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c_hl4ic40u3w3m | In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of r(x) = p(x)/q(x)over the real line is the difference between the number of roots of f(z) located in the right half-plane and those located in the left half-plane. The complex polynomial f(z) is such that f(iy) = q(y) + ip(y).We must also assume that p has degree less than the degree of q. | Cauchy index | [
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c_scxuwnj5gvkt | 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\left(x+{\frac {b-a}{n}}\right)} . | Universal chord theorem | [
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c_35yd2rasda2g | In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems. | Hilbert–Schmidt theorem | [
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c_5lis8a4pdqwu | In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes conditions under which lim t → ∞ f ( t ) = lim s → 0 s F ( s ) {\displaystyle \lim _{t\to \infty }f(t)=\lim _{s\,\to \,0}{sF(s)}} Likewise, if f {\displaystyle f} in discrete time has (unilateral) Z-transform F ( z ) {\displaystyle F(z)} , then a final value theorem establishes conditions under which lim k → ∞ f = lim z → 1 ( z − 1 ) F ( z ) {\displaystyle \lim _{k\to \infty }f=\lim _{z\to 1}{(z-1)F(z)}} An Abelian final value theorem makes assumptions about the time-domain behavior of f ( t ) {\displaystyle f(t)} (or f {\displaystyle f} ) to calculate lim s → 0 s F ( s ) {\displaystyle \lim _{s\,\to \,0}{sF(s)}} . Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F ( s ) {\displaystyle F(s)} to calculate lim t → ∞ f ( t ) {\displaystyle \lim _{t\to \infty }f(t)} (or lim k → ∞ f {\displaystyle \lim _{k\to \infty }f} ) (see Abelian and Tauberian theorems for integral transforms). | Final value theorem | [
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c_28uyklxxco7m | In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein. | Constructive function theory | [
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c_dqwo65pb0gaf | In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕ A = A {\displaystyle A\oplus A=A} . == References == | Idempotent analysis | [
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c_5ev3dhu7bq90 | Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. | Kakutani fixed-point theorem | [
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c_9v2fbcd9isfl | In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. | Kakutani fixed-point theorem | [
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c_3zqp8y9qwl7s | If the codomain has a structure of R {\displaystyle \mathbb {R} } -algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications. | Function of a real variable | [
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c_1qhv2i8119o5 | In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set. | Function of a real variable | [
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c_nyzvymtzjlvn | However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers or the quaternions. The structure R {\displaystyle \mathbb {R} } -vector space of the codomain induces a structure of R {\displaystyle \mathbb {R} } -vector space on the functions. | Function of a real variable | [
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c_ooyjnr72ousd | In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. The closely related FASS curves (approximately space-Filling, self-Avoiding, Simple, and Self-similar curves) can be thought of as finite approximations of a certain type of space-filling curves. | Space-filling curves | [
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c_w1gi2tk9j741 | In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the In. (Here |In| denotes the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue measure 0. | Strong measure zero set | [
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c_rcwego6totmb | Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets. In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture holds. These two results together establish the independence of Borel's conjecture. | Strong measure zero set | [
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c_36k604pzsyof | The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R.This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R for every set A ⊆ R of Lebesgue measure zero.The dual Borel conjecture states that every strongly meagre set is countable. This statement is also independent of ZFC. == References == | Strong measure zero set | [
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c_j9khdlunmqoa | 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 biology. Her work in this area includes pioneering contributions to the theory of combinatorial matrix theory in which she proved connections between the sign pattern of a matrix and its stability, as well as results on matrix decomposition. | Pauline van den Driessche | [
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c_pm1faufk8xvv | 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 quadratic field Q ( − 7 ) {\displaystyle \mathbb {Q} ({\sqrt {-7}})} . This ring is a unique factorization domain. | Kleinian integer | [
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c_2a5150d2itg7 | In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. | Local maxima | [
