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Topics referred to by the same term
mathematics, by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related
Sigma_function
Eighteenth letter of the Greek alphabet
theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function. In applied mathematics, σ(T) denotes the
Sigma
Mathematical function
{\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right).} Gaussian functions are widely used in
Gaussian_function
Arithmetic function related to the divisors of an integer
the number-of-divisors function (OEIS: A000005). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is
Divisor_function
Concept in theoretical computer science
computable function. The score function, Σ : N → N {\displaystyle \Sigma :\mathbb {N} \to \mathbb {N} } , is defined so that Σ ( n ) {\displaystyle \Sigma (n)}
Busy_beaver
Mathematical functions related to Weierstrass's elliptic function
Weierstrass sigma, zeta, and eta functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are
Weierstrass sigma, zeta, and eta functions
Weierstrass_sigma,_zeta,_and_eta_functions
Mapping function
𝜎-additive. Every 𝜎-additive function is additive but not vice versa, as shown below. Suppose that in addition to a sigma algebra A , {\textstyle {\mathcal
Sigma-additive_set_function
Topics referred to by the same term
Weierstrass function may refer to: Weierstrass function (nowhere-differentiable function) Weierstrass sigma, zeta, and eta functions Weierstrass elliptic
Weierstrass_function
Function that is holomorphic on the whole complex plane
sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and
Entire_function
Shorthand used in statistics
probability function, Χ is an observation from a normally distributed random variable, μ (mu) is the mean of the distribution, and σ (sigma) is its standard
68–95–99.7_rule
Algebraic structure of set algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In
Σ-algebra
Business process improvement technique
Six Sigma (6σ) is a set of techniques and tools for process improvement. It was introduced by American engineer Bill Smith while working at Motorola in
Six_Sigma
Probability distribution
{\displaystyle V(x;\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{{\sqrt {2\pi }}\,\sigma }},} where Re[w(z)] is the real part of the Faddeeva function evaluated
Voigt_profile
Kind of mathematical function
with respective σ-algebras Σ X {\displaystyle \Sigma _{X}} and Σ Y {\displaystyle \Sigma _{Y}} . A function f : X → Y {\displaystyle f:X\to Y} is said to
Measurable_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
Topics referred to by the same term
Harish-Chandra's σ function Weierstrass sigma function Sigma additivity Sigma (album) Sigma (DJs), a British drum and bass duo Universal Sigma, a Japanese record
Sigma_(disambiguation)
Measure of variation in statistics
probability density function of f ( x , μ , σ 2 ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 , {\displaystyle f\left(x,\mu ,\sigma ^{2}\right)={\frac {1}{\sigma {\sqrt {2\pi
Standard_deviation
Class of mathematical functions
_{i=1}^{n}{\frac {\sigma (u-a_{i})}{\sigma (u-b_{i})}}\quad c\in \mathbb {C} } where σ {\displaystyle \sigma } is the Weierstrass sigma function and a i , b
Weierstrass_elliptic_function
Method for converting signals between digital and analog
Delta–sigma (ΔΣ; or sigma–delta, ΣΔ) modulation is an oversampling method for encoding signals into low bit depth digital signals at a very high sample-frequency
Delta-sigma_modulation
to the origin (zero point) Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers coprime to
List of mathematical functions
List_of_mathematical_functions
Effect of variables' uncertainties on the uncertainty of a function based on them
_{1}^{2}&\sigma _{12}&\sigma _{13}&\cdots \\\sigma _{21}&\sigma _{2}^{2}&\sigma _{23}&\cdots \\\sigma _{31}&\sigma _{32}&\sigma _{3}^{2}&\cdots
Propagation_of_uncertainty
Probability distribution
cumulative distribution function is F X ( x ) = Φ ( ln x − μ σ ) {\displaystyle F_{X}(x)=\Phi {\left({\frac {\ln x-\mu }{\sigma }}\right)}} where Φ {\displaystyle
Log-normal_distribution
Statistics function
{y-\mu }{\sigma }}} . Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also
Q-function
Function equal to the product of its values on coprime factors
{\displaystyle n} is not square-free σ k ( n ) {\displaystyle \sigma _{k}(n)} : the divisor function, which is the sum of the k {\displaystyle k} -th powers
Multiplicative_function
Class of functions behaving "like" periodic functions
Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Bloch's theorem
Quasiperiodic_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
Form of the utility function
economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes
Exponential_utility
Binary function non degenerative defined between the point of twist of an abelian variety
corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function. Choose an elliptic curve E defined
Weil_pairing
Protein needed for initiation of transcription in prokaryotes
starvation/stationary phase sigma factor σ54 (RpoN) – the nitrogen-limitation sigma factor There are also anti-sigma factors that inhibit the function of sigma factors and
Sigma_factor
Method of solution to differential equations
{\boldsymbol {\sigma }}}.} Suppose that the linear differential operator L is the Laplacian, ∇2, and that there is a Green's function G for the Laplacian
Green's_function
Probability distribution
B( ) is the beta function. If W = μ + σ ( Y − 1 − 1 ) γ , σ > 0 , γ > 0 , {\displaystyle W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,}
Pareto_distribution
Function in statistics
is the inverse of the standard logistic function σ ( x ) = 1 / ( 1 + e − x ) {\displaystyle \textstyle \sigma (x)=1/(1+e^{-x})} , so the logit is defined
Logit
Probability distribution
density function of the Rayleigh distribution is f ( x ; σ ) = x σ 2 e − x 2 / ( 2 σ 2 ) , x ≥ 0 , {\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/(2\sigma
Rayleigh_distribution
Addition of several numbers or other values
also ways to generalize the use of many sigma notations. For example, one writes double summation as two sigma notations with different dummy variables
Summation
Integral transform useful in probability theory, physics, and engineering
{\displaystyle F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,} which is the Fourier transform of the function f ( t ) e − σ t
Laplace_transform
Israeli self-propelled howitzer
The SIGMA 155 is a 155mm self-propelled howitzer manufactured by the Israeli defense company Elbit Systems being introduced into service in the Israel
SIGMA_155
Method used in statistics, pattern recognition, and other fields
discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method
Linear_discriminant_analysis
Generalization of the one-dimensional normal distribution to higher dimensions
k\times k} matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} , such that the characteristic function of X {\displaystyle \mathbf {X} } is φ X ( u ) = exp
Multivariate normal distribution
Multivariate_normal_distribution
Fourier transform of the probability density function
probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Function studied by Ramanujan
n ∈ N {\displaystyle n\in \mathbb {N} } , the divisor function σ k ( n ) {\displaystyle \sigma _{k}(n)} is the sum of the k {\displaystyle k} th powers
Ramanujan_tau_function
Antagonists of a youth's slang
anti-sigma factors include FlgM, DnaK, RssB, and HscC. Inner membrane-bound anti-sigma factors, also called extra-cytoplasmic function (ECF) anti-sigma factors
Anti-sigma_factors
Probability density function
n,{\bar {x}},\sigma )=N\cdot {\begin{cases}\exp(-{\frac {(x-{\bar {x}})^{2}}{2\sigma ^{2}}}),&{\mbox{for }}{\frac {x-{\bar {x}}}{\sigma }}>-\alpha \\A\cdot
Crystal_Ball_function
Family of probability distributions
{\displaystyle \ \xi \ } and σ . {\displaystyle \ \sigma ~.} The probability density function of the standardized distribution is f ( s ; ξ ) = { e
Generalized extreme value distribution
Generalized_extreme_value_distribution
Statistical measure of how far values spread from their average
variable with itself, and it is often represented by σ 2 {\displaystyle \sigma ^{2}} , s 2 {\displaystyle s^{2}} , Var ( X ) {\displaystyle \operatorname
Variance
Probability theory and statistics concept
trivial sigma algebra G = { ∅ , Ω } {\displaystyle {\mathcal {G}}=\{\emptyset ,\Omega \}} , the conditional probability is the constant function P ( A ∣
Conditional probability distribution
Conditional_probability_distribution
Probability distribution
zero. Using the σ {\displaystyle \sigma } parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given
Half-normal_distribution
Family of probability distributions often used to model tails or extreme values
III . The cumulative distribution function of X ∼ GPD ( μ , σ , ξ ) {\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )} ( μ ∈ R {\displaystyle \mu
Generalized Pareto distribution
Generalized_Pareto_distribution
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Mathematical theory of data types
notation σ → τ {\displaystyle \sigma \to \tau } is the type of a function which takes a parameter of type σ {\displaystyle \sigma } and returns a term of type
Type_theory
Business management method
science. While the tools and order used in Six Sigma require a process to be in place and functioning, DFSS has the objective of determining the needs
Design_for_Six_Sigma
Matrices important in quantum mechanics and the study of spin
sigma _{j},\sigma _{k}\right]+\{\sigma _{j},\sigma _{k}\}&=(\sigma _{j}\sigma _{k}-\sigma _{k}\sigma _{j})+(\sigma _{j}\sigma _{k}+\sigma _{k}\sigma
Pauli_matrices
Statistical model
theorem, involves the function σ {\displaystyle \sigma } defined by σ ( h ) = E [ ( X ( t + h ) − X ( t ) ) 2 ] {\displaystyle \sigma (h)={\sqrt {{\mathbb
Gaussian_process
Method of estimating the parameters of a statistical model, given observations
_{2})}{\sigma _{1}\sigma _{2}}}+{\frac {(y_{2}-\mu _{2})^{2}}{\sigma _{2}^{2}}}\right)\right]} In this and other cases where a joint density function exists
Maximum_likelihood_estimation
Probability distribution
function (PDF) is given by f Y ( x ; μ , σ 2 ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 + 1 2 π σ 2 e − ( x + μ ) 2 2 σ 2 {\displaystyle f_{Y}(x;\mu ,\sigma
Folded_normal_distribution
Statement in probability theory
situation when one random variable is a function of another by the inclusion of the σ {\displaystyle \sigma } -algebras generated by the random variables
Doob–Dynkin_lemma
Type of probability distribution
f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\,{\frac {\varphi ({\frac {x-\mu }{\sigma }})}{\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})}}}
Truncated_normal_distribution
Extension of evidence theory to continuous variables of interest
_{1}(\Sigma _{11})^{-1}\\-(\Sigma _{11})^{-1}\\\Sigma _{21}(\Sigma _{11})^{-1}\end{array}}&{\begin{array}{*{20}c}\mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma
Linear_belief_function
Statistical measure of process capability
probability density function Φ ( σ ) = 1 2 π ∫ − σ σ e − t 2 / 2 d t {\displaystyle \Phi (\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-\sigma }^{\sigma }e^{-t^{2}/2}\
Process_capability_index
Solution concept of a non-cooperative game
{\displaystyle {\text{Gain}}_{i}(\sigma ,a)=\max\{0,u_{i}(a,\sigma _{-i})-u_{i}(\sigma _{i},\sigma _{-i})\}.} The gain function represents the benefit a player
Nash_equilibrium
Construct related to weighted sums and averages
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result
Weight_function
Probability distribution
function is f ( x ∣ ν , σ ) = x σ 2 exp ( − ( x 2 + ν 2 ) 2 σ 2 ) I 0 ( x ν σ 2 ) H ( x ) , {\displaystyle f(x\mid \nu ,\sigma )={\frac {x}{\sigma ^{2}}}\exp
Rice_distribution
Probability distribution
) . {\displaystyle \gamma _{2}={\frac {2}{\ \sigma ^{2}\ }}\left(1-\mu \ \sigma \ \gamma _{1}-\sigma ^{2}\right)~.} The entropy is given by: S = ln
Chi_distribution
Mathematical model of ferromagnetism in statistical mechanics
σ ) {\displaystyle Z_{\beta }=\sum _{\sigma }e^{-\beta H(\sigma )}} is the partition function. For a function f {\displaystyle f} of the spins ("observable")
Ising_model
Tool in multivariate statistical analysis
exponential covariance function lim ν → ∞ C ν ( d ) = σ 2 exp ( − d 2 2 ρ 2 ) . {\displaystyle \lim _{\nu \rightarrow \infty }C_{\nu }(d)=\sigma ^{2}\exp \left(-{\frac
Matérn_covariance_function
Continuous stochastic process
\left[\exp \left(2\sigma W_{t}-\sigma ^{2}t\right)\mid {\mathcal {F}}_{s}\right]=e^{\sigma ^{2}(t-s)}\exp \left(2\sigma W_{s}-\sigma ^{2}s\right),\quad
Geometric_Brownian_motion
\|} . The σ {\displaystyle \sigma } function of G {\displaystyle G} is then defined as σ ( x ) = ‖ X ‖ {\displaystyle \sigma (x)=\|X\|} for x = k exp
Harish-Chandra's Schwartz space
Harish-Chandra's_Schwartz_space
Generalizations of the Riemann zeta function
{\displaystyle {\mathfrak {Sh}}_{n,m}=\{\sigma \in S_{m}\mid \sigma (1)<\cdots <\sigma (n),\sigma (n+1)<\cdots <\sigma (m)\}} and S m {\displaystyle S_{m}}
Multiple_zeta_function
Integral of the Gaussian function, equal to sqrt(π)
also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich
Gaussian_integral
Mathematical model in materials science
the function σ 1 − σ 3 2 = σ 1 + σ 3 2 sin ϕ + c cos ϕ {\displaystyle {\cfrac {\sigma _{1}-\sigma _{3}}{2}}={\cfrac {\sigma _{1}+\sigma _{3}}{2}}~\sin
Mohr–Coulomb_theory
Covariance and correlation
_{t_{1}}\right)}}\left(X_{t_{2}}-\mu _{t_{2}}\right)\right]}{\sigma _{X}(t_{1})\sigma _{X}(t_{2})}}} If the function ρ X X {\displaystyle \rho _{XX}} is well-defined
Cross-correlation
physical constants Particular values of the Riemann zeta function Particular values of the gamma function Physical constant Both i and −i are roots of this equation
List of mathematical constants
List_of_mathematical_constants
Analytic function in mathematics
{s+2m+1}{\sigma +2m+1}}T_{m+1,n}(s)\right|,} with σ = Re(s). A modern numerical algorithm is the Odlyzko–Schönhage algorithm. The zeta function occurs in
Riemann_zeta_function
Mathematical concept
result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be
Infinite_product
Function that attains finitely many values
is defined on the space ( X , Σ ) {\displaystyle (X,\Sigma )} , the integral of a simple function f : X → R {\displaystyle f\colon X\to \mathbb {R} } with
Simple_function
x\mapsto g(f(x))} is a measurable function with respect to Σ {\displaystyle \Sigma } and the usual Borel σ {\displaystyle \sigma } -algebra on K . {\displaystyle
Weakly_measurable_function
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
delta (named after Leopold Kronecker), is a function of two variables, usually non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
Kronecker_delta
Type of probability distribution
{\displaystyle s_{1}={\sqrt {\Sigma _{11}}}\qquad s_{2}={\sqrt {\Sigma _{22}}}\qquad \rho =\Sigma _{12}/(s_{1}s_{2})=\Sigma _{21}/(s_{1}s_{2})} then t 2
Hotelling's T-squared distribution
Hotelling's_T-squared_distribution
Property of some mathematical functions
cumulative distribution function at probability level p {\displaystyle p} , σ x 2 , σ y 2 {\displaystyle \sigma _{x}^{2},\sigma _{y}^{2}} are the individual
Subadditivity
Statistical measure of variability
)=P\left(\left|{\frac {X-\mu }{\sigma }}\right|\leq {\frac {\operatorname {MAD} }{\sigma }}\right)=P\left(|Z|\leq {\frac {\operatorname {MAD} }{\sigma }}\right).} Therefore
Median_absolute_deviation
Model of intermolecular interactions
{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right],} where r is the distance between
Lennard-Jones_potential
Probability distribution
\sigma ^{2}}^{-1}(i/K)\right)\right)^{n},} where P {\textstyle P} is the standard logistic function, and Φ μ , σ 2 − 1 {\textstyle \Phi _{\mu ,\sigma ^{2}}^{-1}}
Logit-normal_distribution
Identity in Itô calculus analogous to the chain rule
dX_{t}=\mu _{t}\ dt+\sigma _{t}\ dB_{t},} where Bt is a Wiener process and the functions μ t , σ t {\displaystyle \mu _{t},\sigma _{t}} are deterministic
Itô's_lemma
Uniform distribution on an interval
{\displaystyle \mu } and variance σ 2 , {\displaystyle \sigma ^{2},} the probability density function of the continuous uniform distribution is f ( x ) =
Continuous uniform distribution
Continuous_uniform_distribution
Type of mathematical function
function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear function is a function defined
Piecewise_linear_function
Chaperone protein
The sigma-1 receptor (σ1R), one of two sigma receptor subtypes, is a chaperone protein at the endoplasmic reticulum (ER) that modulates calcium signaling
Sigma-1_receptor
Characteristic property of holomorphic functions
x + σ 2 ∂ y {\displaystyle \nabla \equiv \sigma _{1}\partial _{x}+\sigma _{2}\partial _{y}} . The function f = u + J v {\displaystyle f=u+Jv} is considered
Cauchy–Riemann_equations
Mathematical function for thermoelastic strain energy density
_{z}+2\sigma _{xy}\epsilon _{xy}+2\sigma _{yz}\epsilon _{yz}+2\sigma _{xz}\epsilon _{xz})} A strain energy density function is used to define a hyperelastic
Strain energy density function
Strain_energy_density_function
Mapping of mathematical formulas to a particular meaning
{\mathcal {A}}=(A,\sigma ,I)} consisting of a domain A , {\displaystyle A,} a signature σ , {\displaystyle \sigma ,} and an interpretation function I {\displaystyle
Structure (mathematical logic)
Structure_(mathematical_logic)
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Representation of mechanical stress at every point within a deformed 3D object
sigma _{1}+\sigma _{2}+\sigma _{3}\\I_{2}&=\sigma _{1}\sigma _{2}+\sigma _{2}\sigma _{3}+\sigma _{3}\sigma _{1}\\I_{3}&=\sigma _{1}\sigma _{2}\sigma
Cauchy_stress_tensor
Lower bound on variance of an estimator
V={\frac {\partial }{\partial \sigma ^{2}}}\log \left[L(\sigma ^{2},X)\right]} where L {\displaystyle L} is the likelihood function. Thus in this case, log
Cramér–Rao_bound
Describes the sum of independent normal and exponential random variables
{erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right),} where erfc is the complementary error function defined as erfc ( x ) = 1 −
Exponentially modified Gaussian distribution
Exponentially_modified_Gaussian_distribution
Mathematics formula
_{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{\sigma (i)i}} where sgn {\displaystyle \operatorname {sgn} } is the sign function of
Leibniz formula for determinants
Leibniz_formula_for_determinants
Computational statistics technique
region under the graph of its density function. Note that this property can be extended to N-dimension functions. The algorithm, which was used by John
Rejection_sampling
Partial differential equation in mathematical finance
price of the option as a function of stock price S and time t, r is the risk-free interest rate, and σ {\displaystyle \sigma } is the volatility of the
Black–Scholes_equation
Energy quantum particles contribute to themselves
In quantum field theory, self-energy Σ {\displaystyle \Sigma } is the energy of a particle due to the interaction with its own field. In electrostatics
Self-energy
Mathematical series
}{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}\end{aligned}}} where σa(n) is the divisor function. By specialization to the divisor function d = σ0 we
Dirichlet_series
Result about when a matrix can be diagonalized
operator A as an integral of the coordinate function over the operator's spectrum σ ( A ) {\displaystyle \sigma (A)} with respect to a projection-valued
Spectral_theorem
Type of image blur produced by a Gaussian function
Gaussian function in one dimension is G ( x ) = 1 2 π σ 2 e − x 2 2 σ 2 {\displaystyle G(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}
Gaussian_blur
Class of cell surface receptors
result, contemporary classifications recognize only sigma-1 and sigma-2 receptors. The function of these receptors is poorly understood. Drugs known
Sigma_receptor
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SIGMA FUNCTION
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