In mathematical analysis , a Hermitianfunction is a complex number complex function mathematics function with the property that its complex conjugate is equal to the original function with the variable changed in sign mathematics sign math f x overline f x math for all math x math in the domain of math f math . This definition extends also to functions of two or more variables, e.g., in the case that math f math is a function of two variables it is Hermitian if math f x 1, x 2 overline f x 1, x 2 math for all pairs math x 1, x 2 math in the domain of math f math . From this definition it follows immediately that, if math f math is a Hermitianfunction, then the real part of math f math is an even function the imaginary part of math f math is an odd function Motivation Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follows from basic properties of the Fourier transform The function math f math is real valued if and only if the Fourier transform of math f math is Hermitian. The function math f math is Hermitian if and only if the Fourier transform of math f math is real valued. Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even odd symmetry. This, for example, allows the discrete Fourier transform of a signal which is in general complex to be stored in the same space as the original real signal. If either f or g is Hermitian, then math f star g f g math Where the math star math is correlation , and math math is convolution . Because convolution is commutative we can infer also that If either f or g is Hermitian, then math f star g g star f math , which in general is not true. An example wanted for these two statements above See also Even and odd functions Category Types of functions Category Calculus mathanalysis stub ar bs Hermitijan funkcija es Funci n herm tica fa sr ... more details
wiktionary Hermitianhermitian A number of mathematical entities are named Hermitian , after the mathematician Charles Hermite Hermitian adjoint Hermitian connection , the unique connection on a Hermitian manifold that satisfies specific conditions Hermitian form , a specific sesquilinear form Hermitianfunction , a complex function whose complex conjugate is equal to the original function with the variable changed in sign Hermitian hat wavelet , a low oscillation, complex valued wavelet Hermitian kernel Hermitian manifold structure Hermitian metric , is a smoothly varying positive definite Hermitian form on each fiber of a complex vector bundle Hermitian matrix , a square matrix with complex entries that is equal to its own conjugate transpose Hermitian operator , bounded symmetric operators Hermitian polynomials , a classical orthogonal polynomial sequence that arise in probability Hermitian symmetric space , a K hler manifold which, as a Riemannian manifold, is a Riemannian symmetric space Conjugate transpose Hermitian transpose , the transpose of a matrix and with the complex conjugate of each entry Hermitian variety , a generalisation of quadrics Hermitian wavelet , a family of continuous wavelets See also self adjoint mathdab eo Hermita sv Hermitesk ... more details
Unreferenced date December 2009 Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform . The math n textrm th math Hermitian wavelet is defined as the math n textrm th math derivative of a Gaussian math Psi n t 2n frac n 2 c n H n left frac t sqrt n right e frac 1 2n t 2 math where math H n left x right math denotes the math n textrm th math Hermite polynomial . The normalisation coefficient math c n math is given by math c n left n frac 1 2 n Gamma n frac 1 2 right frac 1 2 left n frac 1 2 n sqrt pi 2 n 2n 1 right frac 1 2 quad n in mathbb Z . math The prefactor math C Psi math in the resolution of the identity of the continuous wavelet transform for this wavelet is given by math C Psi frac 4 pi n 2n 1 math i.e. Hermitian wavelets are admissible for all positive math n math . In computer vision and image processing , Gaussian derivative operators of different orders are frequently used as a basis for expressing various types of visual operations see scale space and N jet . Examples of Hermitian wavelets Starting from a Gaussian function with math mu 0, sigma 1 math math f t pi 1 4 e t 2 2 math the first 3 derivatives read math begin align f t & pi 1 4 te t 2 2 f t & pi 1 4 t 2 1 e t 2 2 f 3 t & pi 1 4 3t t 3 e t 2 2 end align math and their math L 2 math norms math f sqrt 2 2, f sqrt 3 2, f 3 sqrt 30 4 math So the wavelets which are the negative normalized derivatives are math begin align Psi 1 t & sqrt 2 pi 1 4 te t 2 2 Psi 2 t & frac 2 3 sqrt 3 pi 1 4 1 t 2 e t 2 2 Psi 3 t & frac 2 15 sqrt 30 pi 1 4 t 3 3t e t 2 2 end align math DEFAULTSORT Hermitian Wavelet Category Continuous wavelets ... more details
