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- In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } and is closely related to Bessel functions. (wikipedia.org)
- The Weber function (also known as Lommel-Weber function), introduced by H. F. Weber (1879), is a closely related function defined by E ν ( z ) = 1 π ∫ 0 π sin ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta } and is closely related to Bessel functions of the second kind. (wikipedia.org)
- The functions are notated with brackets, as ⟨ x − a ⟩ n {\displaystyle \langle x-a\rangle ^{n}} where n is an integer. (wikipedia.org)
- x − a ⟩ 1 {\displaystyle \langle x-a\rangle ^{1}} is also called the Ramp function. (wikipedia.org)
- Sudan published the lesser-known Sudan function, then shortly afterwards and independently, in 1928, Ackermann published his function φ {\displaystyle \varphi } (the Greek letter phi). (wikipedia.org)
- For example, the function y ( x ) = 4 {\displaystyle y(x)=4} is a constant function because the value of y ( x ) {\displaystyle y(x)} is 4 regardless of the input value x {\displaystyle x} (see image). (wikipedia.org)
- On the other hand, the polynomial f ( x ) = 0 {\displaystyle f(x)=0} is the identically zero function. (wikipedia.org)
- Every set X {\displaystyle X} is canonically isomorphic to the function set X 1 {\displaystyle X^{1}} , or hom set h o m ( 1 , X ) {\displaystyle hom(1,X)} in the category of sets, where 1 is the one-point set. (wikipedia.org)
- Lamb and Batchelor define the stream function ψ ( x , y , t ) {\displaystyle \psi (x,y,t)} - in the point P {\displaystyle P} with two-dimensional coordinates ( x , y ) {\displaystyle (x,y)} and as a function of time t {\displaystyle t} - for an incompressible flow by: ψ = ∫ A P ( u d y − v d x ) . {\displaystyle \psi =\int _{A}^{P}\left(u\,{\text{d}}y-v\,{\text{d}}x\right). (wikipedia.org)
- So the stream function ψ {\displaystyle \psi } is the volume flux through the curve A P {\displaystyle AP} , that is: the integral of the dot product of the flow velocity vector ( u , v ) {\displaystyle (u,v)} and the normal ( + d y , − d x ) {\displaystyle (+{\text{d}}y,-{\text{d}}x)} to the curve element ( d x , d y ) . {\displaystyle ({\text{d}}x,{\text{d}}y). (wikipedia.org)
- An infinitesimal shift δ P = ( δ x , δ y ) {\displaystyle \delta P=(\delta x,\delta y)} of the position P {\displaystyle P} results in a stream function shift: δ ψ = u δ y − v δ x , {\displaystyle \delta \psi =u\,\delta y-v\,\delta x,} which is an exact differential provided ∂ u ∂ x + ∂ v ∂ y = 0. (wikipedia.org)
- Similarly, if we set c equal to e in ƒ5, then we get that f 5 ( x ) = log e x = ln x {\displaystyle f_{5}(x)=\log _{e}x=\ln x} (that is, the natural logarithm) is a transcendental function. (wikipedia.org)
- As a side note, the Green's function as used in physics is usually defined with the opposite sign, instead, that is, L G ( x , s ) = δ ( x − s ) . {\displaystyle LG(x,s)=\delta (x-s). (wikipedia.org)
- If the operator is translation invariant, that is, when L has constant coefficients with respect to x, then the Green's function can be taken to be a convolution operator, that is, G ( x , s ) = G ( x − s ) . {\displaystyle G(x,s)=G(x-s). (wikipedia.org)
- Given a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} , then for any random variable X {\displaystyle X} and any distortion function g {\displaystyle g} we can define a new probability measure Q {\displaystyle \mathbb {Q} } such that for any A ∈ F {\displaystyle A\in {\mathcal {F}}} it follows that Q ( A ) = g ( P ( X ∈ A ) ) . {\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).} Balbás, A. (wikipedia.org)
- It can be defined in terms of the Riemann-Siegel theta-function and the Riemann zeta-function by Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . {\displaystyle Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right). (wikipedia.org)
- For instance, we have, even without the Riemann hypothesis, the Omega theorem that Z ( t ) = Ω ( exp ( 3 4 log t log log t ) ) , {\displaystyle Z(t)=\Omega \left(\exp \left({\frac {3}{4}}{\sqrt {\frac {\log t}{\log \log t}}}\right)\right),} where the notation means that Z ( t ) {\displaystyle Z(t)} divided by the function within the Ω does not tend to zero with increasing t. (wikipedia.org)
- The work function W for a given surface is defined by the difference W = − e ϕ − E F , {\displaystyle W=-e\phi -E_{\rm {F}},} where −e is the charge of an electron, ϕ is the electrostatic potential in the vacuum nearby the surface, and EF is the Fermi level (electrochemical potential of electrons) inside the material. (wikipedia.org)
- The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect( f ), ∫ − ∞ ∞ sinc ( t ) e i 2 π f t d t = rect ( f ) , {\displaystyle \int _{-\infty }^{\infty }\operatorname {sinc} (t)\,e^{i2\pi ft}\,dt=\operatorname {rect} (f)~,} where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. (wikipedia.org)
- In mathematics, the continuum function is κ ↦ 2 κ {\displaystyle \kappa \mapsto 2^{\kappa }} , i.e. raising 2 to the power of κ using cardinal exponentiation. (wikipedia.org)
- In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = I ( y ^ ≠ y ) , {\displaystyle L({\hat {y}},y)=I({\hat {y}}\neq y),\,} where I {\displaystyle I} is the indicator notation. (wikipedia.org)
- Here the decision rule depends on the outcome of X. The risk function is given by: R ( θ , δ ) = E θ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x). (wikipedia.org)
- In mathematics, an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 {\displaystyle R(x_{1},\ldots ,x_{n})=0} , where R {\displaystyle R} is a function of several variables (often a polynomial). (wikipedia.org)
- This implicit equation defines f {\displaystyle f} as a function of x {\displaystyle x} only if − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and one considers only non-negative (or non-positive) values for the values of the function. (wikipedia.org)
- If g is a function of x, then the inverse function of g, called g −1, is the function giving a solution of the equation y = g ( x ) {\displaystyle y=g(x)} for x in terms of y. (wikipedia.org)
- Here g − 1 {\displaystyle g^{-1}} is an implicit function, and more specifically an inverse function. (wikipedia.org)
- While for some functions g, g − 1 {\displaystyle g^{-1}} could be written out explicitly - for instance, if g ( x ) = 2 x , {\displaystyle g(x)=2x,} then g − 1 ( y ) = 1 2 y {\displaystyle g^{-1}(y)={\tfrac {1}{2}}y} - this is often not the case (as in the product log example below). (wikipedia.org)
- For example, an algebraic function in one variable x gives a solution for y of an equation a n ( x ) y n + a n − 1 ( x ) y n − 1 + ⋯ + a 0 ( x ) = 0 {\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0} where the coefficients ai(x) are polynomial functions of x. (wikipedia.org)
- But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as y = f ( x ) , {\displaystyle y=f(x),} where f is the multi-valued implicit function. (wikipedia.org)
- The Airy functions are orthogonal in the sense that ∫ − ∞ ∞ A i ( t + x ) A i ( t + y ) d t = δ ( x − y ) {\displaystyle \int _{-\infty }^{\infty }\mathrm {Ai} (t+x)\mathrm {Ai} (t+y)dt=\delta (x-y)} again using an improper Riemann integral. (wikipedia.org)
- A proper rational function is a rational function in which the degree of P ( x ) {\displaystyle P(x)} is no greater than the degree of Q ( x ) {\displaystyle Q(x)} and both are real polynomials. (wikipedia.org)
- A function that cannot be written in this form, such as f ( x ) = sin ( x ) {\displaystyle f(x)=\sin(x)} , is not a rational function. (wikipedia.org)
- The rational function f ( x ) = x x {\displaystyle f(x)={\frac {x}{x}}} is equal to 1 for all x except 0, where there is a removable singularity. (wikipedia.org)
- Remark: By the Baire Category theorem the set of points of discontinuity of such function F σ {\displaystyle F_{\sigma }} is either meager or else has nonempty interior. (wikipedia.org)
- Let f : D → R {\displaystyle f\colon D\rightarrow \mathbf {R} \quad } be a function defined on a subset D {\displaystyle D} of the set R {\displaystyle \mathbf {R} } of real numbers. (wikipedia.org)
- The equation is used to specify a function g and a parameter λ by the relation g ( x ) = 1 − λ g ( g ( λ x ) ) {\displaystyle g(x)={\frac {1}{-\lambda }}g(g(\lambda x))} with the initial conditions g(0) = 1, g′(0) = 0, and g′′(0) (wikipedia.org)
- In vector calculus, an invex function is a differentiable function ƒ from Rn to R for which there exists a vector valued function g such that f ( x ) − f ( u ) ≧ g ( x , u ) ⋅ ∇ f ( u ) , {\displaystyle f(x)-f(u)\geqq g(x,u)\cdot \nabla f(u),\,} for all x and u. (wikipedia.org)
- In number theory, the Carmichael function of a positive integer n, denoted λ ( n ) {\displaystyle \lambda (n)} , is defined as the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} for every integer a that is coprime to n. (wikipedia.org)
- Carmichael's theorem states that if a is coprime to n, then a λ ( n ) ≡ 1 ( mod n ) , {\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}},} where λ {\displaystyle \lambda } is the Carmichael function defined above. (wikipedia.org)
- The logarithmic or continuously compounded return, sometimes called force of interest, is a function of time defined as follows: δ t = a ′ ( t ) a ( t ) {\displaystyle \delta _{t}={\frac {a'(t)}{a(t)}}\,} which is the rate of change with time of the natural logarithm of the accumulation function. (wikipedia.org)
- For possibly distinct random variables X(s) and Y(t) at different points s and t of some space, the correlation function is C ( s , t ) = corr ( X ( s ) , Y ( t ) ) , {\displaystyle C(s,t)=\operatorname {corr} (X(s),Y(t)),} where corr {\displaystyle \operatorname {corr} } is described in the article on correlation. (wikipedia.org)
- Then there is only one interval, A 0 = R . {\displaystyle A_{0}=\mathbb {R} .} The sign function sgn ( x ) {\displaystyle \operatorname {sgn}(x)} , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. (wikipedia.org)
- The function looks like ϕ ( v i ) = U ( v i ) {\displaystyle \phi (v_{i})=U(v_{i})} , where U {\displaystyle U} is the Heaviside step function. (wikipedia.org)
- Such a function would be of the form ϕ ( v i ) = μ v i {\displaystyle \phi (v_{i})=\mu v_{i}} , where μ {\displaystyle \mu } is the slope. (wikipedia.org)
- Mathematically, this looks like ϕ ( v i ) = U ( v i ) tanh ( v i ) {\displaystyle \phi (v_{i})=U(v_{i})\tanh(v_{i})} , where the hyperbolic tangent function can be replaced by any sigmoid function. (wikipedia.org)
- Two forms of this function are commonly used: ϕ ( v i ) = tanh ( v i ) {\displaystyle \phi (v_{i})=\tanh(v_{i})} whose range is normalized from -1 to 1, and ϕ ( v i ) = ( 1 + exp ( − v i ) ) − 1 {\displaystyle \phi (v_{i})=(1+\exp(-v_{i}))^{-1}} is vertically translated to normalize from 0 to 1. (wikipedia.org)
- 2}+a^{2})^{-1/2}} where c i {\displaystyle c_{i}} is the vector representing the function center and a {\displaystyle a} and σ {\displaystyle \sigma } are parameters affecting the spread of the radius. (wikipedia.org)
- Then given the function f : X → R ∪ { + ∞ } {\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}} , we can define the primal problem by inf x ∈ X f ( x ) . {\displaystyle \inf _{x\in X}f(x). (wikipedia.org)
- If there are constraint conditions, these can be built into the function f {\displaystyle f} by letting f ← f + I c o n s t r a i n t s {\displaystyle f\leftarrow f+I_{\mathrm {constraints} }} where I {\displaystyle I} is the indicator function. (wikipedia.org)
- Given a primal problem (minimize f(x)) and a related perturbation function (F(x,y)) then the Lagrangian L : X × Y ∗ → R ∪ { + ∞ } {\displaystyle L:X\times Y^{*}\to \mathbb {R} \cup \{+\infty \}} is the negative conjugate of F with respect to y (i.e. the concave conjugate). (wikipedia.org)
- In particular if the primal objective is f ( x ) + g ( T x ) {\displaystyle f(x)+g(Tx)} then the perturbation function is given by F ( x , y ) = f ( x ) + g ( T x − y ) {\displaystyle F(x,y)=f(x)+g(Tx-y)} , which is the traditional definition of Fenchel duality. (wikipedia.org)
- The set of all real analytic functions on a given set D {\displaystyle D} is often denoted by C ω ( D ) {\displaystyle C^{\,\omega }(D)} . A function f {\displaystyle f} defined on some subset of the real line is said to be real analytic at a point x {\displaystyle x} if there is a neighborhood D {\displaystyle D} of x {\displaystyle x} on which f {\displaystyle f} is real analytic. (wikipedia.org)
- The set A ∞ ( Ω ) {\displaystyle \scriptstyle A_{\infty }(\Omega )} of all bounded analytic functions with the supremum norm is a Banach space. (wikipedia.org)
- The function Ψ : R → R given by Ψ ( x ) = { exp ( − 1 1 − x 2 ) for x ∈ ( − 1 , 1 ) 0 otherwise {\displaystyle \Psi (x)={\begin{cases}\exp \left(-{\frac {1}{1-x^{2}}}\right)&{\mbox{ for }}x\in (-1,1)\\0&{\mbox{ otherwise}}\end{cases}}} is an example of a bump function in one dimension. (wikipedia.org)
- One considers a compact neighborhood V of K contained in U, so K ⊂ Vo ⊂ V ⊂ U. The characteristic function χ V {\displaystyle \chi _{V}} of V will be equal to 1 on V and 0 outside of V, so in particular, it will be 1 on K and 0 outside of U. This function is not smooth however. (wikipedia.org)
- Such a mollifier can be obtained, for example, by taking the bump function Φ {\displaystyle \Phi } from the previous section and performing appropriate scalings. (wikipedia.org)
- In calculus, a function f {\displaystyle f} defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. (wikipedia.org)
- if f {\displaystyle f} is a monotonic function defined on an interval [ a , b ] {\displaystyle \left[a,b\right]} , then f {\displaystyle f} is Riemann integrable. (wikipedia.org)
- z ) {\displaystyle {\frac {d^{n}}{dz^{n}}}\ {}_{2}F_{1}(a,b;c;z)={\frac {(a)_{n}(b)_{n}}{(c)_{n}}}\ {}_{2}F_{1}(a+n,b+n;c+n;z)} Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. (wikipedia.org)
- b − 1 z ) {\displaystyle M(a,c,z)=\lim _{b\to \infty }{}_{2}F_{1}(a,b;c;b^{-1}z)} so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. (wikipedia.org)
- Formally speaking, a partial function f : N k → N {\displaystyle f:\mathbb {N} ^{k}\rightarrow \mathbb {N} } can be calculated if and only if there exists a computer program with the following properties: If f ( x ) {\displaystyle f(\mathbf {x} )} is defined, then the program will terminate on the input x {\displaystyle \mathbf {x} } with the value f ( x ) {\displaystyle f(\mathbf {x} )} stored in the computer memory. (wikipedia.org)
- Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b {\displaystyle f(z+\omega )=f(z)+az+b\ } are also called quasiperiodic. (wikipedia.org)
- An example of this is the Weierstrass zeta function, where ζ ( z + ω , Λ ) = ζ ( z , Λ ) + η ( ω , Λ ) {\displaystyle \zeta (z+\omega ,\Lambda )=\zeta (z,\Lambda )+\eta (\omega ,\Lambda )\ } for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function. (wikipedia.org)
- A useful example is the function: f ( z ) = sin ( A z ) + sin ( B z ) {\displaystyle f(z)=\sin(Az)+\sin(Bz)} If the ratio A/B is rational, this will have a true period, but if A/B is irrational there is no true period, but a succession of increasingly accurate "almost" periods. (wikipedia.org)
- In mathematics, the family of Debye functions is defined by D n ( x ) = n x n ∫ 0 x t n e t − 1 d t . {\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt. (wikipedia.org)
- quad [\Re \,n> The derivative obeys the relation x D n ′ ( x ) = n ( B ( x ) − D n ( x ) ) , {\displaystyle xD_{n}^{\prime }(x)=n\left(B(x)-D_{n}(x)\right),} where B ( x ) = x / ( e x − 1 ) {\displaystyle B(x)=x/(e^{x}-1)} is the Bernoulli function. (wikipedia.org)
- It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function, which can be demonstrated using the identity sgn ( x ) = 2 H ( x ) − 1 {\displaystyle \operatorname {sgn}(x)=2H(x)-1\,} (where H(x) is the Heaviside step function using the standard H(0) = 1/2 formalism). (wikipedia.org)
- Then, for z ≠ 0, sgn ( z ) = e i arg z , {\displaystyle \operatorname {sgn}(z)=e^{i\arg z}\,,} where arg is the complex argument function. (wikipedia.org)
- In PCF theory the cardinal function p p κ ( λ ) {\displaystyle pp_{\kappa }(\lambda )} is used. (wikipedia.org)
- The Patterson function is also equivalent to the electron density convolved with its inverse: P ( u → ) = ρ ( r → ) ∗ ρ ( − r → ) . {\displaystyle P\left({\vec {u}}\right)=\rho \left({\vec {r}}\right)*\rho \left(-{\vec {r}}\right). (wikipedia.org)
- The cumulative distribution function of T is the function F ( t ) = P ( T ≤ t ) , {\displaystyle F(t)=\operatorname {P} (T\leq t),} where the right-hand side represents the probability that the random variable T is less than or equal to t. (wikipedia.org)
- A realvalued coercive function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is in particular norm-coercive. (wikipedia.org)
- For instance the identity function on R {\displaystyle \mathbb {R} } is norm-coercive but not coercive. (wikipedia.org)
- The function defined by f ( x ) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 {\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}} is flat at x = 0. (wikipedia.org)
- and q ∈ N {\displaystyle q\in \mathbb {N} } , the function is well-defined. (wikipedia.org)
- according to Bézout's identity for some a , b ∈ Z . {\displaystyle a,\;b\in \mathbb {Z} .} Thomae's function has also several interesting, if not astonishing, properties. (wikipedia.org)
- The function's integral is equal to 0 {\displaystyle 0} over any set because the function is equal to zero almost everywhere. (wikipedia.org)
- In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} is a convex set. (wikipedia.org)
- The Lambert series for the Liouville function is ∑ n = 1 ∞ λ ( n ) q n 1 − q n = ∑ n = 1 ∞ q n 2 = 1 2 ( ϑ 3 ( q ) − 1 ) , {\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}={\frac {1}{2}}\left(\vartheta _{3}(q)-1\right),} where ϑ 3 ( q ) {\displaystyle \vartheta _{3}(q)} is the Jacobi theta function. (wikipedia.org)

