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subtraction logarithms or Gaussian logarithms can be utilized to find the logarithms of the sum and difference of a pair of values whose logarithms are known
Gaussian_logarithm
Tool for a fast finite-field arithmetic
\alpha } : Gaussian logarithm Irish logarithm, a similar technique derived empirically by Percy Ludgate Finite field arithmetic Logarithm table Zech,
Zech's_logarithm
Generalization of the one-dimensional normal distribution to higher dimensions
theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Probability distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued
Normal_distribution
Mathematical function
{\textstyle \alpha =-{\tfrac {1}{2c^{2}}}} ) The Gaussian functions are thus those functions whose logarithm is a concave quadratic function. The parameter
Gaussian_function
Gaussian logarithms (also known as addition and subtraction logarithms) Gauss congruence for integer sequences Gauss–Krüger coordinate system Gaussian grid
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Collection of mathematical functions originated by Constantino Tsallis
for the logarithm, in the limit N = 1 1 − q → ∞ {\displaystyle N={\frac {1}{1-q}}\to \infty } . Tsallis entropy Tsallis distribution q-Gaussian q-exponential
Tsallis_statistics
Theorem that tells the maximum rate at which information can be transmitted
archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication
Shannon–Hartley_theorem
Computer representation of real numbers
arithmetic (LI) and symmetric level-index arithmetic (SLI) Gaussian logarithm Zech's logarithm ITU-T G.711 A-law algorithm μ-law algorithm Slide rule Lee
Logarithmic_number_system
Continuous probability distribution
The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as
Normal-inverse Gaussian distribution
Normal-inverse_Gaussian_distribution
1 minus the cosine of an angle
in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 or in a more compact method for sight reduction since
Versine
Generalization of Gaussian distribution
The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a
Q-Gaussian_distribution
Mathematical function, denoted exp(x) or e^x
{\displaystyle \exp(x+y)=\exp x\cdot \exp y} . Its inverse function, the natural logarithm, ln {\displaystyle \ln } or log {\displaystyle \log } , converts
Exponential_function
Surname list
some well-known logarithm tables (and thereby sometimes gets confused with Carl Friedrich Gauss, who introduced Gaussian logarithms earlier on) Christian
Gauss_(surname)
Mathematical constant
In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in
Natural_logarithm_of_2
Application of a function to each point in a data set
unit, it would be common to transform each person's income value by the logarithm function. Guidance for how data should be transformed, or whether a transformation
Data transformation (statistics)
Data_transformation_(statistics)
Measure of the level of acidity or basicity of an aqueous solution
negative decimal logarithm of", and is used in the term pKa for acid dissociation constants, so pH is "the negative decimal logarithm of H+ ion concentration"
PH
Topics referred to by the same term
of several places Logarithm, a mathematical function Log file, a computer file in which events are recorded Laplacian of Gaussian or LoG, an algorithm
Log
Replacing a number with a simpler value
arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed
Rounding
Multivalued function in mathematics
mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation
Lambert_W_function
Mathematical limit applied in statistical physics
{\displaystyle J_{ij}} ) and not its logarithm which we wanted to average, the resulting integral (assuming a Gaussian distribution) is just ∫ d J i j e
Replica_trick
Principal square root of minus 1
the complex plane called the Gaussian integers. The sum, difference, or product of Gaussian integers is also a Gaussian integer: ( a + b i ) + ( c + d
Imaginary_unit
Stochastic process modeling random walk with friction
Ornstein–Uhlenbeck process is a Gaussian, time-homogeneous Markov process. For θ > 0 {\displaystyle \theta >0} , it admits the invariant Gaussian distribution X ∼ N
Ornstein–Uhlenbeck_process
Signal processing computational method
subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each other
Independent component analysis
Independent_component_analysis
Scientific study of digital information
binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. In
Information_theory
Generalization of the standard Boltzmann–Gibbs entropy
Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution. The concept was introduced in 1988 by Constantino Tsallis
Tsallis_entropy