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c_fmduebhpxx67 | As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. | Local maxima | [
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c_o93t6a1dixi5 | In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. | Mean value theorem for divided differences | [
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c_ueebyjc1g5oc | In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as where n is any nonnegative integer. The kernel functions are periodic with period 2 π {\displaystyle 2\pi } . The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i.e., we have where is the kth Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. | Dirichlet kernel | [
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c_87lj9xecycqc | 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)} where u {\displaystyle u} is the displacement vector, Δ 2 {\displaystyle \scriptstyle \Delta _{2}} is the vector laplacian, ∇ {\displaystyle \scriptstyle \nabla } is the gradient, ∇ ⋅ {\displaystyle \scriptstyle \nabla \cdot } is the divergence and ω {\displaystyle \omega } is a Cosserat eigenvalue. The full description of the spectrum and the proof of the completeness of the system of eigenfunctions are also due to Mikhlin, and partly to V.G. Maz'ya in their only joint work. | Solomon Mikhlin | [
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c_eiico05tbyt6 | 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 theorem on perturbations of positive operators in a Hilbert space which let him to obtain an error estimate for the problem of approximating a sloping shell by a plane plate. | Solomon Mikhlin | [
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c_44x0hmcbltls | 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 articles dealing with it. | Solomon Mikhlin | [
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c_ht9zkdaqh9d1 | In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set , which is a complete lattice. | Limit inferior | [
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c_1x4c6t21pxjk | In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma.The lemma is stated as follows: Suppose g is a real-valued continuous function on the interval and S is the set of x in such that there exists a y∈(x,b] with g(y) > g(x). (Note that b cannot be in S, though a may be.) Define E = S ∩ (a,b).Then E is an open set, and it may be written as a countable union of disjoint intervals E = ⋃ k ( a k , b k ) {\displaystyle E=\bigcup _{k}(a_{k},b_{k})} such that g(ak) = g(bk), unless ak = a ∈ S for some k, in which case g(a) < g(bk) for that one k. Furthermore, if x ∈ (ak,bk), then g(x) < g(bk).The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow. | Rising sun lemma | [
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c_xskmiar6tw57 | 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)*(5/12)=5/26 chance that all three randomly chosen numbers are in the same Fano plane set. In this case, there is a 1/5 chance that they are on a line, and hence all three numbers are on one ticket, otherwise each of the three pairs are on three different tickets. | Transylvania lottery | [
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c_lgw8pphziufq | 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 winning. (Javier Martínez, Gloria Gutiérrez & Pablo Cordero et al. 2008, p.85)(Mazur 2010, p.280 problem 15) An upper bound can be given using the Fano plane with a collection of 14 tickets in two sets of seven. | Transylvania lottery | [
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c_x2a1jbftzt7w | In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere. | Rademacher–Menchov theorem | [
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c_472jo8hqam7i | 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 continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if x {\displaystyle x} is some vector such that x = ( x 1 , x 2 , … , x n ) {\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)} in finite dimensional coordinate space, it takes the form: ‖ x ‖ ∞ := max ( | x 1 | , … , | x n | ) . {\displaystyle \|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).} | Chebyshev norm | [
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c_vyukr0vcpidx | 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\,\right\}.} This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. | Chebyshev norm | [
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c_ydxhdt2lmxax | In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier integral operators, oscillatory integral operators, and paradifferential operators. The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point. This gains in importance on manifolds of dimension greater than one. | Microlocalization functor | [
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] |
c_uodmgwqx9ucm | Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration. Integrals in the fourth form that are improper because f ( x ) {\displaystyle f(x)} has a vertical asymptote somewhere on the interval {\displaystyle } may be described as being of the "second" type or kind. Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.In each case above, the improper integral must be rewritten using one or more limits, depending on what is causing the integral to be improper. | Improper integrals | [
