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Mathematical function common in physics
The stretched exponential function f β ( t ) = e − t β {\displaystyle f_{\beta }(t)=e^{-t^{\beta }}} is obtained by inserting a fractional power law into
Stretched exponential function
Stretched_exponential_function
Phenomenon in statistics where highest-ranked data points are outliers
include the power-law distribution, that is a basis for the stretched exponential function, and parabolic fractal distribution. Laherrere and Sornette
King_effect
Special function defined by an integral
In mathematics, the exponential integral Ei {\displaystyle \operatorname {Ei} } is a special function on the complex plane. It is defined as one particular
Exponential_integral
Relaxation model
charge response corresponds to the stretched exponential function or the Kohlrausch–Williams–Watts (KWW) function, for small time arguments. Cole–Cole
Cole–Cole_equation
Statistical distribution
a power law distribution, or less well-known ones such as a stretched exponential function or parabolic fractal distribution, at least approximately for
Rank–size_distribution
Extension of the factorial function
The gamma function is related to the stretched exponential function. For instance, the moments of that function are ⟨ τ n ⟩ ≡ ∫ 0 ∞ t n − 1 e − ( t τ
Gamma_function
Model in electromagnetism
1967. Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less. For
Havriliak–Negami_relaxation
also a characteristic feature of the Fourier transform of the stretched exponential function it has been considered as an approximation of the latter, although
Cole–Davidson_equation
Electrically insulating substance able to be polarised by an applied electric field
asymmetric broadening. Kohlrausch–Williams–Watts function Fourier transform of stretched exponential function. Curie–von Schweidler law This shows the response
Dielectric
Continuous probability distribution
distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other
Weibull_distribution
Topics referred to by the same term
refer to: The Kohlrausch-Williams-Watts function is the Fourier transform of the stretched exponential function Katosi Water Works, water treatment facility
KWW
British biophysicist, material scientist, and academic
is the co-discoverer of the Kohlrausch-Williams-Watts [KWW] stretched-exponential function for modelling molecular-relaxation processes in condensed media
David_C._Watts
Type of electrode
process. The lifetime of oxide cathodes can be evaluated with a stretched exponential function. The survivability of electron emission sources is significantly
Hot_cathode
Distribution of variables which satisfies a stability property under linear combinations
distribution function will be real. In the simplest case β = 0 {\displaystyle \beta =0} , the characteristic function is just a stretched exponential function; the
Stable_distribution
Longest recorded life span
maximum human lifespan to be around 125 years using a modified stretched exponential function for human survival curves. In another study, researchers claimed
Maximum_life_span
German physicist
Kohlrausch introduced the relaxation phenomena, and used the stretched exponential function to explain relaxation effects of a discharging Leyden jar (capacitor)
Rudolf_Kohlrausch
Probability distribution
distributions not only forms an exponential family (EF), but in fact forms a natural exponential family (NEF) with quadratic variance function (NEF-QVF). Many properties
Normal_distribution
List of scientists who are Christians
Science at the University of Manchester: co-discoverer of the KWW stretched-exponential function for relaxation phenomena in condensed media and expert on photopolymerised
List of Christians in science and technology
List_of_Christians_in_science_and_technology
Number with a real and an imaginary part
be regarded as its norm.] However for another inverse function of the complex exponential function (and not the above defined principal value), the branch
Complex_number
System that regulates the formation of blocks on a blockchain
1007/978-3-540-45146-4_25. ISBN 978-3-540-40674-7. Coelho, Fabien (2005). "Exponential memory-bound functions for proof of work protocols". Cryptology ePrint Archive, Report
Proof_of_work
Hash function that is suitable for use in cryptography
secure cryptographic hash functions but do not usually have a strong connection to practical security. For example, an exponential-time algorithm can sometimes
Cryptographic_hash_function
Formula for temperature dependence of rates of chemical reactions
tests of the Arrhenius law. Another common modification is the stretched exponential form[citation needed] k = A exp [ − ( E a R T ) β ] , {\displaystyle
Arrhenius_equation
Theorem of convex functions
particular importance in statistical physics when the convex function is an exponential, giving: eE[X]≤E[eX], where the expected values are with respect
Jensen's_inequality
Mathematical transform that expresses a function of time as a function of frequency
conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. A measurable function f:ℝ→ℂ is called (Lebesgue)
Fourier_transform