In mathematics, the Hermitian connection math nabla math , also called the Chern connection , is the unique connection mathematics connection on a Hermitian manifold that satisfies the following conditions, It preserves the metric mathematics metric g , i.e., math nabla g 0 math . It preserves the complex structure J , i.e., math nabla J 0 math . The torsion is pure in its indices. If the Hermitian manifold admits a symplectic structure , i.e., if it is a K hler manifold , then the Hermitian connection and the Levi Civita connection coincide. References Nakahara, Geometry, Topology, and Physics , Taylor & Francis. Category Complex manifolds Category Structures on manifolds Category Riemannian geometry differential geometry stub ... more details
In mathematics, an Hermitian matrix or self adjoint matrix is a square matrix with complex number complex ... math A math is denoted by math A dagger math , then the Hermitian property can be written concisely as math A A dagger ,. math Hermitian matrices can be understood as the complex extension of real symmetric matrix symmetric matrices . Hermitian matrices are named after Charles Hermite , who demonstrated ... are Hermitian. In theoretical physics such Hermitian matrices usually are multiplied by imaginary ... 20algebra 22 20physics 20 22skew Hermitian 22&pg PA652 v onepage&q&f false ref ref http www.hep.caltech.edu ... at California Institute of Technology ref which results in skew Hermitian matrices see facts below . Properties The entries on the main diagonal top left to bottom right of any Hermitian matrix are necessarily real number real . A matrix that has only real entries is Hermitian if and only if it is a symmetric ... matrix is simply a special case of a Hermitian matrix. Every Hermitian matrix is a normal matrix , and the finite dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalizable ... real entries. This implies that all eigenvectors eigenvalue s of a Hermitian matrix A are real, and that A has ... of C sup n sup consisting of n eigenvectors of A . The sum of any two Hermitian matrices is Hermitian, and the inverse matrix inverse of an invertible Hermitian matrix is Hermitian as well. However, the matrix multiplication product of two Hermitian matrices A and B will only be Hermitian if they commute, i.e., if AB BA . Thus A sup n sup is Hermitian if A is Hermitian and n is an integer. The Hermitian ... matrix math I n math is Hermitian, but math i I n math is not. However the complex Hermitian ... space dimensional vector space of complex n n matrices, the complex Hermitian matrices form a subspace ... as the imaginary unit . If n orthonormal eigenvectors math u 1, dots,u n math of a Hermitian matrix ... of the diagonal matrix math Lambda math . anchor facts Additional facts related to Hermitian ... more details
need not be closed, but the kernel of a continuous operator always is. Hermitian operators A bounded operator A H H is called Hermitian or self adjoint if math A A math which is equivalent to math ... Mathematical concepts Linear algebra Inner product Hilbert space Hermitian operator Norm mathematics ... Dual space Bra ket notation Quantum mechanics Observable s DEFAULTSORT Hermitian Adjoint Category ... more details
In mathematics , a Hermitian manifold is the complex analog of a Riemannian manifold . Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian form Hermitian inner product on each holomorphic tangent space . One can also define a Hermitian manifold as a real manifold ... G structure U n structure on the manifold. By dropping this condition we get an almost Hermitian manifold . On any almost Hermitian manifold we can introduce a fundamental 2 form , or cosymplectic ... are integrable, we have a K hler structure . Formal definition A Hermitian metric on a complex vector ... Hermitian form on each fiber. Such a metric can be written as a smooth section math h in Gamma E otimes ... p sub and math h p zeta, bar zeta 0 math for all nonzero in E sub p sub . A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space . Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent space. On a Hermitian manifold the metric can be written in local holomorphic coordinates z sup sub ... of a positive definite Hermitian matrix . Riemannian metric and associated form A Hermitian ... variously the associated 1,1 form , the fundamental form , or the Hermitian form . In local holomorphic ... align math for all complex tangent vectors u and v . The Hermitian metric h can be recovered from ... u,v end align math for all complex tangent vectors u and v . A Hermitian structure on an almost complex manifold M can therefore be specified by either a Hermitian metric h as above, a Riemannian metric ... vectors u . Note that many authors call g itself the Hermitian metric. Properties Every almost complex manifold admits a Hermitian metric. This follows directly from the analogous statement for Riemannian ... u,v 1 over 2 left g u,v g Ju,Jv right . math Choosing a Hermitian metric on an almost complex manifold ... Hermitian manifold is complex linear frame which is orthonormal with respect to the Hermitian metric ... more details