- This article is about function composition in mathematics. (wikipedia.org)
- In mathematics , function composition is the pointwise application of one function to the result of another to produce a third function. (wikipedia.org)
- In mathematics , the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x . It is denoted by π ( x ) (unrelated to the number π ). (wikipedia.org)
- In pure mathematics there's not too many function studied more than the Riemann zeta function. (scienceblogs.com)
- In various areas of mathematics, the zero set of a function is the set of all its zeros. (wikipedia.org)
- In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. (wikipedia.org)
- Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. (wikipedia.org)
- This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context. (wikipedia.org)
- In mathematics, a constant function is a function whose (output) value is the same for every input value. (wikipedia.org)
- In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. (wikipedia.org)
- In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions. (wikipedia.org)
- In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the argument is one-half. (wikipedia.org)
- In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by sinc(x), has two slightly different definitions. (wikipedia.org)
- In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. (wikipedia.org)
- In mathematics, S-function may refer to: sigmoid function Schur polynomials Slesarenko function(smooth function, S-function) A function in the Laplace transformed 's-domain' In physics, it may refer to: action functional In MATLAB, it may refer to: A type of dynamically linked subroutine for Simulink. (wikipedia.org)
- In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous. (wikipedia.org)
- In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. (wikipedia.org)
- In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. (wikipedia.org)
- In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. (wikipedia.org)
- In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties: f is continuous on [a, b]. (**) there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere. (wikipedia.org)
- In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments. (wikipedia.org)
- Function value may refer to: In mathematics, the value of a function when applied to an argument. (wikipedia.org)
- In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. (wikipedia.org)
- In mathematics, theta functions are special functions of several complex variables. (wikipedia.org)
- In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. (wikipedia.org)
- In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. (wikipedia.org)
- In mathematics, an analytic function is a function that is locally given by a convergent power series. (wikipedia.org)
- In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the Green's function, in the Laplace transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourier series of an integrable function. (wikipedia.org)
- When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics. (wikipedia.org)
- In mathematics, a bump function is a function f : Rn → R on a Euclidean space Rn which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. (wikipedia.org)
- In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. (wikipedia.org)
- In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. (wikipedia.org)
- In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. (wikipedia.org)
- In mathematics - specifically, in large deviations theory - a rate function is a function used to quantify the probabilities of rare events. (wikipedia.org)
- The need for function approximations arises in many branches[examples needed] of applied mathematics, and computer science in particular[why? (wikipedia.org)
- In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. (wikipedia.org)
- In mathematics, a multivalued function is a left-total relation (that is, every input is associated with at least one output). (wikipedia.org)
- In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. (wikipedia.org)
- In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. (wikipedia.org)
- In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. (wikipedia.org)
- In mathematics, especially real analysis, a flat function is a smooth function ƒ : ℝ → ℝ all of whose derivatives vanish at a given point x0 ∈ ℝ. (wikipedia.org)
- In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers (1950, 1951, 1953, 1956) that generalize analytic functions and satisfy a weakened form of the Cauchy-Riemann equations. (wikipedia.org)