Functional relationship between two quantities
1137/070710111. S2CID 9155618. Laherrère, J.; Sornette, D. (1998). "Stretched exponential distributions in nature and economy: "fat tails" with characteristic
Power_law
Kind of numerical parameter of a parametric family of probability distributions
distribution Dagum distribution Erlang distribution ExGaussian distribution Exponential power distribution Fréchet distribution Gamma distribution Generalized
Shape_parameter
Fundamental theorem in probability theory and statistics
more rapidly than t2/n. By the limit of the exponential function (ex=limn→∞(1+xn)n), the characteristic function of Zn equals φZn(t)=(1−t22n+o(t2n))n→e−12t2
Central_limit_theorem
Value that appears most often in a set of data
random variable, the mode is the value x at which the probability mass function P(X) takes its maximum value, i.e., x = argmaxxi P(X = xi). In other words
Mode_(statistics)
Numerical technique
which it breaks down, suffering from unavoidable reflections or even exponential growth. One caveat with perfectly matched layers is that they are only
Perfectly_matched_layer
Discrete probability distribution
applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution:
Poisson_distribution
Mechanical oscillations about an equilibrium point
spring is stretched. The formulas for these values can be found in the references. The major points to note from the solution are the exponential term and
Vibration
Mapping involving integration between function spaces
maps a function from its original function space into another function space via integration, where some of the properties of the original function might
Integral_transform
Trachenko, K.; Zaccone, A. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Periodic vortex array in fluid dynamics
can be solved exactly for a class of two-dimensional flows that decay exponentially without changing shape. He gave the example of a square array of vortices
Taylor–Green_vortex
Theory of cognition
found that “generalization gradients in both series were fitted by exponential functions, supporting Shepard’s law”. Cheng’s research suggested that Shepard’s
Universal law of generalization
Universal_law_of_generalization
Algebraic structure in linear algebra
natural exponential function. The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reflect
Vector_space
Computation model defining an abstract machine
This is because the number of bits required to represent the outcome is exponential in the input size. However, if an algorithm runs in polynomial time in
Turing_machine
Physical system that responds to a restoring force proportional to displacement
2 , {\textstyle \omega _{1}=\omega _{0}{\sqrt {1-\zeta ^{2}}},} the exponential decay of the underdamped harmonic oscillator is given by λ = ω 0 ζ .
Harmonic_oscillator
Fitting an approximating function to data
and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving
Smoothing
Statistical property quantifying how much a collection of data is spread out
variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are
Statistical_dispersion
integrals List of integrals of exponential functions List of integrals of Gaussian functions List of integrals of hyperbolic functions List of integrals of inverse
Lists_of_mathematics_topics
Numerical method for solving physical or engineering problems
variable size h and polynomial degree p to achieve exceptionally fast, exponential convergence rates. The hpk-FEM combines adaptively elements with variable
Finite_element_method
Collection of random variables
a sample function of a stochastic process X {\displaystyle X} is a continuous function of t ∈ T {\displaystyle t\in T} ; a sample function of a stochastic
Stochastic_process
fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, such as the blinking
Chaotic_mixing
Cryptographic hash function
provably secure hash functions. It is based on the concept of the fast Fourier transform (FFT). SWIFFT is not the first hash function based on the FFT, but
SWIFFT
Decrease of the porosity of a sediment under the lithostatic pressure exerted by loading
especially in sedimentary basins, the porosity profiles often show an exponential decrease, called Athy's law as first shown by Athy in 1930. A mathematical
Compaction_(geology)
Probability distribution
distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f ( x ; x 0 , γ
Cauchy_distribution
Tissue in the body that is not hardened by ossification
extensibility. This constitutive model cannot be stretched in uni-axial tension beyond a maximal stretch J m {\displaystyle J_{m}} , which is the positive
Soft_tissue
SPEKE (Simple Password Exponential Key Exchange) is a cryptographic method for password-authenticated key agreement. The protocol consists of little more
SPEKE
Mathematics problem
where the speed of the ant is decreasing exponentially while the length of the rope is increasing exponentially and the ant will also reach the end of the
Ant_on_a_rubber_rope
Cryptanalytic method for unauthorized users to access data
passwords, passphrases and keys have more possible values, making them exponentially more difficult to crack than shorter ones due to the diversity of characters
