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Statistics function
statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, Q ( x ) {\displaystyle Q(x)} is the probability
Q-function
Function in statistics
In statistics, the generalized Marcum Q-function of order ν {\displaystyle \nu } is defined as Q ν ( a , b ) = 1 a ν − 1 ∫ b ∞ x ν exp ( − x 2 + a 2
Marcum_Q-function
Model-free reinforcement learning algorithm
given infinite exploration time and a partly random policy. "Q" refers to the function that the algorithm computes: the expected reward—that is, the
Q-learning
Sigmoid shape special function
minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(an,bn)}N n = 1 for
Error_function
Statistical function that defines the quantiles of a probability distribution
{\mathcal {D}}} is the function Q {\displaystyle Q} such that Pr [ X ≤ Q ( p ) ] = p {\displaystyle \Pr \left[\mathrm {X} \leq Q(p)\right]=p} for any random
Quantile_function
Function in q-analog theory
In q-analog theory, the q {\displaystyle q} -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related
Q-gamma_function
Comparison of two distributions
distribution functions F and G, with associated quantile functions F−1 and G−1 (the inverse function of the CDF is the quantile function), the Q–Q plot draws
Q–Q_plot
Topics referred to by the same term
Look up Q in Wiktionary, the free dictionary. Q, or q, is the seventeenth letter of the English alphabet. Q may also refer to: Q (James Bond), a character
Q_(disambiguation)
Indicator function of rational numbers
Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle \mathbb {Q} } over
Dirichlet_function
Concept in combinatorics (part of mathematics)
sense that lim q → 1 ( q x ; q ) n ( 1 − q ) n = x ( n ) . {\displaystyle \lim _{q\to 1}{\frac {(q^{x};q)_{n}}{(1-q)^{n}}}=x^{(n)}.} The q-Pochhammer symbol
Q-Pochhammer_symbol
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and q ∈ N
Thomae's_function
Probability distribution
Nowak, Robert (August 7, 2003). "The Q-function". Connexions. Barak, Ohad (April 6, 2006). "Q Function and Error Function" (PDF). Tel Aviv University. Archived
Normal_distribution
In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series
Q-theta_function
x ; q ) = x ν ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ∑ k ≥ 0 ( − 1 ) k q k ( k + 1 ) / 2 x 2 k ( q ν + 1 ; q ) k ( q ; q ) k = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ x
Hahn–Exton_q-Bessel_function
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a
Jackson_q-Bessel_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
Reinforcement learning algorithms
{\displaystyle V(s)} , the action-value Q-function Q ( s , a ) , {\displaystyle Q(s,a),} the advantage function A ( s , a ) {\displaystyle A(s,a)} , or
Actor-critic_algorithm
Special functions of several complex variables
generating function, we obtain θ ( 0 , q ) 2 = ( ∑ m q m 2 ) ( ∑ n q n 2 ) = ∑ m , n q m 2 + n 2 {\displaystyle \theta (0,q)^{2}={\Bigl (}\sum _{m}q^{m^{2}}{\Bigr
Theta_function
Function studied by Ramanujan
function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by ∑ n = 1 ∞ τ ( n ) q
Ramanujan_tau_function
Computational physics simulation tool
optics and particularly for tomographic purposes. The Husimi Q distribution (called Q-function in the context of quantum optics) is one of the simplest distributions
Husimi_Q_representation
Symmetric holomorphic function
The q-expansion, where q = e π i τ {\displaystyle q=e^{\pi i\tau }} is the nome, is given by: λ ( τ ) = 16 q − 128 q 2 + 704 q 3 − 3072 q 4 + 11488 q 5
Modular_lambda_function
Noncentral generalization of the chi-squared distribution
gamma function. The Marcum Q-function Q M ( a , b ) {\displaystyle Q_{M}(a,b)} can also be used to represent the cdf. P ( x ; k , λ ) = 1 − Q k 2 ( λ
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Multiplicative function in number theory
Möbius function is ∑ n = 1 ∞ μ ( n ) q n 1 − q n = q , {\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)q^{n}}{1-q^{n}}}=q,} which converges for | q | <
Möbius_function
Mathematical function
mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general
Ramanujan_theta_function
Analytic function on the upper half-plane with a certain behavior under the modular group
eta function is defined as η ( z ) = q 1 / 24 ∏ n = 1 ∞ ( 1 − q n ) , q = e 2 π i z . {\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n})
Modular_form
Q-analog of hypergeometric series
function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is