- The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. (wikipedia.org)
- It is also called the Riemann-Siegel Z-function, the Riemann-Siegel zeta-function, the Hardy function, the Hardy Z-function and the Hardy zeta-function. (wikipedia.org)
- It follows from the functional equation of the Riemann zeta-function that the Z-function is real for real values of t. (wikipedia.org)
- It follows from the fact that the Riemann-Siegel theta-function and the Riemann zeta-function are both holomorphic in the critical strip, where the imaginary part of t is between -1/2 and 1/2, that the Z-function is holomorphic in the critical strip also. (wikipedia.org)
- Moreover, the real zeros of Z(t) are precisely the zeros of the zeta-function along the critical line, and complex zeros in the Z-function critical strip correspond to zeros off the critical line of the Riemann zeta-function in its critical strip. (wikipedia.org)
- Calculation of the value of Z(t) for real t, and hence of the zeta-function along the critical line, is greatly expedited by the Riemann-Siegel formula. (wikipedia.org)
- Ramachandra, K. Lectures on the mean-value and Omega-theorems for the Riemann Zeta-function. (wikipedia.org)
- The Theory of the Riemann Zeta-Function (second revised ed. (wikipedia.org)
- In particular, the discontinuity set has measure zero, from which it follows that a regulated function has a well-defined Riemann integral. (wikipedia.org)
- Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential equation it satisfies. (wikipedia.org)
- A suitable interval may be found through use of a branch cut, a kind of curve that connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer. (wikipedia.org)
- Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function, the Riemann function, or the Stars over Babylon (John Horton Conway's name). (wikipedia.org)
- The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. (wikipedia.org)
- Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. (wikipedia.org)

- k\in \mathbb {Z} \}} These are all examples of multivalued functions that come about from non-injective functions. (wikipedia.org)

- To satisfy the mathematical definition of a function , a production function is customarily assumed to specify the maximum output obtainable from a given set of inputs. (wikipedia.org)
- The production function is central to the marginalist focus of neoclassical economics, its definition of efficiency as allocative efficiency, its analysis of how market prices can govern the achievement of allocative efficiency in a decentralized economy, and an analysis of the distribution of income, which attributes factor income to the marginal product of factor input. (wikipedia.org)
- Most textbooks, especially ones not aimed at college math majors, give a definition of "function" that seems quite intuitive. (scienceblogs.com)
- Hamiltonian function , also called Hamiltonian , mathematical definition introduced in 1835 by Sir William Rowan Hamilton to express the rate of change in time of the condition of a dynamic physical system-one regarded as a set of moving particles. (britannica.com)
- See Definition of indirect-function . (gnu.org)
- The sign of the stream function depends on the definition used. (wikipedia.org)
- Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. (wikipedia.org)
- This definition does not significantly change any of the properties of the Green's function. (wikipedia.org)
- Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1. (wikipedia.org)
- In computer programming, an anonymous function (function literal, lambda abstraction, or lambda expression) is a function definition that is not bound to an identifier. (wikipedia.org)
- In addition, this article discusses the definition for the more general case of functions between two metric spaces. (wikipedia.org)
- A rigorous definition of continuity of real valued function of a real variable is usually given in a first course in calculus in terms of the idea of a limit. (wikipedia.org)
- The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. (wikipedia.org)
- There is very little systematic theory of symmetric non-polynomial functions of n variables, so this sense is little-used, except as a general definition. (wikipedia.org)
- While a function definition specifies how the function does what it does (the "implementation"), a function prototype merely specifies its interface, i.e. what data types go in and come out of it. (wikipedia.org)
- The integer part function is not a step function according to the definition of this article, since it has an infinite number of intervals. (wikipedia.org)
- This is necessary due to the C/C++ separate compilation + linkage model, specifically because the definition (body) of the function must be duplicated in all translation units where it is used, to allow inlining during compiling, which, if the function has external linkage, causes a collision during linking (it violates uniqueness of external symbols). (wikipedia.org)
- without inline qualifier or storage class), the translation unit must contain a definition (no matter whether unqualified, inline, or extern inline) and an externally visible function will be emitted for it. (wikipedia.org)
- Any Taylor series for this function converges not only for x close enough to x0 (as in the definition) but for all values of x (real or complex). (wikipedia.org)
- I.e., if there are any free variables in the function definition. (wikipedia.org)
- One is based on a very simple and intuitive definition a generalized function given by Yu. (wikipedia.org)
- Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. (wikipedia.org)
- Before the precise definition of computable function, mathematicians often used the informal term effectively calculable. (wikipedia.org)
- In agreement with this definition, the remainder of this article presumes that computable functions take finitely many natural numbers as arguments and produce a value which is a single natural number. (wikipedia.org)
- Hash functions are used in hash tables, to quickly locate a data record (e.g., a dictionary definition) given its search key (the headword). (wikipedia.org)
- A third use of null functions is as the definition of a program feature that, if created inadvertently, is almost always deleterious. (wikipedia.org)

- To answer the question, we should start off with Euler's identity, which relates the exponential function with the regular trig functions. (scienceblogs.com)
- Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. (wikipedia.org)
- The natural logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. (wikipedia.org)
- The exponential function is an E-function, in its case cn=1 for all of the n. (wikipedia.org)
- The exponential function is analytic. (wikipedia.org)
- If the time between observed air conditioner failures is approximated using the exponential function, then the exponential curve gives the probability density function, f(t), for air conditioner failure times. (wikipedia.org)

- Derivatives of compositions involving differentiable functions can be found using the chain rule . (wikipedia.org)
- A sigmoid function is a bounded , differentiable, real function that is defined for all real input values and has a non-negative derivative at each point. (wikipedia.org)
- Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. (wikipedia.org)
- A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. (wikipedia.org)
- Most special functions (at least in some range of the complex plane): hypergeometric functions Bessel functions gamma function Typical examples of functions that are not analytic are: The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. (wikipedia.org)
- Any analytic function is smooth, that is, infinitely differentiable. (wikipedia.org)
- in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. (wikipedia.org)
- For the complex numbers, the converse does hold, and in fact any function differentiable once on an open set is analytic on that set (see "analyticity and differentiability" below). (wikipedia.org)
- Volterra's function is differentiable everywhere just as f (as defined above) is. (wikipedia.org)
- The signum function is differentiable with derivative 0 everywhere except at 0. (wikipedia.org)

- Examples of the application of the logistic S-curve to the response of crop yield (barley) to both the soil salinity and depth to water table in the soil are shown in logistic function#In agriculture: modeling crop response . (wikipedia.org)
- In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. (wikipedia.org)
- Simple examples of algebraic functions are the rational functions and the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions. (wikipedia.org)
- In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine, cosine, and exponential functions. (wikipedia.org)
- Typical examples of analytic functions are: All elementary functions: All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. (wikipedia.org)
- Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers. (wikipedia.org)
- The graphs below show examples of hypothetical survival functions. (wikipedia.org)

- In summary, to ensure efficient inference of the regression parameters and the regression function, the heteroscedasticity must be accounted for. (wikipedia.org)
- It's a function that takes two parameters: a more traditional variable z, and a number (lowercase Greek letter nu) denoting the particular subspecies of function g. (scienceblogs.com)
- The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. (wikipedia.org)
- is used as input to all radial basis functions, each with different parameters. (wikipedia.org)
- Type declarations allow functions to require that parameters are of a certain type at call time. (php.net)
- Creates an anonymous function from the parameters passed, and returns a unique name for it. (php.net)
- Using the first array of anonymous functions parameters: 2.3445, M_PI some trig: -1.6291725057799 a hypotenuse: 3.9199852871011 b*a^2 = 4.8103313314525 min(b^2+a, a^2,b) = 8.6382729035898 ln(a)/b = 0.27122299212594 Using the second array of anonymous functions ** 'Twas the night' and 'Twas brilling and the slithy toves' ** Look the same to me! (php.net)
- Investigations of the normal sleep-wake cycle showed that immune parameters like numbers of undifferentiated naïve T cells and the production of pro-inflammatory cytokines exhibit peaks during early nocturnal sleep whereas circulating numbers of immune cells with immediate effector functions, like cytotoxic natural killer cells, as well as anti-inflammatory cytokine activity peak during daytime wakefulness. (springer.com)
- Clapp and Hornberger (1978) derived average values for the parameters of a power-function water retention curve, sorptivity and saturated hydraulic conductivity for different texture classes. (wikipedia.org)
- Thermionic emission In thermionic electron guns, the work function and temperature of the hot cathode are critical parameters in determining the amount of current that can be emitted. (wikipedia.org)
- Functions having more than one parameter may be strict or non-strict in each parameter independently, as well as jointly strict in several parameters simultaneously. (wikipedia.org)
- As an example, the if-then-else expression of many programming languages may be thought of as a function of three parameters. (wikipedia.org)
- In computer programming, a function prototype or function interface is a declaration of a function that specifies the function's name and type signature (arity, data types of parameters, and return type), but omits the function body. (wikipedia.org)
- The function signature determines the number of parameters and their types. (wikipedia.org)
- In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model given data. (wikipedia.org)
- Define a generic function with two parameters object-1 and object-2. (wikipedia.org)

- The function f is equal to the derivative of F almost everywhere , and it is called the probability density function of the distribution of X . (wikipedia.org)
- In general, a sigmoid function is monotonic , and has a first derivative which is bell shaped. (wikipedia.org)
- In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. (wikipedia.org)
- Because a constant function does not change, its derivative is 0. (wikipedia.org)
- A weak derivative is defined that allows singular functions to be used in partial differential equations, etc. (wikipedia.org)
- A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to z exists in the neighbourhood of each and every point in that set, where z is the complex conjugate. (wikipedia.org)
- In measure-theoretic probability theory, the density function is defined as the Radon-Nikodym derivative of the probability distribution relative to a dominating measure, and the likelihood function is this density interpreted as a function of the parameter (vector), not the possible outcomes. (wikipedia.org)
- Finding the maximum of a function often involves taking the derivative of a function and solving for the parameter being maximized, and this is often easier when the function being maximized is a log-likelihood rather than the original likelihood function. (wikipedia.org)
- If f(x) is an E-function then the derivative and integral of f are also E-functions. (wikipedia.org)
- The reciprocal of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose derivative is nowhere zero. (wikipedia.org)
- To construct f2, f ′ is then considered on the smaller interval [0,1/ truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to f1 to produce the function f2. (wikipedia.org)
- If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set C in place of the "fat" (positive-measure) Cantor set S, one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set C instead of the positive-measure set S, and so the resulting function would have an integrable derivative. (wikipedia.org)
- The indefinite integral of a function is the set of functions whose derivative is that function. (wikipedia.org)
- The constant of integration follows from the fact that the derivative of a constant function is 0. (wikipedia.org)
- operatorname {sgn}(x)\cdot x} The signum function is the derivative of the absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x. (wikipedia.org)

- If ν is an integer then Anger functions Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linear combinations of Struve functions. (wikipedia.org)
- The Upsilon function Υ(s) is a meromorphic function of a complex variable s, which has isolated singularities at points,ki where k is non-zero integer. (wikipedia.org)
- After Ackermann's publication of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. (wikipedia.org)
- The Péter-Ackermann function can also be expressed in terms of various other versions of the Ackermann function: the indexed version of Knuth's up-arrow notation (extended to integer indices ≥ -2): A ( m , n ) = 2 ↑ m − 2 ( n + 3 ) − 3. (wikipedia.org)
- As a further useful property, all of the zeros of the normalized sinc function are integer values of x. (wikipedia.org)
- The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimited functions: It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero integer k. (wikipedia.org)
- The functions xk(t) = sinc(t − k) (k integer) form an orthonormal basis for bandlimited functions in the function space L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency fH = 1/2). (wikipedia.org)
- Currying is the process of changing a function so that it takes fewer inputs (in this case, transforming a function that performs division by any integer into one that performs division by a set integer). (wikipedia.org)
- In this case, the function f(x) = tan(x), with the domain of all real x ≠ (2n+1)π/2, n any integer, is continuous. (wikipedia.org)
- The function myfunction expects an integer argument to be on the stack or in a register when it is called. (wikipedia.org)
- By including the function prototype, you inform the compiler that the function myfunction takes one integer argument and you enable the compiler to catch these kinds of errors and make the compilation process run smoothly. (wikipedia.org)
- Divisor function, an arithmetic function giving the number of divisors of an integer. (wikipedia.org)
- For that purpose, one needs a hash function that maps similar keys to hash values that differ by at most m, where m is a small integer (say, 1 or 2). (wikipedia.org)
- 0} which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular function. (wikipedia.org)

- By introducing these transcendental functions and noting the bijection property that implies an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function. (wikipedia.org)
- A common type of implicit function is an inverse function. (wikipedia.org)
- Intuitively, an inverse function is obtained from g by interchanging the roles of the dependent and independent variables. (wikipedia.org)
- Stated another way, the inverse function gives the implicit solution for x of the equation R ( x , y ) = y − g ( x ) = 0. (wikipedia.org)
- A continuous function with a continuous inverse function is called a homeomorphism. (wikipedia.org)
- Such functions do not have an inverse function, but they do have an inverse relation. (wikipedia.org)

- The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values. (wikipedia.org)
- If λ is an algebraic number then the Bessel function Jλ is an E-function. (wikipedia.org)

- For a half iterate of a function, see Functional square root . (wikipedia.org)
- For specialization of brain function generally, see Functional specialization (brain) . (wikipedia.org)
- Reflecting the diversity of this active field in bioinformatics, the chapters in this book discuss a variety of tools and resources such as sequence-, structure-, systems-, and interaction-based function prediction methods, tools for functional analysis of metagenomics data, detecting moonlighting-proteins, sub-cellular localization prediction, and pathway and comparative genomics databases. (springer.com)
- True liver function tests correlate with hepatic functional capacity and include the galactose elimination, caffeine clearance, prothrombin time (PT), albumin, and cholesterol levels. (springer.com)
- Partial unfolding during function, known as cracking ( 2 ), was predicted to be a general property that allows proteins to reduce free-energy barriers and accelerate functional kinetics. (pnas.org)
- Function points are used to compute a functional size measurement (FSM) of software. (wikipedia.org)
- Each of these functional user requirements maps to an end-user business function, such as a data entry for an Input or a user query for an Inquiry. (wikipedia.org)
- In a non-strict functional programming language, strictness analysis refers to any algorithm used to prove the strictness of a function with respect to one or more of its arguments. (wikipedia.org)
- Anonymous functions are ubiquitous in functional programming languages and other languages with first-class functions, where they fulfill the same role for the function type as literals do for other data types. (wikipedia.org)
- and the scaling function that described the covers of the attractor of the logistic map The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. (wikipedia.org)
- other characteristics may be non-functional spandrels, though these in turn may later be co-opted by evolution to serve new functions. (wikipedia.org)
- However, dealing with programs that allow impure and pure functions to be mixed like this can be quite difficult in practice, thus purely functional programming languages do not allow impure functions to be defined. (wikipedia.org)
- Compile time function execution: the evaluation of pure functions at compile time Purely functional data structure Referential transparency (computer science) Lambda calculus Side effect (computer science) Pure procedure Idempotence pure keyword in Fortran annotating pure functions constexpr keyword in C++ annotating pure functions usable at compile-time Peyton Jones, Simon L. (2003). (wikipedia.org)
- In functional analysis a clear formulation is given of the essential feature of an integrable function, namely the way it defines a linear functional on other functions. (wikipedia.org)
- Nested functions are used in many approaches to structured programming, including early ones, such as ALGOL, Simula 67 and Pascal, and also in many modern dynamic languages and functional languages. (wikipedia.org)
- During software development, null functions with the same names and type signatures as a planned functions are often used as stubs-that is, as non-functional placeholders that allow the incomplete body of code to be compiled and tested prior to completion of all planned features. (wikipedia.org)
- if the truth value of the compound statement is determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and the logical connective is said to be truth functional. (wikipedia.org)
- Classical propositional logic is a truth-functional propositional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. (wikipedia.org)
- A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. (wikipedia.org)
- Because a function may be expressed as a composition, a truth-functional logical calculus does not need to have dedicated symbols for all of the above-mentioned functions to be functionally complete. (wikipedia.org)

- Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables. (wikipedia.org)
- An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. (wikipedia.org)
- A constant function such as f(x) = π is a rational function since constants are polynomials. (wikipedia.org)
- While this notion can apply to any type of function whose n arguments have the same domain set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. (wikipedia.org)

- The excitation function typically resembles a Gaussian bell curve. (wikipedia.org)

- The problem now lies in finding the Green's function G that satisfies equation (1). (wikipedia.org)

- In economics , a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. (wikipedia.org)
- One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it. (wikipedia.org)
- For modelling the case of many outputs and many inputs, researchers often use the so-called Shephard's distance functions or, alternatively, directional distance functions,wasd which are generalizations of the simple production function in economics. (wikipedia.org)
- In general, economic output is not a (mathematical) function of input, because any given set of inputs can be used to produce a range of outputs. (wikipedia.org)
- that is, a production function relates physical inputs to physical outputs, and prices and costs are not reflected in the function. (wikipedia.org)
- In the decision frame of a firm making economic choices regarding production-how much of each factor input to use to produce how much output-and facing market prices for output and inputs, the production function represents the possibilities afforded by an exogenous technology. (wikipedia.org)
- The inputs to the production function are commonly termed factors of production and may represent primary factors, which are stocks. (wikipedia.org)
- Thanks to their bijectivity (no collisions, therefore no entropy loss) regardless of the used Boolean functions 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 ciphers, stream ciphers, PRNGs and hash functions. (wikipedia.org)
- The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. (wikipedia.org)
- In computational networks, the activation function of a node defines the output of that node given an input or set of inputs. (wikipedia.org)
- In addition, networks constructed using this model have unstable convergence because neuron inputs along favored paths tend to increase without bound, as this function is not normalizable. (wikipedia.org)
- Register machines Although these models use different representations for the functions, their inputs and their outputs, translations exist between any two models, and so every model describes essentially the same class of functions, giving rise to the opinion that formal computability is both natural and not too narrow. (wikipedia.org)
- Since the original functions do not preserve all the information of their inputs, they are not reversible. (wikipedia.org)
- In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs P and Q. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. (wikipedia.org)

- In short, a virtual function defines a target function to be executed, but the target might not be known at compile time. (wikipedia.org)
- Bang measure - Defines a function metric based on twelve primitive (simple) counts that affect or show Bang, defined as "the measure of true function to be delivered as perceived by the user. (wikipedia.org)
- The implicit function theorem provides conditions under which a relation defines an implicit function. (wikipedia.org)
- The name comes from the fact that any such function defines a perturbation of the initial problem. (wikipedia.org)

- It can be presented as a difference between two Heaviside step functions of amplitude 1. (wikipedia.org)
- The value of H(0) will depend upon the particular convention chosen for the Heaviside step function. (wikipedia.org)
- See Heaviside step function - Analytic approximations. (wikipedia.org)

- In probability theory and statistics , the cumulative distribution function ( CDF ) of a real-valued random variable X , or just distribution function of X , evaluated at x , is the probability that X will take a value less than or equal to x . (wikipedia.org)
- Of great interest in number theory is the growth rate of the prime-counting function. (wikipedia.org)
- Simple functions are sufficiently 'nice' that using them makes mathematical reasoning, theory, and proof easier. (wikipedia.org)
- Propositional functions are useful in set theory for the formation of sets. (wikipedia.org)
- Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types. (wikipedia.org)
- In quantum field theory, Green's functions take the roles of propagators. (wikipedia.org)
- In this case, the Green's function is the same as the impulse response of linear time-invariant system theory. (wikipedia.org)
- The theory of Hardy's Z-function. (wikipedia.org)
- It was introduced by Philip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication", in which he said the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own", and his 1953 book Probability and Information Theory, with Applications to Radar. (wikipedia.org)
- In mathematical optimization, statistics, econometrics, decision theory, machine learning and computational neuroscience, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. (wikipedia.org)
- Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratric loss function. (wikipedia.org)
- In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. (wikipedia.org)
- The most common form of theta function is that occurring in the theory of elliptic functions. (wikipedia.org)
- In quantum field theory there are correlation functions over quantum distributions. (wikipedia.org)
- Any probability distribution which obeys a condition on correlation functions called reflection positivity lead to a local quantum field theory after Wick rotation to Minkowski spacetime. (wikipedia.org)
- tau function (representation theory) in the representation theory of affine Lie algebras and soliton equations. (wikipedia.org)
- They are of interest in transcendental number theory, and are more special than G-functions. (wikipedia.org)
- Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations. (wikipedia.org)
- In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals. (wikipedia.org)
- An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere. (wikipedia.org)
- Sergei Sobolev, working in partial differential equation theory, defined the first adequate theory of generalized functions, from the mathematical point of view, in order to work with weak solutions of partial differential equations. (wikipedia.org)
- An important application of monotonic functions is in probability theory. (wikipedia.org)
- Computable functions are the basic objects of study in computability theory. (wikipedia.org)
- The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions. (wikipedia.org)
- In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem. (wikipedia.org)
- Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics. (wikipedia.org)
- Quasiconvex functions are important also in game theory, industrial organization, and general equilibrium theory, particularly for applications of Sion's minimax theorem. (wikipedia.org)
- Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. (wikipedia.org)
- Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them. (wikipedia.org)
- The functions ℘ take different forms depending on Λ, and a rich theory is developed when one allows Λ to vary. (wikipedia.org)
- The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory. (wikipedia.org)
- Peter Humphries, The distribution of weighted sums of the Liouville function and Pólya's conjecture, Journal of Number Theory 133 (2013), 545-582. (wikipedia.org)