Extension of the factorial function
notation for logarithms. All instances of log ( x ) {\displaystyle \log(x)} without a subscript base should be interpreted as a natural logarithm, also commonly
Gamma_function
Method of interpolation
Kriging (/ˈkriːɡɪŋ/), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances
Kriging
Concept in statistics
In statistics, a Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF
Gaussian_random_field
{\displaystyle \epsilon } -covering, the entropy of T {\displaystyle T} is the logarithm of the minimum number of balls of radius ϵ {\displaystyle \epsilon } required
Dudley's_entropy_integral
Mathematical functions
complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle
Lemniscate_elliptic_functions
Application of mathematical and statistical methods in finance
discrete random walk. Bachelier modeled the time series of changes in the logarithm of stock prices as a random walk in which the short-term changes had a
Mathematical_finance
Method for evaluating indefinite integrals
integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because
Risch_algorithm
Transforming data by taking the logarithm
unit, it would be common to transform each person's income value by the logarithm function. Guidance for how data should be transformed, or whether a transformation
Log transformation (statistics)
Log_transformation_(statistics)
Probabilistic classification algorithm
D)) based on the observation that p(S | D) + p(¬S | D) = 1. Taking the logarithm of all these ratios, one obtains: ln p ( S ∣ D ) p ( ¬ S ∣ D ) = ln
Naive_Bayes_classifier
Difference between logarithm and harmonic series
notation for logarithms. All instances of log ( x ) {\displaystyle \log(x)} without a subscript base should be interpreted as a natural logarithm, also commonly
Euler's_constant
Post-quantum signature scheme
Vadim Lyubashevsky in their 2013 paper "Lattice Signature and Bimodal Gaussians". In cryptography, a digital signature ensures that a message is authentically
BLISS_signature_scheme
Probability distribution
distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally
Log-normal_distribution
Function that can be written as a sum over prime factors
multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 in the OEIS). For example: a0(4) = 2 + 2 = 4 a0(20)
Additive_function
Identity in Itô calculus analogous to the chain rule
non-Gaussian, but the non-Gaussian parts decay faster than the Gaussian part, and at the d t → 0 {\displaystyle dt\to 0} limit, only the Gaussian part
Itô's_lemma
Fundamental theorem in probability theory and statistics
{\displaystyle n\to \infty } . The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random
Central_limit_theorem
Process forming a path from many random steps
of landing on 2. The central limit theorem and the law of the iterated logarithm describe important aspects of the behavior of simple random walks on Z
Random_walk
Continuous probability distribution
variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution (GIG). Its probability density function (see the box) is given
Generalised hyperbolic distribution
Generalised_hyperbolic_distribution
is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. Accelerating change Approximating natural exponents (log base
List_of_exponential_topics
German polymath and scholar (1777–1855)
polygon construction in over 2000 years. He also introduced the concept of Gaussian curvature and proved its key properties, especially with his Theorema Egregium
Carl_Friedrich_Gauss
Type of non-Euclidean geometry
pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they
Hyperbolic_geometry
Number with a real and an imaginary part
unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For
Complex_number
Estimator for quality of a statistical model
are assumed to be i.i.d. Gaussian (with zero mean), then the model has three parameters: b0, b1, and the variance of the Gaussian distributions. Thus, when
Akaike_information_criterion
Notation describing limiting behavior in computational number theory
g. sieves for integer factorization and methods for solving discrete logarithms. The benefit of this notation is that it simplifies the analysis of these
L-notation
Mathematical transform that expresses a function of time as a function of frequency
(}p(x){\bigr )}\,dx} where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case. Fourier's
Fourier_transform
secant cubed Arclength Solid of revolution Shell integration Natural logarithm e (mathematical constant) Exponential function Hyperbolic angle Hyperbolic
List_of_calculus_topics
Concept in information theory
entropy depend on the base of the logarithm, which is usually 2 (i.e., the units are bits). See logarithmic units for logarithms taken in different bases. Related
