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Topics referred to by the same term
Exponential constant may refer to: e (mathematical constant) The growth or decay constant in exponential growth or exponential decay, respectively. This
Exponential_constant
Mathematical function, denoted exp(x) or e^x
complicated expression. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised
Exponential_function
Decrease in value at a rate proportional to the current value
a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: dN(t)dt=−λN(t). The solution
Exponential_decay
Base of natural logarithms
The number e is a mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352
E_(mathematical_constant)
Numerical constant in the Arrhenius equation in chemical kinetics
In chemical kinetics, the pre-exponential factor or A factor is the pre-exponential constant in the Arrhenius equation (equation shown below), an empirical
Pre-exponential_factor
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
Exponential_growth
Probability distribution
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance
Exponential_distribution
Unproven computational hardness assumption
In computational complexity theory, the exponential time hypothesis (ETH) states that satisfiability of 3-CNF Boolean formulas (3-SAT) cannot be solved
Exponential_time_hypothesis
Generates a forecast of future values of a time series
Exponential smoothing or exponential moving average (EMA) is a technique for smoothing time series data using the exponential window function. Whereas
Exponential_smoothing
Characteristic time in a system
response is an exponential decay with time constant τ. The time constant sets the characteristic time scale of this decay: after one time constant, the response
Time_constant
Estimate of time taken for running an algorithm
formally, an algorithm is exponential time if T(n) is bounded by O(2nk) for some constant k. Problems which admit exponential time algorithms on a deterministic
Time_complexity
Special function defined by an integral
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio
Exponential_integral
Family of probability distributions related to the normal distribution
similar – can only be included in the exponential class if the number of Bernoulli trials, n, is treated as a fixed constant – excluded from the free parameter(s)
Exponential_family
Exponential function of an exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x)=abx=a(bx) (where a > 1 and b >
Double_exponential_function
Constant e raised to the power of pi
mathematics, the exponential of pi eπ, also called Gelfond's constant, is the real number e raised to the power π (i.e., the value of the exponential function
Gelfond's_constant
Time for exponential decay to remove half of a quantity
lead-206. Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation.
Half-life
“exponential” is used with two different meanings (linear exponential bounds 2 c n {\displaystyle 2^{cn}} for a constant c, and full exponential bounds
Exponential_hierarchy
Loudspeaker using an acoustic horn
any expansion (exponential, hyperbolic sine, hyperbolic cosine, catenoidal, tractrix, spherical, or a new expansion) horn into a constant directivity horn
Horn_loudspeaker
Property of two varying quantities with a constant ratio
have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization
Proportionality_(mathematics)
Time constant of an RC circuit
following exponentially-decaying voltage curve: VC(t)=V0⋅(e−t/τ) Charging an uncharged capacitor through a series resistor to an applied constant input voltage
RC_time_constant
Numbers expressible as integrals of algebraic functions
among the ones known to be exponential periods: Gauss–Manin connection Jacobian variety L-function Mathematical constant Mixed motives (math) Tannakian
Period_(number_theory)
Functional square root of an exponential
such that f composed with itself results in an exponential function: f(f(x))=abx, for some constants a and b. Hellmuth Kneser first proposed a holomorphic
Half-exponential_function
Complex exponential in terms of sine and cosine
quotient of the trigonometric and exponential expressions is the constant function of one, so they must be equal (the exponential function is never zero, so
Euler's_formula
Fixed number that has received a name
a world record pursuit. Euler's number e, also known as the exponential growth constant, appears in many areas of mathematics, and one possible definition
Mathematical_constant
Type of light-tailed probability distribution
distribution is as a probability distribution whose tails decay at an exponential rate, or faster: a real-valued distribution D {\displaystyle {\cal {D}}}
Subexponential distribution (light-tailed)
Subexponential_distribution_(light-tailed)
Form of the utility function
The exponential utility function is a special case of the hyperbolic absolute risk aversion utility functions. Exponential utility implies constant absolute
Exponential_utility
(mathematical constant) Exponent Exponent bias Exponential (disambiguation) Exponential backoff Exponential decay Exponential dichotomy Exponential discounting
List_of_exponential_topics
Formula for temperature dependence of rates of chemical reactions
increases.[page needed] The Arrhenius equation describes the exponential dependence of the rate constant of a chemical reaction on the absolute temperature as
Arrhenius_equation
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
Major unsolved problem in transcendental number theory
iterated exponential identities for exponential constants, and the exponential subring of the real numbers generated by 1 is the free exponential ring on
Schanuel's_conjecture
Mathematical constant
} the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ)
Ramanujan–Soldner_constant
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
Arithmetic operation
ambiguous, as this can either mean iterated powers or iterated exponentials. The infinite exponential is defined as the limit of the following sequence of finite
Tetration
Special constant related to the exponential integral
functions. It is named after Benjamin Gompertz. It can be defined via the exponential integral as: δ=−eEi(−1)=∫0∞e−x1+xdx. The numerical value of δ is about