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c_5biduppnpbal | If a regular definite integral (which may retronymically be called a proper integral) is worked out as if it is improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in any of the following forms: ∫ a ∞ f ( x ) d x {\displaystyle \int _{a}^{\infty }f(x)\,dx} ∫ − ∞ b f ( x ) d x {\displaystyle \int _{-\infty }^{b}f(x)\,dx} ∫ − ∞ ∞ f ( x ) d x {\displaystyle \int _{-\infty }^{\infty }f(x)\,dx} ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} , where f ( x ) {\displaystyle f(x)} is undefined or discontinuous somewhere on {\displaystyle } The first three forms are improper because the integrals are taken over an unbounded interval. (They may be improper for other reasons, as well, as explained below.) | Improper integrals | [
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] |
c_4tj6joi6y79v | The function f ( x ) {\displaystyle f(x)} can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression). Cases 2–4 are handled similarly. See the examples below. Improper integrals can also be evaluated in the context of complex numbers, in higher dimensions, and in other theoretical frameworks such as Lebesgue integration or Henstock–Kurzweil integration. Integrals that are considered improper in one framework may not be in others. | Improper integrals | [
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c_bngt2fbilcwl | In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , then ∫ R n f ( x ) g ( x ) d x ≤ ∫ R n f ∗ ( x ) g ∗ ( x ) d x {\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx} where f ∗ {\displaystyle f^{*}} and g ∗ {\displaystyle g^{*}} are the symmetric decreasing rearrangements of f {\displaystyle f} and g {\displaystyle g} , respectively.The decreasing rearrangement f ∗ {\displaystyle f^{*}} of f {\displaystyle f} is defined via the property that for all r > 0 {\displaystyle r>0} the two super-level sets E f ( r ) = { x ∈ X: f ( x ) > r } {\displaystyle E_{f}(r)=\left\{x\in X:f(x)>r\right\}\quad } and E f ∗ ( r ) = { x ∈ X: f ∗ ( x ) > r } {\displaystyle \quad E_{f^{*}}(r)=\left\{x\in X:f^{*}(x)>r\right\}} have the same volume ( n {\displaystyle n} -dimensional Lebesgue measure) and E f ∗ ( r ) {\displaystyle E_{f^{*}}(r)} is a ball in R n {\displaystyle \mathbb {R} ^{n}} centered at x = 0 {\displaystyle x=0} , i.e. it has maximal symmetry. | Hardy–Littlewood inequality | [
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c_puw2gp2za5z6 | 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}}}} . | Corner angle | [
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] |
c_ppp4c0c5126h | 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, . . | Corner angle | [
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c_15sm4lpugkb1 | 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 lattice, then x ∗ ( θ ) = arg max x ∈ D f ( x , θ ) {\displaystyle x^{*}(\theta )=\arg \max _{x\in D}f(x,\theta )} is nondecreasing in θ. The result is especially helpful for establishing comparative static results when the objective function is not differentiable. The result is named after Donald M. Topkis. | Topkis's Theorem | [
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c_t8pwbxg0no8w | Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the function. If the function takes − ∞ {\displaystyle -\infty } as a value then − ∞ {\displaystyle -\infty } is necessarily the global minimum value and the minimization problem can be answered; this is ultimately the reason why the definition of "proper" requires that the function never take − ∞ {\displaystyle -\infty } as a value. Assuming this, if the function's domain is empty or if the function is identically equal to + ∞ {\displaystyle +\infty } then the minimization problem once again has an immediate answer. | Proper convex function | [
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c_5gcduzf2mcem | Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases. If the problem is instead a maximization problem (which would be clearly indicated, such as by the function being concave rather than convex) then the definition of "proper" is defined in an analogous (albeit technically different) manner but with the same goal: to exclude cases where the maximization problem can be answered immediately. Specifically, a concave function g {\displaystyle g} is called proper if its negation − g , {\displaystyle -g,} which is a convex function, is proper in the sense defined above. | Proper convex function | [
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c_dl11n5u502yw | In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value − ∞ {\displaystyle -\infty } and also is not identically equal to + ∞ . {\displaystyle +\infty .} In convex analysis and variational analysis, a point (in the domain) at which some given function f {\displaystyle f} is minimized is typically sought, where f {\displaystyle f} is valued in the extended real number line = R ∪ { ± ∞ } . {\displaystyle =\mathbb {R} \cup \{\pm \infty \}.} | Proper convex function | [
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c_gtozwvpyidlh | When evaluating the integral, t is held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t, we then consider t to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration). | Parameter | [