Hermitian varieties are in a sense a generalisation of quadric s, and occur naturally in the theory of polarities . Definition Let K be a field with an involutive automorphism math theta math . Let n be an integer math geq 1 math and V be an n 1 dimensional vectorspace over K . A Hermitian variety H in PG V is a set of points of which the representing vector lines consisting of isotropic points of a non trivial Hermitian sesquilinear form on V . Representation Let math e 0,e 1, ldots,e n math be a basis of V . If a point p in the projective space has homogenous coordinates math X 0, ldots,X n math with respect to this basis, it is on the Hermitian variety if and only if math sum i,j 0 n a ij X i X j theta 0 math where math a i j a j i theta math and not all math a ij 0 math If one construct the Hermitian matrix A with math A i j a i j math , the equation can be written in a compact way math X t A X theta 0 math where math X begin bmatrix X 0 X 1 vdots X n end bmatrix . math Tangent spaces and singularity Let p be a point on the Hermitian variety H . A line L through p is by definition tangent when it is contains only one point p itself of the variety or lies completely on the variety. One can prove that these lines form a subspace, either a hyperplane of the full space. In the latter case, the point is singular. Category Algebraic varieties Algebra stub ... more details
An math n math by math n math complex or real matrix math A a i,j 1 leq i, j leq n math is said to be anti Hermitian , skew Hermitian , or said to represent a skew adjoint linear operator operator , or to be a skew adjoint matrix mathematics matrix , on the complex or real math n math dimensional space math K n math , if its adjoint operator adjoint is the negative of itself math A A math . Note that the adjoint of an operator depends on the scalar product considered on the math n math dimensional complex or real space math K n math . If math cdot cdot math denotes the scalar product on math K n math , then saying math A math is skew adjoint means that for all math u,v in K n math one has math Au v u Av , . math In the particular case of the canonical scalar products on math K n math , the matrix of a skew adjoint operator satisfies math a ij overline a ji math for all math 1 leq i,j leq n math . Imaginary number s can be thought of as skew adjoint since they are like 1 by 1 matrices , whereas real number s correspond to self adjoint operators. See also Skew Hermitian matrix Category Abstract algebra Category Linear algebra ... more details
Unreferenced date December 2006 The Hermitian hat wavelet is a low oscillation , complex valued wavelet . The real and imaginary parts of this wavelet are defined to be the second and first derivative s of a Gauss ian respectively math Psi t frac 2 sqrt 5 pi frac 1 4 1 t 2 it e frac 1 2 t 2 . math The Fourier transform of this wavelet is math hat Psi omega frac 2 sqrt 5 pi frac 1 4 omega 1 omega e frac 1 2 omega 2 . math The Hermitian hat wavelet satisfies the admissibility criterion. The prefactor math C Psi math in the resolution of the identity of the continuous wavelet transform is math C Psi frac 16 5 sqrt pi . math This wavelet was formulated by Szu in 1997 for the numerical estimation of function derivatives in the presence of noise. The technique used to extract these derivative values exploits only the argument phase of the wavelet and, consequently, the relative weights of the real and imaginary parts are unimportant. Category Continuous wavelets ... more details