Gompertz_constant
Time required to double a quantity
rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated
Doubling_time
Logarithm to the base of the mathematical constant e
logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of
Natural_logarithm
Difference between logarithm and harmonic series
the following product involving the three constants e, π and γ: π26eγ=limn→∞logpn∏i=1npipi+1. The exponential also appears in the Weierstrass product formula
Euler's_constant
Recursive mathematical formula
the exponential factorial is bounded above and below by expressions of the form b ↑↑ ( n + c ) {\displaystyle b\uparrow \uparrow (n+c)} with constant c
Exponential_factorial
Map from tangent space to the manifold
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M {\displaystyle T_{p}M} of a Riemannian manifold (or pseudo-Riemannian
Exponential map (Riemannian geometry)
Exponential_map_(Riemannian_geometry)
Mathematical function common in physics
The stretched exponential function fβ(t)=e−tβ is obtained by inserting a fractional power law into the exponential function. In most applications, it is
Stretched exponential function
Stretched_exponential_function
Mathematical model of a dendrite
x = 0 {\displaystyle x=0} (point of current injection), e is the exponential constant (approximate value 2.71828) and V x {\displaystyle V_{x}} is the
Cable_theory
Number, approximately 3.14
the behaviour of the exponential function of a complex variable, described by Euler's formula: eiφ=cosφ+isinφ, where the constant e is the base of the
Pi
Describes approximate behavior of a function
Similarly, logs with different constant bases are equivalent with respect to Big O notation. On the other hand, exponentials with different bases are not
Big_O_notation
Cosmological model
cosmological constant dominance, will increase exponentially. But this expansion is similar for every moment of time (hence the exponential law—the expansion
Big_Rip
Mathematical model for diver decompression
also referred to as the Linear–Exponential model or the Exponential–Linear model. The Mk15 rebreather supplies a constant partial pressure of oxygen of
Thalmann_algorithm
Probability distribution
versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special
Gamma_distribution
Class of numerical methods
Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This
Exponential_integrator
Discount function in which future returns are weighted exponentially
In economics, exponential discounting is a specific form of the discount function, used in the analysis of choice over time (with or without uncertainty)
Exponential_discounting
Randomized mathematical sequence based upon the Fibonacci sequence
Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar
Random_Fibonacci_sequence
Electrical engineering equation
Shockley of Bell Labs, models the exponential current–voltage (I–V) relationship of semiconductor diodes in moderate constant current forward bias or reverse
Shockley_diode_equation
Computational statistics technique
set of piecewise exponential distributions (i.e. segments of one or more exponential distributions, attached end to end). Exponential distributions are
Rejection_sampling
Relative brain size measure
{\displaystyle S} is body weight, and r {\displaystyle r} is the exponential constant. The "encephalization quotient" (EQ) is the coefficient C {\displaystyle
Encephalization_quotient
Type of complex function with growth bounded by an exponential function
function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |z|→∞. When a function
Exponential_type
Constant equal to twice pi
other equivalent ways: notably, τ is equal to the period of the complex exponential function. Like π, τ is irrational and transcendental. While π is used
Tau_(mathematics)
Measure of the effective concentration of a species in a mixture
of standard conditions, R is the gas constant, T is the thermodynamic temperature and e is the exponential constant. Alternatively, this equation can be
Thermodynamic_activity
Monte Carlo distribution shifting technique
Exponential tilting (ET), exponential twisting, or exponential change of measure (ECM) is a distribution-shifting technique used in many parts of mathematics
Exponential_tilting
Mathematical equation linking e, i and π
result explicitly Mathematics portal De Moivre's formula Exponential function Gelfond's constant List of topics named after Leonhard Euler The term "Euler's
Euler's_identity
Frequency swept signal
kept constant, the frequency ratio f(t2)/f(t1) will also be constant. In an exponential chirp, the frequency of the signal varies exponentially as a function
Chirp
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
point 0 is said to have an exponential dichotomy if there exists a (constant) matrix P such that P2 = P and positive constants K, L, α, and β such that
Exponential_dichotomy
Generalization of the exponential function
is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential
C0-semigroup
Sum of the inverses of the positive cubes
Apéry's constant is a mathematical constant, defined as the infinite sum of the reciprocals of the cubes of the positive integers. In symbols, ζ(3)=∑n=1∞1n3
Apéry's_constant
Differential equation that is linear with respect to the unknown function
is a matrix of constants, or, more generally, if A commutes with its antiderivative B=∫Adx, then one may choose U equal the exponential of B. In fact
Linear_differential_equation
Mathematical constant in number theory
In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular
Khinchin's_constant
ideal consisting of power series without constant term. Then, given f(x)∈R0[[x]], its plethystic exponential PE[f] is given by PE[f](x)=exp(∑k=1∞f(xk)k)
Plethystic_exponential
of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the
Exponential_response_formula
Continuous stochastic process
A geometric Brownian motion (GBM), also known as an exponential Brownian motion, is a continuous-time stochastic process in which the logarithm of the
Geometric_Brownian_motion
Influence on an oscillating physical system which reduces or prevents its oscillation