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c_btf95eqtr02d | In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form F ( t ) = ∫ x 0 ( t ) x 1 ( t ) f ( x ; t ) d x . {\displaystyle F(t)=\int _{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.} In this formula, t is the argument of the function F, and on the right-hand side the parameter on which the integral depends. | Parameter | [
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c_d8r8tgiyg3mn | In mathematical analysis, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement. The inequality is named after the mathematicians George Pólya and Gábor Szegő. | Pólya–Szegő inequality | [
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c_xaprntlu9in1 | In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose x ˙ = f ( x ) {\displaystyle {\dot {x}}=f(x)} is a dynamical system, x ( t , x 0 ) {\displaystyle x(t,x_{0})} is a trajectory, and x 0 {\displaystyle x_{0}} is the initial point. Let O := { x ∈ R n ∣ φ ( x ) = 0 } {\displaystyle {\mathcal {O}}:=\left\lbrace x\in \mathbb {R} ^{n}\mid \varphi (x)=0\right\rbrace } where φ {\displaystyle \varphi } is a real-valued function. The set O {\displaystyle {\mathcal {O}}} is said to be positively invariant if x 0 ∈ O {\displaystyle x_{0}\in {\mathcal {O}}} implies that x ( t , x 0 ) ∈ O ∀ t ≥ 0 {\displaystyle x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0} In other words, once a trajectory of the system enters O {\displaystyle {\mathcal {O}}} , it will never leave it again. | Positive invariant set | [
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c_9a1xwybe7kpk | 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 between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K. | Quasiconformal map | [
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] |
c_tc3w9rdaooto | In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by Agranovich and Dynin (1962). | Agranovich–Dynin formula | [
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c_w6mv7xsj3e7x | In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof. | Weierstrass approximation theorem | [
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c_3nub7hbohpbb | His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X {\displaystyle X} are shown to suffice, as is detailed below. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane. | Weierstrass approximation theorem | [
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c_rv1h5rfyh9dc | 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 engine called "Double Negative Gravitational Renderer" a.k.a. "Gargantua", to create a "real" blackhole, used in the final cut. Later on the visual effects team went onto publish a blackhole study | Multimedia | [
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c_bbjatgrhdwxs | 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. | Multimedia | [
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c_p4mwk5hgv5dt | {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.} If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that ( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)} for almost all x {\displaystyle \,x} and ( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) {\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)} for almost all y {\displaystyle \,y} , then T {\displaystyle \,T} extends to a continuous operator T: L 2 → L 2 {\displaystyle T:L^{2}\to L^{2}} with the operator norm ‖ T ‖ L 2 → L 2 ≤ α β . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.} Such functions p ( x ) {\displaystyle \,p(x)} , q ( y ) {\displaystyle \,q(y)} are called the Schur test functions. In the original version, T {\displaystyle \,T} is a matrix and α = β = 1 {\displaystyle \,\alpha =\beta =1} . | Schur test | [
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c_87caax1x4hr3 | In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X , Y {\displaystyle X,\,Y} be two measurable spaces (such as R n {\displaystyle \mathbb {R} ^{n}} ). Let T {\displaystyle \,T} be an integral operator with the non-negative Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} , x ∈ X {\displaystyle x\in X} , y ∈ Y {\displaystyle y\in Y}: T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y . | Schur test | [
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c_psxpsqzage99 | In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations x 1 ′ = f 1 ( x 1 , … , x n ) {\displaystyle x_{1}'=f_{1}(x_{1},\ldots ,x_{n})} x 2 ′ = f 2 ( x 1 , … , x n ) {\displaystyle x_{2}'=f_{2}(x_{1},\ldots ,x_{n})} ⋮ {\displaystyle \vdots } x n ′ = f n ( x 1 , … , x n ) {\displaystyle x_{n}'=f_{n}(x_{1},\ldots ,x_{n})} where x ′ {\displaystyle x'} here represents a derivative of x {\displaystyle x} with respect to another parameter, such as time t {\displaystyle t} . The j {\displaystyle j} 'th nullcline is the geometric shape for which x j ′ = 0 {\displaystyle x_{j}'=0} . The equilibrium points of the system are located where all of the nullclines intersect. In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves. | Nullcline | [
