ⓘ Free online encyclopedia. Did you know? page 305

Cissoid of Diocles

In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a ci ...

Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal su ...

Compact operator on Hilbert space

In functional analysis, the concept of a compact operator in a Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space, Hilbert space, compact operators are precisely the closure of finite rank operato ...

Completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form a x 2 + b x + c {\displaystyle ax^{2}+bx+c} to the form a x − h 2 + k {\displaystyle ax-h^{2}+k} for some values of h and k. Completing ...

Complex conjugate root theorem

In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. It follows fro ...

Continuous mapping theorem

In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergen ...

Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain equals point-wise ...

Cook–Levin theorem

In computational complexity theory, the Cook–Levin theorem, also known as Cooks theorem, states that the Boolean satisfiability problem is NP-complete. That is, any problem in NP can be reduced in polynomial time by a deterministic Turing machine ...

Coupon collector's problem

In probability theory, the coupon collectors problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the pr ...

Crossing number inequality

In the mathematics of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the minimum number of crossings of a given graph, as a function of the number of edges and vertices of the graph. It states that, for gra ...

Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction ...

Darboux's theorem (analysis)

In mathematics, Darbouxs theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of other functions has the intermediate value property: the image of an interval ...

De Moivre's formula

In mathematics, de Moivres formula states that for any real number x and integer n it holds that cos ⁡ x + i sin ⁡ x) n = cos ⁡ n x + i sin ⁡ n x, {\displaystyle {\big }\cosx+i\sinx{\big)}^{n}=\cosnx+i\sinnx,} where i is the imaginary unit i 2 = ...

Delta method

In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.

Difference of two squares

In mathematics, the difference of two squares is a squared number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = a + b a − b {\displaystyle a^{2}-b^{2}=a+ba-b} in elementa ...

Dimension theorem for vector spaces

In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite, and defines the dimension of the vector space. Formally, the dimension ...

Dini's theorem

In the mathematical field of analysis, Dinis theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform.

Divisibility rule

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or bas ...

Dominated convergence theorem

In measure theory, Lebesgues dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L 1 norm. Its power and utility are two of the primary theor ...

Doob decomposition theorem

In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale an ...

Egorov's theorem

In measure theory, an area of mathematics, Egorovs theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after ...

Eisenstein's criterion

In mathematics, Eisensteins criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers - that is, for it to not be factorizable into the product of non-constant polynomials with r ...

Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...

Elias Bassalygo bound

Let C {\displaystyle C} be a q {\displaystyle q} -ary code of length n {\displaystyle n}, i.e. a subset of ^{n}\:\ \Delta x,y\leqslant \rho n\right\}} be the Hamming ball of radius ρ n {\displaystyle \rho n} centered at y {\displaystyle y}. Let V ...

Erdos–Anning theorem

The Erdos–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdos and Norman H. Anning, who published a proof of it in 1945.

Erdos–Ko–Rado theorem

In combinatorics, the Erdos–Ko–Rado theorem of Paul Erdos, Chao Ko, and Richard Rado is a theorem on intersecting set families. The theorem is as follows. Suppose that A is a family of distinct subsets of { 1, 2., n } {\displaystyle \{1.2.,n\}} s ...

Erdos–Szekeres theorem

In mathematics, the Erdos–Szekeres theorem is a finitary result that makes precise one of the corollaries of Ramseys theorem. While Ramseys theorem makes it easy to prove that every infinite sequence of distinct real numbers contains a monotonica ...

Euclid–Euler theorem

The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes. It states that every even perfect number has the form 2 p −1, where 2 p − 1 is a prime number. The theorem is named after Euclid and Leonhard Eu ...

Euclid's lemma

In number theory, Euclids lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and since this is divisible by 19, the lemma implies that one or both ...

Euclid's theorem

Euclids theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.

Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, a ...

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological spaces shape or structure regardless of the way it is bent. It is ...

Euler's criterion

In number theory, Eulers criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 mod p if there is an integer x such ...

Euler's formula

Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Eulers formula states that for any r ...

Euler's theorem

In number theory, Eulers theorem states that if n and a are coprime positive integers, then a φ n ≡ 1 mod n {\displaystyle a^{\varphi n}\equiv 1{\pmod {n}}} where φ n {\displaystyle \varphi n} is Eulers totient function. The notation is explained ...

Euler's theorem in geometry

In geometry, Eulers theorem states that the distance d between the circumcentre and incentre of a triangle is given by d 2 = R − 2 r {\displaystyle d^{2}=RR-2r} or equivalently 1 R − d + 1 R + d = 1 r, {\displaystyle {\frac {1}{R-d}}+{\frac {1}{R ...

Exterior angle theorem

The exterior angle theorem is Proposition 1.16 in Euclids Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute ge ...

Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval such that: f c ≥ f x ≥ f d for all x ∈ } by the value f a {\displaystyle fa}. If e &gt, a {\displaystyle e&gt, a} is another point ...

Fary's theorem

In mathematics, Farys theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow ...

Fatou–Lebesgue theorem

In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these funct ...

Fermat point

In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because ...

Fermat's theorem (stationary points)

In mathematics, Fermats theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point. Fermats theorem is a theorem in real analysis, named ...

Feynman–Kac formula

The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes. When Mark Kac and Richard Feynman were both on Cornell faculty, Kac attended a lecture ...

Five lemma

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also wo ...

Fixed-point lemma for normal functions

The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points. It was first proved by Oswald Veblen in 1908.

Frattini's argument

In group theory, a branch of mathematics, Frattinis argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. The ...

Frechet inequalities

In probabilistic logic, the Frechet inequalities, also known as the Boole–Frechet inequalities, are rules implicit in the work of George Boole and explicitly derived by Maurice Frechet that govern the combination of probabilities about logical pr ...

Freivalds' algorithm

Freivalds algorithm is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices A {\displaystyle A}, B {\displaystyle B}, and C {\displaystyle C}, a general problem is to verify whether A × B = C {\dis ...

Frobenius theorem (real division algebras)

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every s ...

Froda's theorem

In mathematics, Darboux–Frodas theorem, named after Alexandru Froda, a Romanian mathematician, describes the set of discontinuities of a monotone real-valued function of a real variable. Usually, this theorem appears in literature without a name. ...