Brute-force_attack
Numerical optimization algorithm
this point is better than the best current point, then we can try stretching exponentially out along this line. On the other hand, if this new point isn't
Nelder–Mead_method
Type of cryptographic software obfuscation
obfuscator to a collision-resistant hash function family, even with a trapdoor permutation, except with an exponential loss of security. Black-box obfuscation
Indistinguishability obfuscation
Indistinguishability_obfuscation
Simple model of a polymer
polymer. The polymer's orientation correlation function can then be solved for, and it follows an exponential decay with decay constant 1/P: ⟨ t ^ ( s ) ⋅
Worm-like_chain
Nanotechnology simulation of human organ function
are stretched. Since a vacuum is connected to the side channels, a decrease in pressure will cause the middle channel to expand, thus stretching the porous
Organ-on-a-chip
Scientific instrument used to measure fluid flow (rheology)
at an exponentially increasing velocity while measuring force and diameter as a function of time and position. By deforming at an exponentially increasing
Rheometer
is exponential with the length L: τ=eL. Here, the diffusion is against a linear potential.[citation needed] The solution for the Green's function Gij(t;L)
Heterogeneous random walk in one dimension
Heterogeneous_random_walk_in_one_dimension
Concept in calculus of variations
a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends. In the calculus
Functional_derivative
Mathematical model of the time dependence of a point in space
be solved in terms of simple functions such as exponentials and simple trigonometric functions (i.e. complex exponentials), and the behavior of all orbits
Dynamical_system
American scientist
decay; with trap states giving rise to broadband emission and a stretched exponential decay with long characteristic lifetimes. The nonexponential decay
Lauren_Rohwer
expansion of external argument For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there
External_ray
also lead to other statistics such as power-law distributions or stretched exponentials. One needs to note here that the word super here is short for superposition
Superstatistics
Mathematical description of mixing substances
with exponential decay rate" if ρt < e−δt for some δ > 0. For a stationary Markov process, the coefficients ρt may either decay at an exponential rate
Mixing_(mathematics)
Continuous probability distribution
0 {\displaystyle \alpha >0} and ν = 1 {\displaystyle \nu =1} ; Stretched Exponential distribution, when α > 0 {\displaystyle \alpha >0} and ν = 1 / α
Kaniadakis_Gamma_distribution
property. Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations –
List of logarithmic identities
List_of_logarithmic_identities
Theory of rapid universe expansion
inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the very early universe. This enormous expansion
Cosmic_inflation
interpolation Newton's method: finds zeros of functions with calculus Ridder's method: 3-point, exponential scaling Secant method: 2-point, 1-sided Hybrid
List_of_algorithms
Tendency of AC current flow in a conductor's outer layer
density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing eddy
Skin_effect
String-searching algorithm
− m)⌋ comparisons, with ε = 1⁄2(Fc+2 − 1)−1 ∈ O(φ−c) going to zero exponentially quickly as c increases. The algorithm is considered fairly efficient
Two-way string-matching algorithm
Two-way_string-matching_algorithm
probability is sub-linear then the network's degree distribution is stretched exponential and hubs are much smaller than in a scale-free network. If this
Non-linear preferential attachment
Non-linear_preferential_attachment
Idea that small causes can have large effects
demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence
Butterfly_effect
Physics theorem argued by G. H. Derrick
was obtained by R. H. Hobart in 1963. A stronger statement, linear (or exponential) instability of localized stationary solutions to the nonlinear wave
Derrick's_theorem
Family of cryptographic hash functions
In cryptography, the fast syndrome-based hash functions (FSB) are a family of cryptographic hash functions introduced in 2003 by Daniel Augot, Matthieu
Fast_syndrome-based_hash
Continuous probability distribution
in machine learning. The κ-distributions are written as function of the κ-deformed exponential, taking the form f i = exp κ ( − β E i + β μ ) {\displaystyle
Kaniadakis_distribution
Biomechanical paradigm explaining how muscles work
tension and therefore is stretched a finite amount. Because the overall length of the muscle is kept constant, the stretching of the series element can
Hill's_muscle_model
Method of exchanging cryptographic keys
agreement Diffie–Hellman key establishment Diffie–Hellman key negotiation Exponential key exchange Diffie–Hellman protocol Diffie–Hellman handshake Diffie
Diffie–Hellman_key_exchange
Unidentified phenomenon at Hessdalen valley in Norway
bremsstrahlung process is flat up to a cutoff frequency, νcut, and falls off exponentially at higher frequencies. This sequence of events forms the typical spectrum
Hessdalen_lights