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c_hqcg9ef3sika | 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 converge uniformly to the diagonal of the square. However, each staircase has length two, while the length of the diagonal is the square root of 2, so the sequence of staircase lengths does not converge to the length of the diagonal. Martin Gardner calls this "an ancient geometrical paradox". | Staircase paradox | [
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c_qe4optiqci0w | 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 converge to the correct length can be explained by the fact that some of their vertices do not lie on the diagonal. In higher dimensions, the Schwarz lantern provides an analogous example showing that polyhedral surfaces that converge pointwise to a curved surface do not necessarily converge to its area, even when the vertices all lie on the surface.As well as highlighting the need for careful definitions of arc length in mathematics education, the paradox has applications in digital geometry, where it motivates methods of estimating the perimeter of pixelated shapes that do not merely sum the lengths of boundaries between pixels. | Staircase paradox | [
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c_9oeg4aeoqeyb | In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f: I → ( M , d ) {\displaystyle f:I\to (M,d)} , where ( M , d ) {\displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is ‖ f ‖ p -var = ( sup D ∑ t k ∈ D d ( f ( t k ) , f ( t k − 1 ) ) p ) 1 / p {\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}} where D ranges over all finite partitions of the interval I. The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then g ∘ f {\displaystyle g\circ f} has finite p α {\displaystyle {\frac {p}{\alpha }}} -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions. | P-variation | [
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c_vhzl46wlpgc9 | In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain. Let f = ∑ a n X n ∈ R ] , {\displaystyle f=\sum a_{n}X^{n}\in R],} and suppose S {\displaystyle S} is a commutative associative algebra over R {\displaystyle R} , I {\displaystyle I} is an ideal in S {\displaystyle S} such that the I-adic topology on S {\displaystyle S} is complete, and x {\displaystyle x} is an element of I {\displaystyle I} . Define: f ( x ) = ∑ n ≥ 0 a n x n . | Power series ring | [
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] |
c_bry3orcebb9d | {\displaystyle f(x)=\sum _{n\geq 0}a_{n}x^{n}.} This series is guaranteed to converge in S {\displaystyle S} given the above assumptions on x {\displaystyle x} . Furthermore, we have ( f + g ) ( x ) = f ( x ) + g ( x ) {\displaystyle (f+g)(x)=f(x)+g(x)} and ( f g ) ( x ) = f ( x ) g ( x ) . | Power series ring | [
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c_7fg6shi206hk | {\displaystyle (fg)(x)=f(x)g(x).} Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved. Since the topology on R ] {\displaystyle R]} is the ( X ) {\displaystyle (X)} -adic topology and R ] {\displaystyle R]} is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients (so that they belong to the ideal ( X ) {\displaystyle (X)} ): f ( 0 ) {\displaystyle f(0)} , f ( X 2 − X ) {\displaystyle f(X^{2}-X)} and f ( ( 1 − X ) − 1 − 1 ) {\displaystyle f((1-X)^{-1}-1)} are all well defined for any formal power series f ∈ R ] . | Power series ring | [
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c_en4nfi6mvxym | 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. The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable. | Radó's theorem (Riemann surfaces) | [
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c_a03w2o1mz77b | 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 made to maintain this), and dynamic replication, where the portfolio does not have the same cash flows, but has the same "Greeks" as the reference asset, meaning that for small (properly, infinitesimal) changes to underlying market parameters, the price of the asset and the price of the portfolio change in the same way. Dynamic replication requires continual adjustment, as the asset and portfolio are only assumed to behave similarly at a single point (mathematically, their partial derivatives are equal at a single point). Given an asset or liability, an offsetting replicating portfolio (a "hedge") is called a static hedge or dynamic hedge, and constructing such a portfolio (by selling or purchasing) is called static hedging or dynamic hedging. | Replicating portfolio | [
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c_grnhiladhd66 | 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, unless they are claims against the same counterparties, there is credit risk. Further, dynamic replication is invariably imperfect, since actual price movements are not infinitesimal – they may in fact be large – and transaction costs to change the hedge are not zero. | Replicating portfolio | [
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c_vvp24ggdjzn2 | 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 short elementary proof. | Baer–Suzuki theorem | [
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] |
c_rijutjngt4bo | 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. | Margrabe's formula | [
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c_c8wxfu1gdhcj | 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. | Volatility risk premium | [