Basic_hypergeometric_series
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Branch of mathematics
The q-Taylor expansion allows for the definition of q-analogs of all of the usual functions, such as the sine function, whose q-derivative is the q-analog
Quantum_calculus
Order-preserving mathematical function
( q i ) {\displaystyle (q_{i})} of the rational numbers, the monotonically increasing function f ( x ) = ∑ q i ≤ x a i {\displaystyle f(x)=\sum _{q_{i}\leq
Monotonic_function
Point to which functions converge in analysis
rules. q + ∞ = ∞ if q ≠ − ∞ q × ∞ = { ∞ if q > 0 − ∞ if q < 0 q ∞ = 0 if q ≠ ∞ and q ≠ − ∞ ∞ q = { 0 if q < 0 ∞ if q > 0 q ∞ = { 0 if 0 < q < 1
Limit_of_a_function
Mathematical function
the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad |q|<1.} Named
Euler_function
Ratio of polynomial functions
P} and Q {\displaystyle Q} are polynomial functions of x {\displaystyle x} and Q {\displaystyle Q} is not the zero function. The domain of f {\displaystyle
Rational_function
Extension of the factorial function
gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master theorem Spouge's
Gamma_function
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Special mathematical function defined as sin(x)/x
needed] x n = q − q − 1 − 2 3 q − 3 − 13 15 q − 5 − 146 105 q − 7 − ⋯ , {\displaystyle x_{n}=q-q^{-1}-{\frac {2}{3}}q^{-3}-{\frac {13}{15}}q^{-5}-{\frac
Sinc_function
Mathematical finite graph-associated function
that for regular graphs the zeta function is a rational function. If G {\displaystyle G} is a q + 1 {\displaystyle q+1} -regular graph with adjacency
Ihara_zeta_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
Meromorphic function on the complex plane
q ( Q ) = 1 {\displaystyle \textstyle q(\mathbb {Q} )=1} , so that the complete Riemann zeta function takes the form Λ ( Q , s ) := γ ( Q , s ) L ( Q
L-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
Function whose domain is the positive integers
tau function, is defined by its generating function identity: ∑ n ≥ 1 τ ( n ) q n = q ∏ n ≥ 1 ( 1 − q n ) 24 . {\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod
Arithmetic_function
Mathematical function
Im(τ) > 0, let q = e2πiτ; then the eta function is defined by, η ( τ ) = e π i τ 12 ∏ n = 1 ∞ ( 1 − e 2 n π i τ ) = q 1 24 ∏ n = 1 ∞ ( 1 − q n ) . {\displaystyle
Dedekind_eta_function
Arithmetic function related to the divisors of an integer
involving the divisor function is: ∑ n = 1 ∞ q n σ a ( n ) = ∑ n = 1 ∞ ∑ j = 1 ∞ n a q j n = ∑ n = 1 ∞ n a q n 1 − q n = ∑ n = 1 ∞ Li − a ( q n ) {\displaystyle
Divisor_function
Polynomial function of degree 4
In algebra, a quartic function is a function of the form f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , {\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,} where
Quartic_function
Class of mathematical functions
Jacobi's theta functions: ℘ ( z , τ ) = ( π θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( π z , q ) θ 1 ( π z , q ) ) 2 − π 2 3 ( θ 2 4 ( 0 , q ) + θ 3 4 ( 0 , q ) ) {\displaystyle
Weierstrass_elliptic_function
Number of integers coprime to and less than n
generating function is ∑ n = 1 ∞ φ ( n ) q n 1 − q n = q ( 1 − q ) 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}}
Euler's_totient_function
Iterative method for finding maximum likelihood estimates in statistical models
the function: F ( q , θ ) := E q [ log L ( θ ; x , Z ) ] + H ( q ) , {\displaystyle F(q,\theta ):=\operatorname {E} _{q}[\log L(\theta ;x,Z)]+H(q),}
Expectation–maximization algorithm
Expectation–maximization_algorithm
Solutions of Legendre's differential equation
function P on the Wolfram functions site. Legendre function Q on the Wolfram functions site. Associated Legendre function P on the Wolfram functions site
Legendre_function
Statistical method of dividing data into equal-sized intervals for analysis
distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of the cumulative
Quantile
Mathematical function
{(q^{a}+q^{2p-a})(q^{a+p}+q^{p-a})}{1-q^{3p}+{\cfrac {q^{p}(q^{a}+q^{3p-a})(q^{a+2p}+q^{p-a})}{1-q^{5p}+{\cfrac {q^{2p}(q^{a}+q^{4p-a})(q^{a+3p}+q
Jacobi_elliptic_functions
Q-analog of the ordinary derivative
forms of q-derivative, see Chung et al. (1994). The q-derivative of a function f(x) is defined as ( d d x ) q f ( x ) = f ( q x ) − f ( x ) q x − x . {\displaystyle
Q-derivative
Formal power series
special functions and enumerate partition functions. In particular, we recall that the partition function p(n) is generated by the reciprocal infinite q-Pochhammer
Generating_function