In linear algebra , a square matrix with Complex number complex entries is said to be skew Hermitian or antihermitian if its conjugate transpose is equal to its negative. ref harvtxt Horn Johnson 1985 , 4.1.1 harvtxt Meyer 2000 , 3.2 ref That is, the matrix A is skew Hermitian if it satisfies the relation math A dagger A, math where math dagger math denotes the conjugate transpose of a matrix. In component form, this means that math a i,j overline a j,i , math for all i and j , where a sub i , j sub is the i , j th entry of A , and the overline denotes complex conjugate complex conjugation . Skew Hermitian matrices can be understood as the complex versions of real Skew symmetric matrix skew symmetric ... Horn Johnson 1985 , 4.1.2 ref All skew Hermitian var n var var n var matrices form the u var n ... Norm mathematics norm . Example For example, the following matrix is skew Hermitian math begin bmatrix 0 & 2 i 2 i & 0 end bmatrix math Properties The eigenvalues of a skew Hermitian matrix are all purely imaginary. Furthermore, skew Hermitian matrices are normal matrix normal . Hence they are diagonalizable ... Horn Johnson 1985 , 2.5.2, 2.5.4 ref All entries on the main diagonal of a skew Hermitian matrix have ... purely imaginary . ref harvtxt Meyer 2000 , Exercise 3.2.5 ref If A, B are skew Hermitian, then aA bB is skew Hermitian for all real number real scalar mathematics scalars a and b . ref name HJ85S411 harvtxt Horn Johnson 1985 , 4.1.1 ref If A is skew Hermitian, then both iA and iA are Hermitian matrix Hermitian . ref name HJ85S411 If A is skew Hermitian, then A sup k sup is Hermitian if k is an even integer and skew Hermitian if k is an odd integer. An arbitrary square matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew Hermitian matrix B ref name HJ85S412 math ... Hermitian, then e sup A sup is unitary matrix unitary . The space of skew Hermitian matrices forms the Lie algebra u n of the Lie group U n . See also Hermitian matrix Normal matrix Skew symmetric ... more details
distinguish Moore determinant over a finite field In mathematics, the Moore determinant is a determinant defined for Hermitian matrix Hermitian matrices over a quaternion algebra , introduced by harvs txt authorlink E. H. Moore last Moore year 1922 . See also Dieudonn determinant References Citation last1 Moore first1 E. H. author1 link E. H. Moore title On the determinant of an hermitian matrix with quaternionic elements. Definition and elementary properties with applications. doi 10.1090 S0002 9904 1922 03536 7 year 1922 journal Bulletin of the American Mathematical Society issn 0002 9904 volume 28 issue 4 pages 161 162 Category Matrices ... more details
wiktionary functionFunction may refer to Diatonic function , a term in music theory Function E 40 song , a 2012 song by American rapper E 40 featuring YG rapper YG , iAmSu & Problem Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. An organised event such as a party or meeting See also Functionalism disambiguation Function hall Functional disambiguation Functionality in polymer chemistry see Structural unit Functor disambiguation bg bs Funkcija vor ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko id Fungsi it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon nn Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
In mathematics, S function may refer to sigmoid function Schur polynomials In physics, it may refer to Action physics action functional mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally vulnerable to guess and determine attacks, well chosen bitwise transposition mathematics transposition ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
In mathematics, a Hermitian symmetric space is a K hler manifold M which, as a Riemannian manifold , is a Riemannian symmetric space . Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian metric Hermitian . The complex structure is automatically preserved by the isometry group H of the metric, and so M is a homogeneous complex manifold. Some examples are complex vector space s and complex projective space s, with their usual Hermitian metric s and Fubini Study metric s, and the complex unit ball s with suitable metrics so that they become complete and Riemannian symmetric. The compact space compact Hermitian symmetric spaces are projective variety projective varieties , and admit a strictly larger Lie group G of biholomorphism s with respect to which they are homogeneous in fact, they are generalized flag manifold s, i.e., G is semisimple Lie group semisimple and the stabilizer of a point is a parabolic subgroup P of G . Among complex generalized flag manifolds G P , they are characterized as those for which the Nilradical of a Lie algebra nilradical of the Lie algebra of P is abelian. The non compact Hermitian symmetric spaces can be realized as bounded domain mathematics domains in complex vector spaces. Hermitian symmetric spaces are used in the construction of the holomorphic discrete series representation s of semisimple Lie groups. Compact Hermitian symmetric spaces The irreducible compact Hermitian symmetric spaces H K are classified as follows. class wikitable G H K width 10 complex ... In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces ... Hermitian symmetric spaces As with symmetric spaces in general, each compact Hermitian symmetric ... , for which every biholomorphism is an isometry. This realizes &Omega as a Hermitian symmetric ... Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds , World Scientific, 1989 ... more details