Differential_entropy
Statistical distance measure
two discrete probability distributions, given that one uses the base 2 logarithm: 0 ≤ J S D ( P ∥ Q ) ≤ 1 {\displaystyle 0\leq {\rm {JSD}}(P\parallel Q)\leq
Jensen–Shannon_divergence
Quadric surface with one axis of symmetry and no center of symmetry
\left({\frac {R+Q}{P}}\right),} where ln x means the natural logarithm of x, i.e. its logarithm to base e. The volume of the dish, the amount of liquid it
Paraboloid
Cryptography secured against quantum computers
integer factorization problem, the discrete logarithm problem, or the elliptic-curve discrete logarithm problem. All of these problems could be easily
Post-quantum_cryptography
Representation of a type of random process
ε t {\displaystyle \varepsilon _{t}} is a Gaussian process then X t {\displaystyle X_{t}} is also a Gaussian process. In other cases, the central limit
Autoregressive_model
Number divisible only by 1 and itself
prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved
Prime_number
Type of mathematical function
0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, log f ( θ x + ( 1 − θ
Logarithmically concave function
Logarithmically_concave_function
Computer hardware technology that uses quantum mechanics
polynomial-time quantum algorithms for integer factorization and the discrete logarithm problem. A sufficiently large quantum computer could therefore break widely
Quantum_computing
Class of inequalities
inequalities involving the norm of a function f {\displaystyle f} , its logarithm, and its gradient ∇ f {\displaystyle \nabla f} . These inequalities were
Logarithmic Sobolev inequalities
Logarithmic_Sobolev_inequalities
Danish actuary and mathematician (1850–1916)
Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been
Jørgen_Pedersen_Gram
Probability distribution and special case of gamma distribution
unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables
Chi-squared_distribution
Lobes of far field radiation pattern of antenna
the radiation pattern has a canonical form of The function inside the logarithm is known as the Sinc function. Simple substitutions of various values
Sidelobes
Function representing the number of primes less than or equal to a given number
notation for logarithms. All instances of log ( x ) {\displaystyle \log(x)} without a subscript base should be interpreted as a natural logarithm, also commonly
Prime-counting_function
Probability distribution
frequency data on a log-log graph, with the axes being the logarithm of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent
Zipf's_law
Algorithm for computing greatest common divisors
was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic
Euclidean_algorithm
German mathematician and algorithms researcher (b.1936)
Invariance Principle for the Law of the Iterated Logarithm. Strassen's law of the iterated logarithm has been widely cited and led to a 1966 presentation
Volker_Strassen
Notion in statistics
where the Fisher information appears as the covariance of the fitted Gaussian. Statistical systems of a scientific nature (physical, biological, etc
Fisher_information
Statistical transform
output random number. The basic form requires two multiplications, 1/2 logarithm, 1/2 square root, and one trigonometric function for each normal variate
Box–Muller_transform
Operation in calculus
The case n = −1 required the invention of a function, the hyperbolic logarithm, achieved by quadrature of the hyperbola in 1647. Further steps were made
Integral
Fourth standardized moment in statistics
second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which
Kurtosis
Statistical method
regression method. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian (normal) distribution
Bootstrapping_(statistics)
In mathematics, invariant of square matrices
determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant
Determinant
Metric for fit of statistical models
as for the chi-square test, ln {\textstyle \ln } denotes the natural logarithm, and the sum is taken over all non-empty bins. Furthermore, the total
Goodness_of_fit
Mathematical term for squaring a plane figure
Saint-Vincent and A. A. de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came
Quadrature_(mathematics)
2D graphic with logarithmic scales on both axes
logarithm, though most commonly base 10 (common logs) are used. Given a monomial equation y = a x k , {\displaystyle y=ax^{k},} taking the logarithm of
Log–log_plot
Method in statistics
a differentiable function of a random variable which is asymptotically Gaussian. More generally, the delta method applies to Hadamard directionally differentiable
Delta_method
Concept in information theory
