Search results
Results From The WOW.Com Content Network
and : are continuous functions on topological spaces, and . being topologically semiconjugate to means, by definition, that is a surjection such that =.. and being topologically conjugate means, by definition, that they are topologically semiconjugate and is furthermore injective, then bijective, and its inverse is continuous too; i.e. is a homeomorphism; further, is termed a topological ...
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
(where the indicates the complex conjugate) for all in the domain of . In physics , this property is referred to as PT symmetry . This definition extends also to functions of two or more variables, e.g., in the case that f {\displaystyle f} is a function of two variables it is Hermitian if
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
For the cases where has full row or column rank, and the inverse of the correlation matrix ( for with full row rank or for full column rank) is already known, the pseudoinverse for matrices related to can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the ...
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, [1] [2] or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle —between them.
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation , Fenchel transformation , or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel ).