volatility surface well. The model is a Heston process for stochastic volatility with an added Merton log-normal jump. It assumes the following correlated processes: d S = μ S d t + ν S d Z 1 ... | https://en.wikipedia.org/wiki/Stochastic_volatility_jump |
In mathematical finance, the term is also used as an informal synonym for the principle of no-arbitrage. This principle states that a combination of securities that has the same cash-flows as another security must have the same net price in equilibrium. | https://en.wikipedia.org/wiki/No_such_thing_as_a_free_lunch |
In mathematical finance, the volatility risk premium is a measure of the extra amount investors demand in order to hold a volatile security, above what can be computed based on expected returns. It can be defined as the compensation for inherent volatility risk divided by the volatility beta. | https://en.wikipedia.org/wiki/Volatility_risk_premium |
In mathematical finite group theory, Thompson's original uniqueness theorem (Feit & Thompson 1963, theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof of the uniq... | https://en.wikipedia.org/wiki/Thompson_uniqueness_theorem |
In mathematical finite group theory, a Thompson factorization, introduced by Thompson (1966), is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of p-subgroups for some prime p. | https://en.wikipedia.org/wiki/Thompson_factorization |
In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher. | https://en.wikipedia.org/wiki/Aschbacher_block |
In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type (Gorenstein 1982, definition 1.45). As the name suggests, many of the groups of Lie type over the field with 2 elements are groups of GF(2)-type. Also 16 ... | https://en.wikipedia.org/wiki/Groups_of_GF(2)_type |
The groups of each of these 8 types were classified by various authors. They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases, including the majority of the sporadic simple groups. Smith (1980) gave a survey of this work. Smit... | https://en.wikipedia.org/wiki/Groups_of_GF(2)_type |
In mathematical finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic. According to Thompson (1968, p.386), the p-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an ext... | https://en.wikipedia.org/wiki/Group_of_symplectic_type |
In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by Thompson (1971), is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimal polynomial (x − 1)2. Thomp... | https://en.wikipedia.org/wiki/Quadratic_pair |
In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman (1964, 1971). Several of the sporadic simple groups were discovered as rank 3 permutation groups. | https://en.wikipedia.org/wiki/Rank_3_permutation_group |
In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups. | https://en.wikipedia.org/wiki/N-group_(finite_group_theory) |
In mathematical finite group theory, an exceptional character of a group is a character related in a certain way to a character of a subgroup. They were introduced by Suzuki (1955, p. 663), based on ideas due to Brauer in (Brauer & Nesbitt 1941). | https://en.wikipedia.org/wiki/Exceptional_character |
In mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements of the conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a s... | https://en.wikipedia.org/wiki/Baer–Suzuki_theorem |
In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another resul... | https://en.wikipedia.org/wiki/Brauer–Fowler_theorem |
In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup H with support on a subset K of H to class functions on a group G (Collins 1990, 6.1). It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their pr... | https://en.wikipedia.org/wiki/Dade_isometry |
In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} of G L 5 ( F 2 ) {\displaystyle \mathrm {GL} _{5}(\mathbb {F} _{2})} by its natural module... | https://en.wikipedia.org/wiki/Dempwolff_group |
Huppert (1967, p.124) showed that any extension of G L n ( F q ) {\displaystyle \mathrm {GL} _{n}(\mathbb {F} _{q})} by its natural module F q n {\displaystyle \mathbb {F} _{q}^{n}} splits if q > 2 {\displaystyle q>2} , and Dempwolff (1973) showed that it also splits if n {\displaystyle n} is not 3, 4, or 5, and in eac... | https://en.wikipedia.org/wiki/Dempwolff_group |
G L 3 ( F 2 ) {\displaystyle 2^{3\,. }\mathrm {GL} _{3}(\mathbb {F} _{2})} is a maximal subgroup of the Chevalley group G 2 ( F 3 ) {\displaystyle G_{2}(\mathbb {F} _{3})} . The nonsplit extension 2 4 . | https://en.wikipedia.org/wiki/Dempwolff_group |
G L 4 ( F 2 ) {\displaystyle 2^{4\,. }\mathrm {GL} _{4}(\mathbb {F} _{2})} is a maximal subgroup of the sporadic Conway group Co3. The nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} is a maximal subgroup of the Thompson sporadic group Th. | https://en.wikipedia.org/wiki/Dempwolff_group |
In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973, 1974) in a 464-page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups.Finite simple groups of section 2 that rank at least 5, have S... | https://en.wikipedia.org/wiki/Gorenstein–Harada_theorem |
In mathematical finite group theory, the L-balance theorem was proved by Gorenstein & Walter (1975). The letter L stands for the layer of a group, and "balance" refers to the property discussed below. | https://en.wikipedia.org/wiki/L-balance_theorem |
In mathematical finite group theory, the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup. | https://en.wikipedia.org/wiki/Puig_subgroup |
In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson (Held 1969, p.279), gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955). | https://en.wikipedia.org/wiki/Thompson_order_formula |
In mathematical finite group theory, the Thompson subgroup J ( P ) {\displaystyle J(P)} of a finite p-group P refers to one of several characteristic subgroups of P. John G. Thompson (1964) originally defined J ( P ) {\displaystyle J(P)} to be the subgroup generated by the abelian subgroups of P of maximal rank. More o... | https://en.wikipedia.org/wiki/Thompson_subgroup |
In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson un... | https://en.wikipedia.org/wiki/Thompson_transitivity_theorem |
In mathematical finite group theory, the classical involution theorem of Aschbacher (1977a, 1977b, 1980) classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. Berkman (2001) extended the classical in... | https://en.wikipedia.org/wiki/Classical_involution_theorem |
In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian p-groups, but is general enough to include most "small" p-groups. Regular p-groups were introduced by Phillip Hall (1934). | https://en.wikipedia.org/wiki/Regular_p-group |
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