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Mathematical function used in cryptography
In cryptography, a T-function is a bijective mapping that updates every bit of the state in a way that can be described as x i ′ = x i + f ( x 0 , ⋯ ,
T-function
In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 )
Owen's_T_function
Extension of the factorial function
}t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function:
Gamma_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
normalized boxcar function) is defined as rect ( t T ) = Π ( t T ) = { 0 , if | t | > T 2 1 2 , if | t | = T 2 1 , if | t | < T 2 . {\displaystyle
Rectangular_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Probability distribution
cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. For t > 0 , F ( t ) = ∫ − ∞ t f ( u ) d u =
Student's_t-distribution
Hyperbolic analogues of trigonometric functions
hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin
Hyperbolic_functions
Economic model which weighs rewards based on when they are received
the discount function f(t) having a negative first derivative and with ct (or c(t) in continuous time) defined as consumption at time t, total utility
Discount_function
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Maximized objective function of an optimization problem
value function represents the optimal payoff of the system over the interval [ t , t 1 ] {\displaystyle [t,t_{1}]} when started at the time- t {\displaystyle
Value_function
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
actuarial mathematics, the accumulation function a(t) is a function of time t expressing the ratio of the value at time t (future value) and the initial investment
Accumulation_function
mimic function changes a file A {\displaystyle A} so it assumes the statistical properties of another file B {\displaystyle B} . That is, if p ( t , A )
Mimic_function
probability density function, Φ ( x ) = ∫ − ∞ x φ ( t ) d t = 1 2 [ 1 + erf ( x 2 ) ] {\displaystyle \Phi (x)=\int _{-\infty }^{x}\varphi (t)\,dt={\frac
List of integrals of Gaussian functions
List_of_integrals_of_Gaussian_functions
Type of white blood cell
On the other hand, CD4+ T cells function as "helper cells." Unlike CD8+ killer T cells, the CD4+ helper T (TH) cells function by further activating memory
T_cell
Mathematical function
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
Z_function
Genetically engineered T cell
activating functions into a single receptor. CAR T cell therapy is a cell therapy that uses T cells engineered with CARs to treat cancer. T cells are modified
CAR_T_cell
Special mathematical function
mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special
Polylogarithm
Scorer's functions can also be defined in terms of Airy functions: G i ( x ) = B i ( x ) ∫ x ∞ A i ( t ) d t + A i ( x ) ∫ 0 x B i ( t ) d t , H i ( x
Scorer's_function
Mathematical function relating circular and hyperbolic functions
{gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s
Gudermannian_function
Effect in signal processing
The Fourier transform of a function of time, s ( t ) {\displaystyle s(t)} , is a complex-valued function of frequency, S ( f ) {\displaystyle S(f)} ,
Spectral_leakage
Model of thermodynamic properties
specified temperature T and pressure P. Common departure functions include those for enthalpy, entropy, and internal energy. Departure functions are used to calculate
Departure_function
Class of mathematical functions
Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at
Subharmonic_function
First known wavelet basis
wavelet function ψ ( t ) {\displaystyle \psi (t)} can be described as ψ ( t ) = { 1 0 ≤ t < 1 2 , − 1 1 2 ≤ t < 1 , 0 otherwise. {\displaystyle \psi (t)={\begin{cases}1\quad
Haar_wavelet
Medical condition
T cell deficiency is a deficiency of T cells, caused by decreased function of individual T cells. It causes an immunodeficiency of cell-mediated immunity
T_cell_deficiency
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Symbol representing a mathematical concept
Similarly, if T {\displaystyle T} is some term in the language, F ( T ) {\displaystyle F(T)} is also a term. As such, the interpretation of a function symbol
Function_symbol
Periodic distribution ("function") of "point-mass" Dirac delta sampling
as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T ( t ) := ∑ k = − ∞ ∞ δ ( t − k T ) {\displaystyle
Dirac_comb
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Special case of the polylogarithm
dilogarithm function is sometimes defined as ∫ 1 v ln t 1 − t d t = Li 2 ( 1 − v ) . {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname
Dilogarithm
Theorem of convex functions
convex function (for t ∈ [0,1]), t f ( x 1 ) + ( 1 − t ) f ( x 2 ) , {\displaystyle tf(x_{1})+(1-t)f(x_{2}),} while the graph of the function is the convex
Jensen's_inequality
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Mathematical function
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}}\
Debye_function
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
mathematics the synchrotron functions are defined as follows (for x ≥ 0): First synchrotron function F ( x ) = x ∫ x ∞ K 5 3 ( t ) d t {\displaystyle F(x)=x\int
Synchrotron_function
Zeta-like functions approximate arbitrary holomorphic functions
universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate
Zeta_function_universality
Dynamical system whose system function is not directly dependent on time
time-dependent output function y ( t ) {\displaystyle y(t)} , and a time-dependent input function x ( t ) {\displaystyle x(t)} , the system will
Time-invariant_system
Parameter in atmospheric modeling
The Exner function is a parameter used in atmospheric modeling. Depending on the application, the Exner function may be defined as Π = c p ( p p 0 ) R
Exner_function
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Integral transform useful in probability theory, physics, and engineering
transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex variable s {\displaystyle
Laplace_transform
mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by Heatley (1943), Weisstein, as T ( m , n , r ) =
Toronto_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Concept in mathematics
In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of
K-function
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Complex complementary error function