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c_8uaognnplxym | In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical form for them. | Canonizant | [
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c_pry42kdmesja | In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurface that vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning that they have a common point where they meet to unusually high order. | Osculant | [
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c_6vpxb3tvbtvd | 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 involution theorem to groups of finite Morley rank. A classical involution t of a finite group G is an involution whose centralizer has a subnormal subgroup containing t with quaternion Sylow 2-subgroups. | Classical involution theorem | [
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c_8cgw3qvm2pxp | In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. | Univalence axiom | [
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c_kqnqpt1xgj7k | Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally. This kind of variability is unavoidable when a field is in rapid flux. | Univalence axiom | [
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c_7v9pxc0djz0i | If V {\displaystyle V} is a set of strings, then V ∗ {\displaystyle V^{*}} is defined as the smallest superset of V {\displaystyle V} that contains the empty string ε {\displaystyle \varepsilon } and is closed under the string concatenation operation. If V {\displaystyle V} is a set of symbols or characters, then V ∗ {\displaystyle V^{*}} is the set of all strings over symbols in V {\displaystyle V} , including the empty string ε {\displaystyle \varepsilon } .The set V ∗ {\displaystyle V^{*}} can also be described as the set containing the empty string and all finite-length strings that can be generated by concatenating arbitrary elements of V {\displaystyle V} , allowing the use of the same element multiple times. If V {\displaystyle V} is either the empty set ∅ or the singleton set { ε } {\displaystyle \{\varepsilon \}} , then V ∗ = { ε } {\displaystyle V^{*}=\{\varepsilon \}} ; if V {\displaystyle V} is any other finite set or countably infinite set, then V ∗ {\displaystyle V^{*}} is a countably infinite set. As a consequence, each formal language over a finite or countably infinite alphabet Σ {\displaystyle \Sigma } is countable, since it is a subset of the countably infinite set Σ ∗ {\displaystyle \Sigma ^{*}} . The operators are used in rewrite rules for generative grammars. | Kleene star | [
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c_5xuxzguq6qba | In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid construction. The application of the Kleene star to a set V {\displaystyle V} is written as V ∗ {\displaystyle V^{*}} . It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions". | Kleene star | [
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c_fis02p4wqglx | In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H 2 ( G , Z ) {\displaystyle H_{2}(G,\mathbb {Z} )} of a group G. It was introduced by Issai Schur (1904) in his work on projective representations. | Schur cover | [
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c_mz23ddu2plqu | In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. | Universal Horn theory | [
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c_16srhpgg7rf6 | 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 uniqueness theorem. | Thompson transitivity theorem | [
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c_4quex9pcq49c | 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. | Thompson factorization | [
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c_rji790yn9m4y | 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). | Regular p-group | [
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c_3uju06rykpib | 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 identity matrix and ρ ( g h ) = ρ ( g ) ρ ( h ) for all g , h ∈ G . {\displaystyle \rho (gh)=\rho (g)\rho (h){\text{ for all }}g,h\in G.} Quaternionic representations of associative and Lie algebras can be defined in a similar way. | Pseudoreal representation | [
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c_jis98rhm1t02 | 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 imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). | Pseudoreal representation | [
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c_7z495wkn9rdn | 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 due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. | Equivalent Martingale Measure | [
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c_k9515ivgpvy5 | In mathematical group theory, a tame group is a certain kind of group defined in model theory. Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that (K, T) is of finite Morley rank in its full language. A group G is then called a tame group if no bad field is interpretable in G. | Tame group | [
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c_5nd6rbtrxlby | In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970. | Root data | [
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c_k3b44wxsv67t | In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied without function symbols. | Two-variable logic with counting | [