In mathematics , especially in the fields of group theory and group representation representation theory of groups , a class function is a function mathematics function f on a group mathematics group G , such that f is constant on the conjugacy class es of G . In other words, f is invariant under the conjugation map on G . Such functions play a basic role in representation theory . The character group theory character of a linear representation of G over a field mathematics field K is always a class function with values in K . The class functions form the center algebra center of the group ring K G . Here a class function f is identified with the element math sum g in G f g g math . Inner products The set of class functions of a finite group G with values in a field K form a K vector space . If the characteristic algebra characteristic of the field does not divide the order of G , then there is an inner product defined on this space defined by math langle phi , psi rangle frac 1 G sum g in G phi g psi g 1 math where G denotes the order of G . The set of Character theory irreducible characters of G forms an orthogonal basis , and if K is a splitting field for G , for instance if K is algebraically closed , then the irreducible characters form an orthonormal basis . In the case of a compact group and K C the field of complex number s, the notion of Haar measure allows one to replace the finite sum above with an integral math langle phi, psi rangle int G phi t psi t 1 , dt. math When restricted to real linear combinations of characters, the inner product is a degenerate form non degenerate Hermitian form Hermitian bilinear form . References Jean Pierre Serre , Linear representations of finite groups , Graduate Texts in Mathematics 42 , Springer Verlag, Berlin, 1977. Category Group theory fr Fonction centrale ja zh ... more details
Merge Functional calculus date July 2011 In mathematics , a matrix function is a function mathematics function which maps a matrix mathematics matrix to another matrix. Extending scalar functions to matrix functions There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function are defined may differ. Power series If the real function math f math has the Taylor expansion math f x f 0 f 0 cdot x f 0 cdot frac x 2 2 cdots math then a matrix function can be defined by substituting math x math by a matrix the powers become matrix power s, the additions become matrix sums and the multiplications become scaling operations. If the real series converges for math x r math , then the corresponding matrix series will converge for matrix argument A if math A r math for some matrix norm math cdot math which satisfies math AB leq A cdot B math . Jordan decomposition main Jordan normal form If the matrix A is diagonalizable matrix diagonalizable , then we can find a matrix P and a diagonal matrix D such that math A P cdot D cdot P 1 math . Applying the power series definition to this decomposition, we find that math f A math ... f lambda 0 end bmatrix . math This definition can be used to extend the domain of the matrix function ... as a sum of a diagonalizable and a nilpotent part. Hermitian matrices A Hermitian matrix has all real .... Cauchy s integral formula states that for any analytic function f defined on a set D , it holds ... functions can be extended to matrix functions of Hermitian matrix Hermitian matrices ref cite ... Springer volume 169 ref . Operator monotone If function math f math is called operator monotone ... function monotone function in the scalar case. Operator concave convex A function math f math is called ... is analogous to a concave function concave scalar function . An operator convex function can be defined ... more details