}|f(x)|^{2\alpha }\,dx\right)^{1/2\alpha }.} Squaring both sides and taking the logarithm, we get 1 β log ( ∫ R | g ( y ) | 2 β d y ) ≤ 1 2 log ( 2 α ) 1 /
Entropic_uncertainty
Method for specifying point positions
The log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference
Coordinate_system
Distribution function associated with the empirical measure of a sample
mean-zero Gaussian process G F = B ∘ F {\textstyle G_{F}=B\circ F} , where B is the standard Brownian bridge. The covariance structure of this Gaussian process
Empirical distribution function
Empirical_distribution_function
Number n where the highest prime factor of (n^2 + 1) is at least 2n
905. It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven. Because the Størmer
Størmer_number
Number, approximately 3.14
uncertainty principle only for the Gaussian function. Equivalently, π is the unique constant making the Gaussian normal distribution e−πx2 equal to its
Pi
Mathematical methods used in Bayesian inference and machine learning
the logarithm of q μ ∗ ( μ ) {\displaystyle q_{\mu }^{*}(\mu )} , we can see that q μ ∗ ( μ ) {\displaystyle q_{\mu }^{*}(\mu )} itself is a Gaussian distribution
Variational_Bayesian_methods
process Gauss–Markov process Gaussian process Gaussian random field Gaussian isoperimetric inequality Large deviations of Gaussian random functions Girsanov's
List_of_probability_topics
Methods of calculating definite integrals
hyperbola by Saint-Vincent and de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came
Numerical_integration
Probability distribution
variables. The concepts are sometimes ambiguously termed as in "product of Gaussians". The product is one type of algebra for random variables: Related to
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
Branch of statistics to estimate models based on measured data
consists of an unknown constant A {\displaystyle A} with additive white Gaussian noise (AWGN) w [ n ] {\displaystyle w[n]} with zero mean and known variance
Estimation_theory
Formula in number theory
easily seen when K = Q(i). In this case, the ring of integers in K is the Gaussian integers. An elementary manipulation shows that the residue of the Dedekind
Class_number_formula
Class of statistical models
termed an exponential-response model (or log-linear model, since the logarithm of the response is predicted to vary linearly). Similarly, a model that
Generalized_linear_model
this becomes the Laplacian of a Gaussian function. For reasons of computational complexity, the Laplacian of a Gaussian function is often approximated
Log_Gabor_filter
practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; and of course the bit - a new way of seeing the most fundamental
History_of_information_theory
Concept in cryptography
difficulty to compute discrete logarithms in a carefully chosen finite field, and the difficulty of computing discrete logarithms in a carefully chosen elliptic
Ring learning with errors key exchange
Ring_learning_with_errors_key_exchange
Difference between two successive prime numbers
notation for logarithms. All instances of log ( x ) {\displaystyle \log(x)} without a subscript base should be interpreted as a natural logarithm, also commonly
Prime_gap
Probability distribution on the circle
where ϕ ( q ) {\displaystyle \phi (q)\,} is the Euler function. The logarithm of the density of the wrapped normal distribution may be written: ln
Wrapped_normal_distribution
Extensions of the concept of randomness
= 0 {\displaystyle u'=0} and u ′ = u {\displaystyle u'=u} . Take the logarithm of p ( u ′ ) p ( u − u ′ ) {\displaystyle p(u')p(u-u')} and write: Δ (
Seven_states_of_randomness
Statistical transformation
\over 1-r}\right)=\operatorname {artanh} (r),} where "ln" is the natural logarithm function and "artanh" is the inverse hyperbolic tangent function. If (X
Fisher_transformation
Fixed number that has received a name
It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, and Cauchy distributions in probability
Mathematical_constant
Numerical integration method
effect is available for peak-like functions, such as Gaussian, Exponentially modified Gaussian and other functions with derivatives at integration limits
Trapezoidal_rule
Regression analysis
parameters a and b and with multiplicative error term U. If we take the logarithm of both sides, this becomes ln ( y ) = ln ( a ) + b x + u , {\displaystyle
Nonlinear_regression
Computational problem possibly useful for post-quantum cryptography
cryptography in the future just as the integer factorization and discrete logarithm problem have served as the base for public key cryptography since the
Ring_learning_with_errors
GAUSSIAN LOGARITHM
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