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c_zmcv9slvsx6y | In mathematical folklore, the no free lunch theorem (sometimes pluralized) of David Wolpert and William G. Macready appears in the 1997 "No Free Lunch Theorems for Optimization. "This mathematical result states the need for a specific effort in the design of a new algorithm, tailored to the specific problem to be optimized. Kimeme allows the design and experimentation of new optimization algorithms through the new paradigm of memetic computing, a subject of computational intelligence which studies algorithmic structures composed of multiple interacting and evolving modules (memes). | Kimeme | [
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c_cv8qqt1io5y4 | 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. | Puig subgroup | [
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c_hdn6e4ykd1f2 | In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures. | 74 knot | [
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c_y58rpzmf3125 | 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. | Rank 3 permutation group | [
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c_5x4pb7efh1g0 | 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. For a European call or put on an underlying stock paying no dividends, the equation is: ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 {\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0} where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and σ {\displaystyle \sigma } is the volatility of the stock. The key financial insight behind the equation is that, under the model assumption of a frictionless market, one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently “eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula. | Black-Scholes equation | [
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c_1r01xh9b2kir | 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 expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. | Kelly criterion | [
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c_98tishu6utuq | 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 than the Kelly criterion (e.g., half). | Kelly criterion | [
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c_zl2kb29xoimr | 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 proof of the odd order theorem, and was used by Peterfalvi (2000) in his revision of the character theory of the odd order theorem. | Dade isometry | [
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c_s7r6gstcus2a | 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} } where F is the field of scalars. A representation of a group G preserves ω if ω ( g ⋅ v , g ⋅ w ) = ω ( v , w ) {\displaystyle \omega (g\cdot v,g\cdot w)=\omega (v,w)} for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if ω ( ξ ⋅ v , w ) + ω ( v , ξ ⋅ w ) = 0 {\displaystyle \omega (\xi \cdot v,w)+\omega (v,\xi \cdot w)=0} for all ξ in g and v, w in V. Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp(V,ω) or its Lie algebra sp(V,ω) If G is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius–Schur indicator. | Symplectic representation | [
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c_vevaozxmv6ym | In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection. The simple C-groups were determined by Suzuki (1965), and his classification is summarized by Gorenstein (1980, 16.4). The classification of C-groups was used in Thompson's classification of N-groups. The simple C-groups are the projective special linear groups PSL2(p) for p a Fermat or Mersenne prime the projective special linear groups PSL2(9) the projective special linear groups PSL2(2n) for n≥2 the projective special linear groups PSL3(q) for q a prime power the Suzuki groups Sz(22n+1) for n≥1 the projective unitary groups PU3(q) for q a prime power | C-group | [
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c_asq6ef548j4t | 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. | Local volatility | [
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c_8rje2dvhpq81 | 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 {\displaystyle r_{t}} is the instantaneous risk free rate, giving an average local direction to the dynamics, and W t {\displaystyle W_{t}} is a Wiener process, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility σ t {\displaystyle \sigma _{t}} . In the simplest model i.e. the Black–Scholes model, σ t {\displaystyle \sigma _{t}} is assumed to be constant, or at most a deterministic function of time; in reality, the realised volatility of an underlying actually varies with time and with the underlying itself. When such volatility has a randomness of its own—often described by a different equation driven by a different W—the model above is called a stochastic volatility model. | Local volatility | [
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c_7z8duzb4yhp5 | "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 d t + σ ( S t , t ) S t d W t . {\displaystyle dS_{t}=(r_{t}-d_{t})S_{t}\,dt+\sigma (S_{t},t)S_{t}\,dW_{t}.} This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options. | Local volatility | [
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Subsets and Splits