Special function occurring in problems possessing elliptic symmetry
mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q cos ( 2 x
Mathieu_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
= 0 n q ℓ ( a b q ℓ , a c q ℓ , a d q ℓ ; q ) n − ℓ × ( q − n , a b c d q n − 1 ; q ) ℓ ( q ; q ) ℓ ∏ j = 0 ℓ − 1 ( 1 − 2 a q j cos θ + a 2 q 2 j )
Askey–Wilson_polynomials
Wigner distribution function in physics as opposed to in signal processing
Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of ψ {\displaystyle \psi } is
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Mathematical constants
! ! ! ! 4 n Γ ( n + 1 q ) = Γ ( 1 q ) ( q n − ( q − 1 ) ) ! ( q ) q n Γ ( n + p q ) = Γ ( p q ) 1 q n ∏ k = 1 n ( k q + p − q ) {\displaystyle {\begin{aligned}\Gamma
Particular values of the gamma function
Particular_values_of_the_gamma_function
Mathematical concept
Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible algebraic curve over
Global_field
Function used in Lagrangian mechanics
quadratic functions q ↦ R ( q ˙ ) = 1 2 q ˙ ⋅ V q ˙ {\displaystyle q\mapsto R({\dot {q}})={\frac {1}{2}}{\dot {q}}\cdot \mathbb {V} {\dot {q}}} to dissipation
Rayleigh_dissipation_function
Generalisation of the generalised hypergeometric function pFq(z)
the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935): p Ψ q [ ( a 1 , A 1 ) ( a 2 , A 2 )
Fox–Wright_function
Function defined by a hypergeometric series
(q)_{n}={\begin{cases}1&n=0\\q(q+1)\cdots (q+n-1)&n>0\end{cases}}} The series terminates if either a or b is a nonpositive integer, in which case the function reduces to
Hypergeometric_function
Function returning minus 1, zero or plus 1
In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether
Sign_function
the function itself is bounded in the UHP (so we can use Q ( z ) = 1 {\displaystyle Q(z)=1} ), and if a is negative then the function equals 1/Q(z) with
Bounded_type_(mathematics)
Function for Heun's differential equation
In mathematics, the local Heun function H ℓ ( a , q ; α , β , γ , δ ; z ) {\displaystyle H\ell (a,q;\alpha ,\beta ,\gamma ,\delta ;z)} is the solution
Heun_function
Information-theoretic measure
Let P {\displaystyle P} and Q {\displaystyle Q} be probability density functions of p {\displaystyle p} and q {\displaystyle q} with respect to r {\displaystyle
Cross-entropy
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Generalization of the Meijer G-function and the Fox–Wright function
Meijer G-function G p , q m , n ( a 1 , … , a p b 1 , … , b q | z ) = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j − s ) ∏ j = 1 n Γ ( 1 − a j + s ) ∏ j = m + 1 q Γ ( 1
Fox_H-function
Quantum mechanical model
Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have Q n ( x
Quantum_harmonic_oscillator
Multivariate functions can be written using univariate functions and summing
functions ϕ q , p {\displaystyle \phi _{q,p}} are continuous and universal, that is, independent of f {\displaystyle f} , while the outer functions Φ
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Number of partitions of an integer
Leonhard Euler. The formulation of Euler's generating function is a special case of a q {\displaystyle q} -Pochhammer symbol and is similar to the product
Partition function (number theory)
Partition_function_(number_theory)
Family of solutions to related differential equations
functions. Anger function Bessel polynomials Bessel–Clifford function Bessel–Maitland function Fourier–Bessel series Hahn–Exton q-Bessel function Hankel transform
Bessel_function
One-way cryptographic tool
totient function of n {\displaystyle n} ) is the trapdoor: f ( x ) = x e mod n . {\displaystyle f(x)=x^{e}\mod n.} If the factorization of n = p q {\displaystyle
Trapdoor_function
Mathematical technique for manipulating signals
amplitude modulation rely heavily on I/Q. The term alternating current applies to a voltage vs. time function that is sinusoidal with a frequency f. When
In-phase and quadrature components
In-phase_and_quadrature_components
Function in economics
production function. For the simple case of a good that is produced with two inputs, the function is of the form q = Min ( z 1 a , z 2 b ) {\displaystyle q
Leontief_production_function
Resonator damping parameter
depending on their function and design. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q near 1⁄2. Clocks
Q_factor
Mathematical function
time. The logistic function, with maximum growth rate at time M {\displaystyle M} , is the case where Q = ν = 1 {\displaystyle Q=\nu =1} . A particular
Generalised_logistic_function
Q-analog in combinatorial mathematics
mathematics, a q-exponential is a q-analog of the exponential function, namely the eigenfunction of a q-derivative. There are many q-derivatives, for
Q-exponential
Type of infinitesimal in calculus
differential d Q {\displaystyle dQ} for some differentiable function Q {\displaystyle Q} in an orthogonal coordinate system (hence Q {\displaystyle Q} is a multivariable
Exact_differential
Function in thermodynamics and statistical physics
partition function is defined as Z = 1 h 3 ∫ e − β H ( q , p ) d 3 q d 3 p , {\displaystyle Z={\frac {1}{h^{3}}}\int e^{-\beta H(q,p)}\,d^{3}q\,d^{3}p,}
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
create a distributed point function for P a , 1 ( x ) {\displaystyle P_{a,1}(x)} and send the resulting two keys q {\displaystyle q} and r {\displaystyle r}
Distributed_point_function
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
Function in number theory given by Srinivasa Ramanujan
function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum
Ramanujan's_sum
another function and Dqg denotes its q-derivative, we can formally write ∫ f ( x ) D q g d q x = ( 1 − q ) x ∑ k = 0 ∞ q k f ( q k x ) D q g ( q k x ) =
Jackson_integral
Generalization of the Euler gamma function and the Barnes G-function
related to the q-gamma function, and triple gamma functions Γ 3 {\displaystyle \Gamma _{3}} are related to the elliptic gamma function. For ℜ a i > 0
Multiple_gamma_function
Programming language for quantum algorithms
Computer programming portal Free and open-source software portal Q# (pronounced Q sharp) is a domain-specific programming language used for expressing
Q_Sharp
The Stone–Geary utility function takes the form U = ∏ i ( q i − γ i ) β i {\displaystyle U=\prod _{i}(q_{i}-\gamma _{i})^{\beta _{i}}} where U {\displaystyle
Stone–Geary_utility_function
Mathematical table used in logic
definitions of each of the 6 possible 2-input logic gate functions of two Boolean variables P and Q: For binary operators, a condensed form of truth table
Truth_table
Formulation of classical mechanics using momenta
T ( q , q ˙ , t ) − V ( q , q ˙ , t ) ) ∂ q ˙ i q ˙ i ) − ( T ( q , q ˙ , t ) − V ( q , q ˙ , t ) ) = ∑ i = 1 n ( ∂ T ( q , q ˙ , t ) ∂ q ˙ i q ˙ i −
Hamiltonian_mechanics
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Computational operation
function, ⌊ ⌋ {\displaystyle \lfloor \,\rfloor } is the floor function (rounding down), and a | n | ∈ Q {\displaystyle {\frac {a}{|n|}}\in \mathbb {Q}
Modulo
deviations rate function. If P and Q are probability distributions on the real line, such that P is absolutely continuous with respect to Q, i.e. P << Q, and whose
Kullback's_inequality
Type of mathematical function
equations of L-functions. Let χ {\displaystyle \chi } be a primitive character modulo q {\displaystyle q} , with q > 1 {\displaystyle q>1} . There are
Dirichlet_L-function
Function valued in a vector space; typically a real or complex one
partial derivative of a vector function a with respect to a scalar variable q is defined as ∂ a ∂ q = ∑ i = 1 n ∂ a i ∂ q e i {\displaystyle {\frac {\partial
Vector-valued_function
Ranking function used by search engines
functions with slightly different components and parameters. One of the most prominent instantiations of the function is as follows. Given a query Q,
Okapi_BM25
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
Probability of shared birthdays
n = 1 + Q(M), where Q ( M ) = ∑ k = 1 M M ! ( M − k ) ! M k . {\displaystyle Q(M)=\sum _{k=1}^{M}{\frac {M!}{(M-k)!M^{k}}}.} The function Q ( M ) = 1
Birthday_problem
Average uncertainty in variable's states
probabilities: lim q → 0 + H 2 ( 1 − q , q ) = 0 {\displaystyle \lim _{q\to 0^{+}}\mathrm {H} _{2}(1-q,q)=0} . It was shown that any function H {\displaystyle
Entropy_(information_theory)
Generalization of the hypergeometric function
G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1): G p , q m , n ( a 1 , … , a p b 1 , … , b q |
Meijer_G-function
Mathematical term
arithmetic function f: ∑ n ≥ 1 f ( n ) q n 1 + q n = ∑ n ≥ 1 f ( n ) q n 1 − q n − ∑ n ≥ 1 2 f ( n ) q 2 n 1 − q 2 n = L f ( q ) − 2 ⋅ L f ( q 2 ) . {\displaystyle
Lambert_series
In computability theory, a semicomputable function is a partial function f : Q → R {\displaystyle f:\mathbb {Q} \rightarrow \mathbb {R} } that can be approximated
Semicomputable_function
mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) =
Local_zeta_function
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