hertz. Time constant: τ=1/λ, the time for the amplitude to decrease by the factor of e. Half-life is the time it takes for the exponential amplitude envelope
Damping
Compounding sum paid for the use of money
up interest in Wiktionary, the free dictionary. Credit card interest Exponential growth Fisher equation Interest rate Rate of return Rate of return on
Compound_interest
S-shaped curve
limit as x tends to −∞ is 0, and the limit as x tends to +∞ is L. The exponential function with negated argument (e−x) is used to define the standard logistic
Logistic_function
Spiking neuron model
rate for constant input, and the linear response to fluctuations, even in the presence of input noise. A didactive review of the exponential integrate-and-fire
Exponential integrate-and-fire
Exponential_integrate-and-fire
Hyperbolic analogues of trigonometric functions
defined with the exponential function eu. In the figure A=(e−u,eu), B=(eu, e−u), OA+OB=OC . Hyperbolic sine: the odd part of the exponential function, that
Hyperbolic_functions
Growth of bacterial colonies
bacteria ) and their exponential phase growth is often not ever a constant rate, but instead a slowly decaying rate, a constant stochastic response to
Bacterial_growth
Transcendental number(s) with all positive integers in order
In mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist
Champernowne_constant
Finite sum formed using the exponential function
mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function,
Exponential_sum
Study of the universe
modelling of the universe explored the possibility that the cosmological constant, introduced by Einstein in his 1917 paper, may result in an expanding universe
Cosmology
that both sides have the same constant term. Similarly, one can verify a product expansion for the Artin–Hasse exponential: Ep(x)=∏(p,n)=1(1−xn)−μ(n)/n
Artin–Hasse_exponential
Probability of survival beyond any specified time
decreases with time. The Weibull distribution extends the exponential distribution to allow constant, increasing, or decreasing hazard rates. There are several
Survival_function
Polynomial function of degree at most one
the function and of the line (see below). If the slope is a=0, this is a constant function f(x)=b defining a horizontal line, which some authors exclude
Linear_function_(calculus)
Example of the learning curve effect on performance
occur at a constant rate according to this function; our learning is hindered. The exponential function shows that learning increases at a constant rate in
Power_law_of_practice
Analytic function that does not satisfy a polynomial equation
related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. By introducing these
Transcendental_function
whose running time is exponential with a low base. Approximation-preserving reduction APX is the class of problems with some constant-factor approximation
Exact_algorithm
Observation in physical cosmology
coordinate. Though the Hubble constant H0 is constant at any given moment in time, the Hubble parameter H, of which the Hubble constant is the current value,
Hubble's_law
Statistical models for network analysis
Exponential family random graph models (ERGMs) are a set of statistical models used to study the structure and patterns within networks, such as those
Exponential family random graph models
Exponential_family_random_graph_models
Linear graph commonly used in chemical kinetics
constant A {\displaystyle A} = pre-exponential factor E a {\displaystyle E_{\text{a}}} = (molar) activation energy R {\displaystyle R} = gas constant
Arrhenius_plot
Product of numbers from 1 to n
In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra
Factorial
Chart indicator of moving average convergence/divergence
their time constants. It also can be seen to approximate the derivative as if it were calculated and then filtered by a single low pass exponential filter
MACD
Distribution of new data marginalized over the posterior
Most, but not all, common families of distributions are exponential families. Exponential families have a large number of useful properties. One of
Posterior predictive distribution
Posterior_predictive_distribution
Class of statistical models
varying over a wide range, constant input changes lead to geometrically (i.e. exponentially) varying, rather than constantly varying, output changes. As
Generalized_linear_model
Principal square root of minus 1
The imaginary unit, usually denoted by i, is a mathematical constant that is a solution to the quadratic equation x2 = −1, which is not solved by any real
Imaginary_unit
Mathematical function, inverse of an exponential function
inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups;
Logarithm
Arithmetic operation
exponents must be constant. As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. The theoretical
Exponentiation
Time interval in science
In science, e-folding is the time interval in which an exponentially growing quantity increases or decreases by a factor of e; it is the base-e analog
E-folding
Physical phenomenon
the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium state to thermodynamic
Spin–lattice_relaxation
In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. It permits
Shift_theorem
For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative
List of integrals of trigonometric functions
List_of_integrals_of_trigonometric_functions
Special function in the physical sciences
equationd2ydx2−ky=0 is oscillatory for k<0 and exponential for k>0, the Airy functions are oscillatory for x<0 and exponential for x>0. In fact, the Airy equation
Airy_function
Complexity class of approximable problems
approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems
APX
Integral transform useful in probability theory, physics, and engineering
Laplace transform for f an entire function of exponential type, meaning that |f(z)|≤AeB|z| for some constants A and B. The generalized Borel transform allows
Laplace_transform
formula[clarification needed] and is a special case of a general formula for the exponential function: ex/y=1+2x2y−x+x26y+x210y+x214y+x218y+⋱ The number e can be
List_of_representations_of_e
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