The Faddeeva function or Kramp function is a scaled complex complementary error function, w ( z ) := e − z 2 erfc ( − i z ) = erfcx ( − i z ) = e
Faddeeva_function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Function of propagation delay and Doppler frequency
function is given by χ ( τ , f ) = ∫ − ∞ ∞ s ( t ) s ∗ ( t − τ ) e i 2 π f t d t {\displaystyle \chi (\tau ,f)=\int _{-\infty }^{\infty }s(t)s^{*}(t-\tau
Ambiguity_function
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Function in condensed matter physics
scattering function is the spatial Fourier transform of the van Hove function G ( r → , t ) {\displaystyle G({\vec {r}},t)} : F ( k → , t ) ≡ ∫ G ( r → , t ) exp
Dynamic_structure_factor
Evaluation of a function on its argument
In mathematics, function application (or evaluation) is the act of taking a function and an input from its domain to obtain the corresponding value from
Function_application
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the
Theta_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Mathematical function
function is a generalization of the reciprocal gamma function of the Laplace transform. Formally, it can be defined as ν ( x ) ≡ ∫ 0 ∞ x t d t Γ ( t +
Nu_function
Special mathematical function defined as sin(x)/x
In mathematics, physics and engineering, the sinc function (/ˈsɪŋk/ SINK), denoted by sinc(x), is defined as either sinc ( x ) = sin x x . {\displaystyle
Sinc_function
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
Function that only depends on time
for each value of t. In the more general case, any nonhomogeneous source function in any variable can be described as a forcing function, and the resulting
Forcing function (differential equations)
Forcing_function_(differential_equations)
non-tangential maximal function takes a function F defined on the upper-half plane R + n + 1 := { ( x , t ) : x ∈ R n , t > 0 } {\displaystyle \mathbf
Maximal_function
Asymmetric sigmoid function
or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes
Gompertz_function
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Mathematical finite graph-associated function
mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate
Ihara_zeta_function
Mathematical function
Riemann–Siegel theta function is defined in terms of the gamma function as θ ( t ) = arg ( Γ ( 1 4 + i t 2 ) ) − log π 2 t {\displaystyle \theta (t)=\arg \left(\Gamma
Riemann–Siegel_theta_function
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Electrical engineering concept
real-valued function s(t), it is determined from the function's analytic representation, sa(t): φ ( t ) = arg { s a ( t ) } = arg { s ( t ) + j s ^ ( t ) }
Instantaneous phase and frequency
Instantaneous_phase_and_frequency
Indian scientist
equations and Ramanujan’s T-function. Shorey's contribution to irreducibility of Laguerre polynomials is extensive. Shorey, T.N. "On gaps between numbers
Tarlok_Nath_Shorey
can be formulated in terms of N(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T. In 1942 Atle Selberg investigated
Selberg's zeta function conjecture
Selberg's_zeta_function_conjecture
Practice and study of secure communication techniques
cryptographic hash function is computed, and only the resulting hash is digitally signed. Cryptographic hash functions are functions that take a variable-length
Cryptography
Nowhere analytic, infinitely differentiable function
the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). This function satisfies
Fabius_function
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
One-way cryptographic tool
In theoretical computer science and cryptography, a trapdoor function is a function that is easy to compute in one direction, yet difficult to compute
Trapdoor_function
Mathematical function
Kummer's function is defined by Λ n ( z ) = ∫ 0 z log n − 1 | t | 1 + t d t . {\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt
Kummer's_function
Integral transform and linear operator
singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given
Hilbert_transform
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Branch of functional analysis
holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which
Holomorphic functional calculus
Holomorphic_functional_calculus
In control systems theory, the describing function (DF) method, developed by Nikolay Mitrofanovich Krylov and Nikolay Bogoliubov in the 1930s, and extended
Describing_function
Theory of cryptography
In cryptography, a sponge function or sponge construction is any of a class of algorithms with finite internal state that take an input bit stream of any
Sponge_function
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
Extension of superfactorials to the complex numbers
the double gamma function, is log G ( 1 + z ) = z 2 log ( 2 π ) + ∫ 0 ∞ d t t [ 1 − e − z t 4 sinh 2 t 2 + z 2 2 e − t − z t ] {\displaystyle \log
Barnes_G-function
_{+}.} It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if s u p p f ^ ⊆ R − . {\displaystyle
Progressive_function
of functions on the unit interval of the following form: { t ↦ r n ( t ) = sgn ( sin 2 n + 1 π t ) ; t ∈ [ 0 , 1 ] , n ∈ N } . {\displaystyle \{t\mapsto
Rademacher_system
Special case of the short-time Fourier transform
analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by
Gabor_transform
characteristic function φ ( t ) {\displaystyle \varphi (t)} . The empirical characteristic function (ECF) defined as φ n ( t ) = 1 n ∑ j = 1 n e i t X j , {\displaystyle
Empirical characteristic function
Empirical_characteristic_function
Concept in the analysis of dynamical systems
ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of
Lyapunov_function
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Cryptography algorithm
internal IV using the pseudorandom function S2V. S2V is a keyed hash based on CMAC, and the input to the function is: Additional authenticated data (zero
Block cipher mode of operation
Block_cipher_mode_of_operation
Field of medical research
Gain-of-function research (GoF research or GoFR) is medical research that genetically alters an organism in a way that may enhance the biological functions of
Gain-of-function_research
Growth of quantities at rate proportional to the current amount
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size
Exponential_growth
Linear map or polynomial function of degree one
the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a
Linear_function
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