- Anonymous functions originate in the work of Alonzo Church in his invention of the lambda calculus in 1936, before electronic computers, in which all functions are anonymous. (wikipedia.org)
- In several programming languages, anonymous functions are introduced using the keyword lambda, and anonymous functions are often referred to as lambdas or lambda abstractions. (wikipedia.org)
- The class of computable functions can be defined in many equivalent models of computation, including Turing machines μ-recursive functions Lambda calculus Post machines (Post-Turing machines and tag machines). (wikipedia.org)

- A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. (wikipedia.org)
- Formally, an analytic function ƒ(z) of one real or complex variable z is transcendental if it is algebraically independent of that variable. (wikipedia.org)
- For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the Fourier-Bros-Iagolnitzer transform. (wikipedia.org)
- The Fourier transform of a bump function is a (real) analytic function, and it can be extended to the whole complex plane: hence it cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see Paley-Wiener theorem). (wikipedia.org)

- Language functions such as grammar, vocabulary and literal meaning are typically lateralized to the left hemisphere, especially in right-handed individuals. (wikipedia.org)
- Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. (wikipedia.org)
- In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. (wikipedia.org)
- Piecewise defined functions (functions given by different formulas in different regions) are typically not analytic where the pieces meet. (wikipedia.org)
- Typically the function subexpression is simply a function identifier. (wikipedia.org)
- Typically a generic function is an instance of a class that inherits both from function and standard-object. (wikipedia.org)
- Typically, the domain of a hash function (the set of possible keys) is larger than its range (the number of different table indices), and so it will map several different keys to the same index. (wikipedia.org)
- Still, in a half-full table, a good hash function will typically narrow the search down to only one or two entries. (wikipedia.org)

- For many applications, the natural logarithm of the likelihood function, called the log-likelihood, is more convenient to work with. (wikipedia.org)

- The normalized sinc function is the Fourier transform of the rectangular function with no scaling. (wikipedia.org)
- Tau function may refer to: Ramanujan tau function, giving the Fourier coefficients of the Ramanujan modular form. (wikipedia.org)

- The indefinite integral of many algebraic functions is transcendental. (wikipedia.org)
- Algebraic functions play an important role in mathematical analysis and algebraic geometry. (wikipedia.org)
- The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list). (wikipedia.org)

- since polynomial functions are continuous , the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions). (wikipedia.org)
- However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution ). (wikipedia.org)

- : 29 The general form of the variance function is presented under the exponential family context, as well as specific forms for Normal, Bernoulli, Poisson, and Gamma. (wikipedia.org)
- The capital Greek letter gamma is itself a function . (scienceblogs.com)
- Some liver function tests, like the gamma GT (gamma-glytamyl transpeptidase), are very sensitive and can be raised by something as simple. (netdoctor.co.uk)
- Some liver function tests, like the gamma GT (gamma-glytamyl transpeptidase), are very sensitive and can be raised by something as simple as taking paracetamol regularly and perhaps a bit too often, or as complicated as showing damage caused by too much alcohol. (netdoctor.co.uk)
- Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental. (wikipedia.org)
- Those that are not, such as the gamma and the zeta functions, are called transcendentally transcendental or hypertranscendental functions. (wikipedia.org)
- end{aligned}}} Here, Γ denotes the Gamma function. (wikipedia.org)

- Formally this isn't quite right - the essence of a function is in the set of ordered pairs {x, f(x)} and not in the specific rule that connects them. (scienceblogs.com)
- Formally, a simple function is a finite linear combination of indicator functions of measurable sets. (wikipedia.org)
- Although not formally recognized and named until 1989, the concept of the pedotransfer function has long been applied to estimate soil properties that are difficult to determine. (wikipedia.org)
- Effect systems, among other things, allow one to reason precisely and formally about the purity of certain expressions even in the presence of higher-order functions etc. (wikipedia.org)
- Formally, an elliptic function is a function f meromorphic on ℂ for which there exist two non-zero complex numbers ω1 and ω2 with ω1/ω2 ∉ ℝ, such that f(z) = f(z + ω1) and f(z) = f(z + ω2) for all z ∈ ℂ. (wikipedia.org)

- For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (wikipedia.org)

- More research is needed to know how cannabis affects the immune system and its ability to function normally. (cdc.gov)
- Although the communication between sleep regulatory networks in the central nervous system and the cells and tissues of the immune system is basically bidirectional, in this review, we will focus on the role of sleep for proper functioning of the immune system. (springer.com)
- Basophils are white blood cells from the bone marrow that play a role in keeping the immune system functioning correctly. (medicalnewstoday.com)

- In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction. (wikipedia.org)
- R) forms an infinite-dimensional real Banach algebra: finite linear combinations and products of regulated functions are again regulated functions. (wikipedia.org)
- A step function takes only a finite number of values. (wikipedia.org)
- The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute the function. (wikipedia.org)

- Variance functions play a very important role in parameter estimation and inference. (wikipedia.org)
- This function frees the CERT_CONTEXT referenced by non- NULL values of this parameter. (microsoft.com)
- while (pCertContext= CertEnumCertificatesInStore( hStoreHandle, pCertContext)) // on the first call to the function, // this parameter is NULL // on all subsequent calls, // this parameter is the last pointer // returned by the function { //---------------------------------------------------------------- // Do whatever is needed for a current certificate. (microsoft.com)
- You can retrieve this salt value by using the CryptGetKeyParam function with the dwParam parameter set to KP_SALT. (microsoft.com)
- The function returns the previous value of the Destination parameter. (microsoft.com)
- If expression contains an aggregate function that specifies a scope other than the default, the scope parameter for the Previous function must be a containing scope for the scope specified in the aggregate function call. (microsoft.com)
- The functions Level , InScope , Aggregate and Previous cannot be used in the expression parameter. (microsoft.com)
- Specifying the recursive parameter for any aggregate function is not supported. (microsoft.com)
- It is obtained by taking the expected value with respect to the probability distribution, Pθ, of the observed data, X. This is also referred to as the risk function of the decision rule δ and the parameter θ. (wikipedia.org)
- In a prototype, parameter names are optional (and in C/C++ have function prototype scope, meaning their scope ends at the end of the prototype), however, the type is necessary along with all modifiers (e.g. if it is a pointer or a const parameter). (wikipedia.org)
- Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. (wikipedia.org)
- Let X be a discrete random variable with probability mass function p depending on a parameter θ. (wikipedia.org)
- Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter θ. (wikipedia.org)
- If an argument is passed by reference, any parameter mutation will alter the value of the argument outside the function, which will render the function impure. (wikipedia.org)
- Here we define a method for the generic function collide which is specialized for the classes asteroid (first parameter object-1) and spaceship (second parameter object-2). (wikipedia.org)
- A nested function can access other local functions, variables, constants, types, classes, etc. that are in the same scope, or in any enclosing scope, without explicit parameter passing, which greatly simplifies passing data into and out of the nested function. (wikipedia.org)
- The function F is nested within E. Note that E's parameter x is visible also in F (as F is a part of E) while both x and y are invisible outside E and F respectively. (wikipedia.org)
- In general, the parameter estimates for a given model are chosen so as to make the discrepancy function for that model as small as possible. (wikipedia.org)
- The discrepancy function is a continuous function of the elements of S, the sample covariance matrix, and Σ(θ), the "reproduced" estimate of S obtained by using the parameter estimates and the structural model. (wikipedia.org)

- Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. (wikipedia.org)

- A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve . (wikipedia.org)
- Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams ). (wikipedia.org)
- Sigmoid functions have domain of all real numbers , with return value monotonically increasing most often from 0 to 1 or alternatively from −1 to 1, depending on convention. (wikipedia.org)
- A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons . (wikipedia.org)
- Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic distribution , the normal distribution , and Student's t probability density functions . (wikipedia.org)
- Some sigmoid functions compared. (wikipedia.org)
- When a specific mathematical model is lacking, a sigmoid function is often used. (wikipedia.org)
- Wikimedia Commons has media related to Sigmoid functions . (wikipedia.org)
- Unit step function Crenel function Simple function Piecewise defined function Sigmoid function Step detection Bachman, Narici, Beckenstein. (wikipedia.org)
- All problems mentioned above can be handled by using a normalizable sigmoid activation function. (wikipedia.org)

- The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these. (wikipedia.org)
- The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain. (wikipedia.org)
- Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. (wikipedia.org)

- A discrepancy function is a mathematical function which describes how closely a structural model conforms to observed data. (wikipedia.org)

- Every constant function whose domain and codomain are the same is idempotent. (wikipedia.org)
- That is, a null function is an identity function whose domain and codomain are both the state space S of the program, and for which: f(s) = s for all elements s in S. Less rigorous definitions may also be encountered. (wikipedia.org)