Concept involving online bot activity
maneuvered ourselves into. Decoupled, proliferated, and in a state of exponential metastasis. — Thomas Sommerer Facebook includes an option to provide
Dead_Internet_theory
Algorithm for manipulating images
attempt to improvef(x,y) by testing a sequence of candidate offsets at an exponentially decreasing distance from v0 ui=v0+wαiRi where Ri is a uniform random
PatchMatch
Practice and study of secure communication techniques
Theo (2021): Super Secreto – The Third Epoch of Cryptography: Multiple, exponential, quantum-secure and above all, simple and practical Encryption for Everyone
Cryptography
Property of materials with both viscous and elastic characteristics under deformation
tensor as function of time t, p is the pressure 𝐈 is the unity tensor M is the memory function showing, usually expressed as a sum of exponential terms for
Viscoelasticity
Simple polynomial map exhibiting chaotic behavior
time: exponential convergence to zero convergence to a non-zero fixed value (see Exponential function or Characterizations of the exponential function point
Logistic_map
Management system for electrical power distribution
11747849B2 In recent years, utilization of electrical energy increased exponentially and customer requirement and quality definitions of power were changed
Distribution management system
Distribution_management_system
Academic field
"sub-linear" and the degree distribution of the network tends to a stretched exponential distribution. If α > 1 {\displaystyle \alpha >1} , NLPA is referred
Network_science
Earliest theoretical cosmological structures
paradigm, the exponential growth of the scale factor during inflation caused quantum fluctuations of the inflaton field to be stretched to macroscopic
Primordial_fluctuations
Region in spacetime from which nothing can escape
prevent an event horizon from existing, depending on the acceleration function chosen. The observer never touches the horizon and never passes a location
Event_horizon
Signal processing technique
the phase of the chirped signal (that is, the argument of the complex exponential), is the quadratic: ϕ ( t ) = 2 π ( ( f 0 − Δ f 2 ) t + Δ f 2 T t 2 )
Pulse_compression
Fluid breakup of a falling stream
stream dominates that of the curvature of the wave, such components grow exponentially with time. By decomposing an arbitrary disturbance on the free surface
Plateau–Rayleigh_instability
Mathematical concept
equilibrium then consists of those nearby orbits that neither decay nor grow exponentially quickly. Mathematically, the first step when studying equilibrium points
Center_manifold
Mathematics of smooth surfaces
that they provide a mathematical model for the shape of soap films when stretched across a wire frame ruled surfaces, which are surfaces that have at least
Differential geometry of surfaces
Differential_geometry_of_surfaces
Machine learning technique useful for dimensionality reduction
winner selection. The homogeneous Gaussian neighborhood function is replaced with the matrix exponential. Thus one can specify the orientation either in the
Self-organizing_map
are concerned with flows having chaotic particle paths, and associated exponential rates of mixing (Ottino 1989). At the dynamic level, the fact that vortex
Topological_fluid_dynamics
Sense of self-orientation
sum of 3 decaying exponentials: r(t)=K[1+Aexp(−at)+Bexp(−bt)+Cexp(−ct)]u(t) where r(t) is the firing rate and u(t) is a step function of force. The corresponding
Proprioception
Process of converting plaintext to ciphertext
Theo (2021): SUPER SECRETO – The Third Epoch of Cryptography: Multiple, exponential, quantum-secure and above all, simple and practical Encryption for Everyone
Encryption
Explanation for the rates of electron transfer reactions
stretched bonds, (b) In the Δ G ‡ = 0 {\displaystyle \Delta G^{\ddagger }=0} case the equilibrium configuration of the initial state is the stretched
Marcus_theory
Method of analysing cardiac performance
acute alterations in contractility or afterload. Tau represents the exponential decay of the ventricular pressure during isovolumic relaxation. Several
Pressure–volume loop analysis in cardiology
Pressure–volume_loop_analysis_in_cardiology
Cryptographic algorithm
to the threat of post-quantum cryptography. Quantum computers would exponentially increase the speed at which these ciphers can be decoded; notably, Grover's
Symmetric-key_algorithm
Wave where gravity is the main restoring force
linear instability, the growth rate of the disturbance in this phase is exponential in time. This Miles–Phillips Mechanism process can continue until an
Gravity_wave
History and future of the universe
cosmology and the Big Bang. During inflation, the universe expanded at an exponential rate. Once this ended, cosmic expansion subsequently decelerated to much
Chronology_of_the_universe
Concepts from linear algebra
rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk
Eigenvalues_and_eigenvectors
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STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
STRETCHED EXPONENTIAL-FUNCTION
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