In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi eigenvalue algorithm Jacobi iteration method . The Jacobi eigenvalue algorithm Jacobi iteration method is also explained in Introduction to Linear Algebra by harvtxt Strang 1993 . Derivation The complex unitary matrices unitary Rotation matrix rotation matrices R sub pq sub can be used for Jacobi eigenvalue algorithm Jacobi iteration of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously. Similar to the Givens rotation Givens rotation matrices , R sub pq sub are defined as math begin align R pq m,n & delta m,n & qquad m,n ne p,q, 10pt R pq p,p & frac 1 sqrt 2 e i theta , 10pt R pq q,p & frac 1 sqrt 2 e i theta , 10pt R pq p,q & frac 1 sqrt 2 e i theta , 10pt R pq q,q & frac 1 sqrt 2 e i theta end align math Each rotation matrix, R sub pq sub , will modify only the p th and q th rows or columns of a matrix M if it is applied from left or right, respectively math begin align R pq M m,n & begin cases M m,n & m ne p,q 8pt frac 1 sqrt 2 M p,n e i theta M q,n e i theta & m p 8pt frac 1 sqrt 2 M p,n e i theta M q,n e i theta & m q end cases 8pt MR pq dagger m,n & begin cases M m,n & n ne p,q frac 1 sqrt 2 M m,p e i theta M m,q e i theta & n p 8pt frac 1 sqrt 2 M m,p e i theta M m,q e i theta & n q end cases end align math A Hermitian matrix , H is defined by the conjugate transpose symmetry property math H dagger H Leftrightarrow H i,j H j,i math By definition, the complex conjugate of a complex unitary matrices unitary Rotation matrix rotation matrix, R is its inverse and also a complex unitary matrices unitary Rotation matrix rotation matrix math begin align R dagger pq & R 1 pq 6pt Rightarrow R dagger dagger pq ... transformation math T math of a Hermitian matrices Hermitian matrix H is also a Hermitian matrices Hermitian matrix similar to H math begin align T & equiv R pq H R dagger pq , & & 6pt T dagger & R pq ... more details
form an important class of function mathematics functions used in complex analysis . On a K hler manifold , plurisubharmonic functions form a subset of the subharmonic function s. However, unlike subharmonic ... in full generality on complex space s. Formal definition A function mathematics function math f colon ... if it is semi continuous function upper semi continuous , and for every complex number complex line math a b z mid z in mathbb C subset mathbb C n math with math a, b in mathbb C n math the function math z mapsto f a bz math is a subharmonic function on the set math z in mathbb C mid a b z in G ... space math X math as follows. An semi continuity upper semi continuous function math f colon X ... varphi colon Delta to X math the function math f circ varphi colon Delta to mathbb R cup infty math is subharmonic function subharmonic , where math Delta subset mathbb C math denotes the unit disk ... , then math f math is plurisubharmonic, if and only if the hermitian matrix math L f lambda ij math ... is positive semidefinite. Equivalently, a math C 2 math function f is plurisubharmonic if and only ... f math is a plurisubharmonic function and math c 0 math a positive real number, then the function ... functions, then the sum math f 1 f 2 math is a plurisubharmonic function. Plurisubharmonicity is a local property , i.e. a function is plurisubharmonic if and only if it is plurisubharmonic in a neighborhood ... increasing, convex function then math phi circ f math is plurisubharmonic. If math f 1 math and math f 2 math are plurisubharmonic functions, then the function math f x max f 1 x ,f 2 x math ... function can be obtained as a limit of monotonically decreasing sequence of smooth plurisubharmonic ... function subharmonic , for any K hler manifold K hler metric . Therefore, plurisubharmonic ..., Tohoku Math. J. 49 1942 , 15 52. ref A continuous function math f M mapsto Bbb R math is called ... function f is called strongly plurisubharmonic if the form math sqrt 1 partial bar ... more details
In mathematics, omega function or function may refer to Pearson Cunningham function Lambert W function Wright Omega function Mathematical disambiguation ... more details
Barnes G function , related to the Gamma function Meijer G function , a generalization of the hypergeometric function Siegel G function , a class of functions in transcendence theory mathdab ... more details
In mathematics, by sigma function one can mean one of the following The Divisor function sum of divisors function sub a sub n , an arithmetic function Weierstrass sigma function , related to elliptic functions Kronecker s sigma function Rado s sigma function, see busy beaver See also sigmoid function . mathdab de Teilersumme fr Fonction sigma ... more details
Function test or functional test can refer to Thyroid function tests Liver function tests Lung function test Functional testing disambig ... more details
wiktionary Comparison function may refer to Nachbin s theorem a theorem in the area of complex analysis Class kappa function a special function used control theory Class kappa ell function a special function used control theory . Disambig ... more details
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