- Graph showing ratio of the prime-counting function π ( x ) to two of its approximations, x /ln x and Li( x ). (wikipedia.org)
- So what I've plotted in the first graph is just the function we're integrating to get g(z). (scienceblogs.com)
- The ramp function is a unary real function, whose graph is shaped like a ramp. (wikipedia.org)
- A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis. (wikipedia.org)
- such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. (wikipedia.org)
- A graph of the cumulative probability of failures up to each time point is called the cumulative distribution function, or CDF. (wikipedia.org)

- After all, most of the time you first meet the hyperbolic trig functions in intro calculus, where their rather odd definitions are presented and then used as test beds for blindly applying newly-learned differentiation rules. (scienceblogs.com)
- Mark-II: ISO/IEC 20968:2002 Software engineering - Ml II Function Point Analysis - Counting Practices Manual NESMA: ISO/IEC 24570:2005 Software engineering - NESMA function size measurement method version 2.1 - Definitions and counting guidelines for the application of Function Point Analysis 2. (wikipedia.org)
- There are several different definitions of continuity of a function. (wikipedia.org)
- In C++, function prototypes are also used in class definitions. (wikipedia.org)
- tau )\end{aligned}}} The above definitions of the Jacobi theta functions are by no means unique. (wikipedia.org)
- With these definitions, the study of correlation functions is similar to the study of probability distributions. (wikipedia.org)
- Lexically nested function definitions are a form of information hiding and are useful for dividing procedural tasks into subtasks which are only meaningful locally. (wikipedia.org)

- In contrast, mechanical work and heat are process quantities or path functions, because their values depend on the specific transition (or path) between two equilibrium states. (wikipedia.org)
- Whereas the CALCULATE function requires as its first argument an expression that returns a single value, the CALCULATETABLE function takes a table of values. (microsoft.com)
- This function can return one of these values. (microsoft.com)
- If you do not need to count cells that contain logical values or text (in other words, if you want to count only cells that contain numbers), use the COUNT or COUNTX functions. (microsoft.com)
- We integrate a function of x and evaluate that function with the appropriate z values to get g(z). (scienceblogs.com)
- The function compares two specified 64-bit values and exchanges with another 64-bit value based on the outcome of the comparison. (microsoft.com)
- To operate on 32-bit values, use the InterlockedCompareExchangeAcquire function. (microsoft.com)
- The following time series forecasting functions can be used to predict future values based on historical data. (microsoft.com)
- A Propositional Function, or a predicate, in a variable x is a sentence p(x) involving x that becomes a proposition when we give x a definite value from the set of values it can take. (wikipedia.org)
- Considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate (or volumetric flux) through a line connecting the two points. (wikipedia.org)
- Interpreting positive values as true and negative values as false, an R-function is transformed into a "companion" Boolean function (the two functions are called friends). (wikipedia.org)
- It is an even function, and real analytic for real values. (wikipedia.org)
- In some programming languages, anonymous functions are commonly implemented for very specific purposes such as binding events to callbacks, or instantiating the function for particular values, which may be more efficient, more readable, and less error-prone than calling a more-generic named function. (wikipedia.org)
- Given any function f in n variables with values in an abelian group, a symmetric function can be constructed by summing values of f over all permutations of the arguments. (wikipedia.org)
- Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. (wikipedia.org)
- He found a method to show that the values taken by certain E-functions were algebraically independent.This was a result which established the algebraic independence of classes of numbers rather than just linear independence. (wikipedia.org)
- The function may return multiple result values and these conditions must apply to all returned values for the function to be considered pure. (wikipedia.org)
- Generic functions correspond roughly to what Smalltalk terms methods, with the notable exception that, in Smalltalk, the receiver's class is the sole determinant of which body of code is called: the types or values of the arguments are irrelevant (single dispatch). (wikipedia.org)
- The values returned by a hash function are called hash values, hash codes, digests, or simply hashes. (wikipedia.org)
- If time can only take discrete values (such as 1 day, 2 days, and so on), the distribution of failure times is called the probability mass function (pmf). (wikipedia.org)
- Larger values of the discrepancy function indicate a poor fit of the model to data. (wikipedia.org)
- In computer science, a null function (or null operator) is subroutine that returns no data values and leaves the program state unchanged. (wikipedia.org)
- Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. (wikipedia.org)
- Which one gets used depends on the number of arguments provided when the new Bill object is created (none, or two): Bill(double tip, double total) : tip(tip), total(total) { } Now a function that creates a new Bill object could pass two values into the constructor and set the data members in one step. (wikipedia.org)
- In logic, a truth function is a function that accepts truth values as input and produces a truth value as output, i.e., the input and output are all truth values. (wikipedia.org)

- Compositions of two real functions, absolute value and a cubic function , in different orders show a non-commutativity of the composition. (wikipedia.org)
- Here's a straightforward function of two variables, x and y: Its domain is all real x and y, with the single exception of x = y = 0, which would make the denominator 0. (scienceblogs.com)
- Here's a very simple function: You give it a positive real number, it gives you the square root. (scienceblogs.com)
- to create real-valued or complex-valued transfer function models, or to convert dynamic system models to transfer function form. (mathworks.com)
- In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line, similar to a step function. (wikipedia.org)
- In On the Infinite, David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert's personal secretary and former student, who actually proved the hypothesis in his paper On Hilbert's Construction of the Real Numbers. (wikipedia.org)
- The domain of this function is the set of all real numbers ℝ. (wikipedia.org)
- Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function. (wikipedia.org)
- The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π). (wikipedia.org)
- As such, the step functions form an algebra over the real numbers. (wikipedia.org)
- There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. (wikipedia.org)
- The Flavors syntax to send a message is: (send object :message) With New Flavors, it was decided the message should be a real function and the usual function calling syntax should be used: (message object) message now is a generic function, an object and function in its own right. (wikipedia.org)
- It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded and closed support. (wikipedia.org)
- As in the case with real functions, the restricted range may be called principal branch of the function. (wikipedia.org)
- yet we arrive at the collision with the reserved names for upper and lower semicontinuous real-valued function). (wikipedia.org)
- This real-valued function f(x) of the real variable x is defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and q ∈ N coprime 0 if x is irrational. (wikipedia.org)
- Quasiconvexity and quasiconcavity extend to functions with multiple arguments the notion of unimodality of functions with a single real argument. (wikipedia.org)
- In calculus, a function series is a series, where the summands are not just real or complex numbers but functions. (wikipedia.org)

- Moreover, important formulas like Paul Lévy 's inversion formula for the characteristic function also rely on the "less than or equal" formulation. (wikipedia.org)
- In studying the function of a behavioral characteristic of an animal, a researcher seeks to understand how natural selection favours the behaviour. (britannica.com)
- The work function is not a characteristic of a bulk material, but rather a property of the surface of the material (depending on crystal face and contamination). (wikipedia.org)
- In practice, one directly controls EF by the voltage applied to the material through electrodes, and the work function is generally a fixed characteristic of the surface material. (wikipedia.org)

- All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. (wikipedia.org)
- Both Sudan and Ackermann are credited with discovering total computable functions (termed simply "recursive" in some references) that are not primitive recursive. (wikipedia.org)
- 2 it extends these basic operations in a way that can be compared to the hyperoperations: (Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann's function that are specifically designed for that purpose-such as Goodstein's hyperoperation sequence. (wikipedia.org)
- Unlike named nested functions, they cannot be recursive without the assistance of a fixpoint operator (also termed an anonymous fixpoint or anonymous recursion) or binding them to a name. (wikipedia.org)
- In computer science, the Tak function is a recursive function, named after Ikuo Takeuchi (竹内郁雄). (wikipedia.org)
- The best known way to optimize tarai is to use mutually recursive helper function as follows. (wikipedia.org)
- Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions. (wikipedia.org)
- Due to simple recursive scope rules, a nested function is itself invisible outside of its immediately enclosing function, but can see (access) all local objects (data, functions, types, etc.) of its immediately enclosing function as well as of any function(s) which, in turn, encloses that function. (wikipedia.org)

- It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions . (wikipedia.org)
- From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part. (wikipedia.org)
- Every cumulative distribution function F is non-decreasing and right-continuous , which makes it a càdlàg function. (wikipedia.org)
- Every function with these four properties is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable. (wikipedia.org)
- If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists). (wikipedia.org)
- Another name for the survival function is the complementary cumulative distribution function. (wikipedia.org)
- In survival analysis, the cumulative distribution function gives the probability that the survival time is less than or equal to a specific time, t. (wikipedia.org)

- The print() function outputs one or more strings. (w3schools.com)

- Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. (wikipedia.org)
- Otherwise, a function is said to be a discontinuous function. (wikipedia.org)
- By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps at each point in time when money is deposited or withdrawn, so the function M(t) is discontinuous. (wikipedia.org)
- A function is said to be discontinuous (or to have a discontinuity) at some point when it is not continuous there. (wikipedia.org)
- Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. (wikipedia.org)
- A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. (wikipedia.org)

- model object can represent SISO or MIMO transfer functions in continuous time or discrete time. (mathworks.com)
- that you can use in a rational expression to create a discrete-time transfer function model. (mathworks.com)
- The likelihood function is usually defined differently for discrete and continuous probability distributions. (wikipedia.org)
- This provides a likelihood function for any probability model with all distributions, whether discrete, absolutely continuous, a mixture or something else. (wikipedia.org)

- It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere. (wikipedia.org)

- Invex functions were introduced by Hanson as a generalization of convex functions. (wikipedia.org)
- A slight generalization of invex functions called Type 1 invex functions are the most general class of functions for which the Karush-Kuhn-Tucker conditions are necessary and sufficient for a global minimum. (wikipedia.org)
- All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. (wikipedia.org)

- In the case of a continuous distribution , it gives the area under the probability density function from minus infinity to x . (wikipedia.org)

- The functions are defined as: where: δ(x) is the Dirac delta function, also called the unit impulse. (wikipedia.org)
- A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space ℝn, at a point s, is any solution of where δ is the Dirac delta function. (wikipedia.org)

- Forcing function can mean: In differential calculus, a Forcing function (differential equations) In interaction design, a behavior-shaping constraint, a means of preventing undesirable user input usually made by mistake. (wikipedia.org)
- Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations. (wikipedia.org)
- In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. (wikipedia.org)
- This property of a Green's function can be exploited to solve differential equations of the form If the kernel of L is non-trivial, then the Green's function is not unique. (wikipedia.org)

- However, to define a quasi-likelihood, one need only specify a relationship between the mean and the variance of the observations to then be able to use the quasi-likelihood function for estimation. (wikipedia.org)
- The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency , a key focus of economics. (wikipedia.org)
- Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions. (wikipedia.org)
- Pure virtual functions can also be used where the method declarations are being used to define an interface - similar to what the interface keyword in Java explicitly specifies. (wikipedia.org)
- Distortion functions are used to define distortion risk measures. (wikipedia.org)
- Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above. (wikipedia.org)
- However, some authors also define step functions with an infinite number of intervals. (wikipedia.org)

- they established field capacity, permanent wilting point, available water content, and air capacity as a function of textural class, and as well as deriving continuous functions estimating these soil-water properties. (wikipedia.org)
- These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola xy = 1 by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola. (wikipedia.org)
- Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale. (wikipedia.org)

- In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity. (wikipedia.org)
- The manual said that the function had no "closed-form analytic antiderivative" but nonetheless the calculator could integrate it numerically. (scienceblogs.com)
- Evolutionary biology is faced with two major explanatory problems: form and function. (britannica.com)
- to transfer-function form. (mathworks.com)
- A platelet function test is a blood test that is used to determine how well the platelets clump together to form blood clots. (cdc.gov)
- The Airy function also underlies the form of the intensity near an optical directional caustic, such as that of the rainbow. (wikipedia.org)
- Anonymous functions are a form of nested function, in allowing access to variables in the scope of the containing function (non-local variables). (wikipedia.org)
- This concept of function as opposed to form (respectively Aristotle's ergon and morphê) was central in biological explanations in classical antiquity, and in more modern times formed part of the Cuvier-Geoffroy debate. (wikipedia.org)
- Form and Function: A Contribution to the History of Animal Morphology. (wikipedia.org)
- Following form and function: A philosophical archaeology of life science. (wikipedia.org)
- In its simplest form, this function is binary-that is, either the neuron is firing or not. (wikipedia.org)
- The final model, then, that is used in multilayer perceptrons is a sigmoidal activation function in the form of a hyperbolic tangent. (wikipedia.org)
- In particular E-functions form a ring. (wikipedia.org)

- There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. (wikipedia.org)
- One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. (wikipedia.org)
- See Jacobi theta functions (notational variations) for further discussion. (wikipedia.org)
- There are two families of 'canonical' elliptic functions: those of Jacobi and those of Weierstrass. (wikipedia.org)

- Predicate (mathematical logic) Boolean-valued function Formula (logic) Sentence (logic) Open sentence Tiles, Mary (2004). (wikipedia.org)
- Engineering function points - Elements (variable names) and operators (e.g., arithmetic, equality/inequality, Boolean) are counted. (wikipedia.org)

- Because arithmetic operations 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. (wikipedia.org)
- Another important property of T-functions based on arithmetic operations is predictability of their period, which is highly attractive to cryptographers. (wikipedia.org)

- The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). (wikipedia.org)
- These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing (−1)k into the series, resulting in alternating series. (wikipedia.org)
- The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. (wikipedia.org)

- Used in a loop, this function can retrieve in sequence all certificates in a certificate store. (microsoft.com)
- It is important to document, preferably in a single file with a self-describing title (such as "README-testing.txt"), the main ideas behind the function testing, including the prerequisites, the setup for the server and the client, the overall (or detailed) sequence of the scripts to follow, how to check for success/failures of the scripts, how to perform the cleanup, and to restart the testing. (ibm.com)
- Any non-negative measurable function is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. (wikipedia.org)
- X) by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. (wikipedia.org)
- A function can perform input or output and still be pure if the sequence of operations on the relevant Input/Output devices is modeled explicitly as both an argument and a result, and I/O operations are taken to fail when the input sequence does not describe the operations actually taken since the program began execution. (wikipedia.org)
- the first allows different calls to an I/O-performing function to return different results on account of the sequence arguments having changed. (wikipedia.org)
- This turns out to be the case: given a sequence of probability measures (μδ)δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I(x) = J(x) for all x ∈ X. It is possible to convert a weak large deviation principle into a strong one if the measures converge sufficiently quickly. (wikipedia.org)

- by means of numerical method, In the interval [−10000, the Griewank function has 6365 critical points. (wikipedia.org)
- This function will be defined to be 0 outside of the interval [0, 1/ (wikipedia.org)
- We then translate this function to the interval [3/8, 5/so that the resulting function, which we call f1, is nonzero only on the middle interval of the complement of the Smith-Volterra-Cantor set. (wikipedia.org)

- Transfer functions are a frequency-domain representation of linear time-invariant systems. (mathworks.com)
- R-functions are used in computer graphics and geometric modeling in the context of implicit surfaces and the function representation. (wikipedia.org)
- A picture function is a mathematical representation of a two-dimensional image as a function of two spatial variables. (wikipedia.org)

- This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. (wikipedia.org)
- This freedom allows the designer to choose the update functions or S-boxes that satisfy all other cryptographic criteria and even choose arbitrary or key-dependent update functions (see family keying). (wikipedia.org)
- Through the superposition principle for linear operator problems, the convolution of a Green's function with an arbitrary function f (x) on that domain is the solution to the inhomogeneous differential equation for f (x). (wikipedia.org)
- Most languages provide a generic sort function that implements a sort algorithm that will sort arbitrary objects. (wikipedia.org)
- This function usually accepts an arbitrary function that determines how to compare whether two elements equal or if one is greater or less than the other. (wikipedia.org)
- V. Egorov (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions. (wikipedia.org)
- A hash function is any function that can be used to map data of arbitrary size to data of fixed size. (wikipedia.org)

- The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of input. (wikipedia.org)
- A function describes what something does, not what its 'purpose' is. (wikipedia.org)
- The function f(x,y) describes the intensity of the point at coordinates (x,y). (wikipedia.org)
- The Shkarofsky function is a physics formula which describes the behavior of microwaves. (wikipedia.org)

- Radial basis function networks have many uses, including function approximation , time series prediction , classification , and system control . (wikipedia.org)
- For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB approximation, when the potential may be locally approximated by a linear function of position. (wikipedia.org)
- In general, a function approximation problem asks us to select a function among a well-defined class[clarification needed] that closely matches ("approximates") a target function in a task-specific way. (wikipedia.org)

- Liver function tests (also known as a liver panel) are blood tests that measure different enzymes, proteins, and other substances made by the liver. (medlineplus.gov)
- Is there anything else I need to know about liver function tests? (medlineplus.gov)
- If any of your liver function tests were not normal, your provider may need more tests to confirm or rule out a specific diagnosis. (medlineplus.gov)
- Although frequently called liver function tests, they are more properly considered markers of liver damage than function. (springer.com)
- Chaiwala S., Wu G.Y. (1998) True Liver Function Tests. (springer.com)
- There are a number of liver function tests, so without knowing which one - or ones - are abnormal I can only hazard a guess at a diagnosis. (netdoctor.co.uk)
- Other liver function tests can be abnormal if certain drugs are making the chemical factory work too hard, or if the liver is being damaged by drugs. (netdoctor.co.uk)
- Liver function tests are common tests that are used to see how well the liver is working. (medlineplus.gov)

- I first met this function sometime in the year 2001 in the manual for a graphing calculator. (scienceblogs.com)

- Also, Green's functions in general are distributions, not necessarily proper functions. (wikipedia.org)
- In other words, distributions cannot be multiplied (except for very special cases): unlike most classical function spaces, they are not an algebra. (wikipedia.org)